math Namespace Reference

Mathematical functions and data types. More...

Namespaces
	compile
	Compile-time mathematical functions.
	hash
	Hash functions.
	literals
	User-defined math literals.
	matrix_types
	Matrix types.
	noise
	Noise functions.
	numbers
	Mathematical constants.
	operators
	Mathematical operators.
	polynomial
	Polynomial functions.
	quadrature
	Numerical integration functions.
	quaternion_types
	Quaternion types.
	types
	Linear algebra data types.
	vector_types
	Vector types.

Classes
struct	matrix
	n by m column-major matrix. More...
class	moving_average
	Calculates a moving average. More...
struct	quaternion
	Quaternion composed of a real scalar part and imaginary vector part. More...
struct	se3
	SE(3) proper rigid transformation (rototranslation). More...
struct	transform
	SRT transformation. More...
struct	vector
	n-dimensional vector. More...

Enumerations
enum class	rotation_sequence : std::uint8_t {   zxz = 0b100010 , xyx = 0b000100 , yzy = 0b011001 , zyz = 0b100110 ,   xzx = 0b001000 , yxy = 0b010001 , xyz = 0b100100 , yzx = 0b001001 ,   zxy = 0b010010 , xzy = 0b011000 , zyx = 0b000110 , yxz = 0b100001 }
	Rotation sequences of proper Euler and Tait-Bryan angles. More...

Functions
template<class T >
constexpr T	degrees (T radians) noexcept
	Converts an angle from radians to degrees. More...
template<class T >
constexpr T	radians (T degrees) noexcept
	Converts an angle given in degrees to radians. More...
template<class T >
constexpr T	wrap_degrees (T degrees)
	Wraps an angle to [-180, 180]. More...
template<class T >
constexpr T	wrap_radians (T radians)
	Wraps an angle to [-pi, pi]. More...
template<std::integral T>
constexpr vec3< T >	rotation_axes (rotation_sequence sequence) noexcept
	Returns the indices of the axes of a rotation sequence. More...
template<std::floating_point T>
T	fract (T x)
	Returns the fractional part of a floating-point number. More...
template<typename T , typename S = float>
constexpr T	lerp (const T &x, const T &y, S a) noexcept
	Linearly interpolates between x and y. More...
template<typename T >
constexpr T	lerp_angle (T x, T y, T a)
	Linearly interpolates between two angles, x and y. More...
template<typename T , typename S = float>
T	log_lerp (const T &x, const T &y, S a)
	Logarithmically interpolates between x and y. More...
template<class T >
constexpr T	map (T x, T from_min, T from_max, T to_min, T to_max) noexcept
	Remaps a number from one range to another. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	add (const matrix< T, N, M > &a, const matrix< T, N, M > &b) noexcept
	Adds two matrices. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	add (const matrix< T, N, M > &a, T b) noexcept
	Adds a matrix and a scalar. More...
template<class T , std::size_t N>
constexpr T	determinant (const matrix< T, N, N > &m) noexcept
	Calculates the determinant of a square matrix. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	componentwise_mul (const matrix< T, N, M > &a, const matrix< T, N, M > &b) noexcept
	Performs a component-wise multiplication of two matrices. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	div (const matrix< T, N, M > &a, const matrix< T, N, M > &b) noexcept
	Divides a matrix by a matrix. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	div (const matrix< T, N, M > &a, T b) noexcept
	Divides a matrix by a scalar. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	div (T a, const matrix< T, N, M > &b) noexcept
	Divides a scalar by a matrix. More...
template<class T , std::size_t N>
constexpr matrix< T, N, N >	inverse (const matrix< T, N, N > &m) noexcept
	Calculates the inverse of a square matrix. More...
template<class T >
constexpr mat4< T >	look_at_rh (const vec3< T > &position, const vec3< T > &target, const vec3< T > &up)
	Constructs a right-handed viewing transformation matrix. More...
template<class T >
constexpr std::tuple< mat4< T >, mat4< T > >	look_at_rh_inv (const vec3< T > &position, const vec3< T > &target, const vec3< T > &up)
	Constructs a right-handed viewing transformation matrix and its inverse. More...
template<class T , std::size_t N, std::size_t M, std::size_t P>
constexpr matrix< T, P, M >	mul (const matrix< T, N, M > &a, const matrix< T, P, N > &b) noexcept
	Multiplies two matrices. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	mul (const matrix< T, N, M > &a, T b) noexcept
	Multiplies a matrix by a scalar. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >::column_vector_type	mul (const matrix< T, N, M > &a, const typename matrix< T, N, M >::row_vector_type &b) noexcept
	Calculates the product of a matrix and a row vector. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >::row_vector_type	mul (const typename matrix< T, N, M >::column_vector_type &a, const matrix< T, N, M > &b) noexcept
	Calculates the product of a column vector and a matrix. More...
template<class T >
mat3< T >	rotate (T angle, const vector< T, 3 > &axis)
	Constructs a rotation matrix. More...
template<class T >
mat3< T >	rotate_x (T angle)
	Produces a matrix which rotates Cartesian coordinates about the x-axis by a given angle. More...
template<class T >
mat3< T >	rotate_y (T angle)
	Produces a matrix which rotates Cartesian coordinates about the y-axis by a given angle. More...
template<class T >
mat3< T >	rotate_z (T angle)
	Produces a matrix which rotates Cartesian coordinates about the z-axis by a given angle. More...
template<class T >
constexpr mat4< T >	scale (const vec3< T > &v)
	Constructs a scale matrix. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	sub (const matrix< T, N, M > &a, const matrix< T, N, M > &b) noexcept
	Subtracts a matrix from another matrix. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	sub (const matrix< T, N, M > &a, T b) noexcept
	Subtracts a scalar from matrix. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >	sub (T a, const matrix< T, N, M > &b) noexcept
	Subtracts a matrix from a scalar. More...
template<class T , std::size_t N>
constexpr T	trace (const matrix< T, N, N > &m) noexcept
	Calculates the trace of a square matrix. More...
template<class T >
constexpr mat4< T >	translate (const vec3< T > &v)
	Constructs a translation matrix. More...
template<class T , std::size_t N, std::size_t M>
constexpr matrix< T, M, N >	transpose (const matrix< T, N, M > &m) noexcept
	Calculates the transpose of a matrix. More...
template<class T >
T	horizontal_fov (T v, T r)
	Calculates a horizontal FoV given a vertical FoV and aspect ratio. More...
template<class T >
T	vertical_fov (T h, T r)
	Calculates a vertical FoV given a horizontal FoV and aspect ratio. More...
template<class T >
constexpr mat4< T >	ortho (T left, T right, T bottom, T top, T near, T far) noexcept
	Constructs an orthographic projection matrix which will transform the near and far clipping planes to [-1, 1], respectively. More...
template<class T >
constexpr std::tuple< mat4< T >, mat4< T > >	ortho_inv (T left, T right, T bottom, T top, T near, T far) noexcept
	Constructs an orthographic projection matrix which will transform the near and far clipping planes to [-1, 1], respectively, along with its inverse. More...
template<class T >
constexpr mat4< T >	ortho_half_z (T left, T right, T bottom, T top, T near, T far) noexcept
	Constructs an orthographic projection matrix which will transform the near and far clipping planes to [0, 1], respectively. More...
template<class T >
constexpr std::tuple< mat4< T >, mat4< T > >	ortho_half_z_inv (T left, T right, T bottom, T top, T near, T far) noexcept
	Constructs an orthographic projection matrix which will transform the near and far clipping planes to [0, 1], respectively, along with its inverse. More...
template<class T >
mat4< T >	perspective (T vertical_fov, T aspect_ratio, T near, T far)
	Constructs a perspective projection matrix which will transform the near and far clipping planes to [-1, 1], respectively. More...
template<class T >
std::tuple< mat4< T >, mat4< T > >	perspective_inv (T vertical_fov, T aspect_ratio, T near, T far)
	Constructs a perspective projection matrix which will transform the near and far clipping planes to [-1, 1], respectively, along with its inverse. More...
template<class T >
mat4< T >	perspective_half_z (T vertical_fov, T aspect_ratio, T near, T far)
	Constructs a perspective projection matrix which will transform the near and far clipping planes to [0, 1], respectively. More...
template<class T >
std::tuple< mat4< T >, mat4< T > >	perspective_half_z_inv (T vertical_fov, T aspect_ratio, T near, T far)
	Constructs a perspective projection matrix which will transform the near and far clipping planes to [0, 1], respectively, along with its inverse. More...
template<class T >
mat4< T >	inf_perspective_half_z_reverse (T vertical_fov, T aspect_ratio, T near)
	Constructs a perspective projection matrix, with an infinite far plane, which will transform the near and far clipping planes to [1, 0], respectively. More...
template<class T >
std::tuple< mat4< T >, mat4< T > >	inf_perspective_half_z_reverse_inv (T vertical_fov, T aspect_ratio, T near)
	Constructs a perspective projection matrix, with an infinite far plane, which will transform the near and far clipping planes to [1, 0], respectively, along with its inverse. More...
template<class T >
constexpr quaternion< T >	add (const quaternion< T > &a, const quaternion< T > &b) noexcept
	Adds two quaternions. More...
template<class T >
constexpr quaternion< T >	add (const quaternion< T > &a, T b) noexcept
	Adds a quaternion and a scalar. More...
template<class T >
constexpr quaternion< T >	conjugate (const quaternion< T > &q) noexcept
	Calculates the conjugate of a quaternion. More...
template<class T >
constexpr T	dot (const quaternion< T > &a, const quaternion< T > &b) noexcept
	Calculates the dot product of two quaternions. More...
template<class T >
constexpr quaternion< T >	div (const quaternion< T > &a, const quaternion< T > &b) noexcept
	Divides a quaternion by another quaternion. More...
template<class T >
constexpr quaternion< T >	div (const quaternion< T > &a, T b) noexcept
	Divides a quaternion by a scalar. More...
template<class T >
constexpr quaternion< T >	div (T a, const quaternion< T > &b) noexcept
	Divides a scalar by a quaternion. More...
template<class T >
T	inv_length (const quaternion< T > &q)
	Calculates the inverse length of a quaternion. More...
template<class T >
T	length (const quaternion< T > &q)
	Calculates the length of a quaternion. More...
template<class T >
constexpr quaternion< T >	lerp (const quaternion< T > &a, const quaternion< T > &b, T t) noexcept
	Performs linear interpolation between two quaternions. More...
template<class T >
quaternion< T >	look_rotation (const vec3< T > &forward, vec3< T > up)
	Creates a unit quaternion rotation using forward and up vectors. More...
template<class T >
constexpr quaternion< T >	mul (const quaternion< T > &a, const quaternion< T > &b) noexcept
	Multiplies two quaternions. More...
template<class T >
constexpr quaternion< T >	mul (const quaternion< T > &a, T b) noexcept
	Multiplies a quaternion by a scalar. More...
template<class T >
constexpr vec3< T >	mul (const quaternion< T > &q, const vec3< T > &v) noexcept
	Rotates a vector by a unit quaternion. More...
template<class T >
constexpr vec3< T >	mul (const vec3< T > &v, const quaternion< T > &q) noexcept
	Rotates a vector by the inverse of a unit quaternion. More...
template<class T >
constexpr quaternion< T >	negate (const quaternion< T > &q) noexcept
	Negates a quaternion. More...
template<class T >
quaternion< T >	nlerp (const quaternion< T > &a, const quaternion< T > &b, T t)
	Performs normalized linear interpolation between two quaternions. More...
template<class T >
quaternion< T >	normalize (const quaternion< T > &q)
	Normalizes a quaternion. More...
template<class T >
quaternion< T >	angle_axis (T angle, const vec3< T > &axis)
	Creates a rotation from an angle and axis. More...
template<class T >
quaternion< T >	rotation (const vec3< T > &from, const vec3< T > &to, T tolerance=T{1e-6})
	Constructs a quaternion representing the minimum rotation from one direction to another direction. More...
template<class T >
quaternion< T >	rotate_towards (const vec3< T > &from, const vec3< T > &to, T max_angle)
	Constructs a quaternion representing an angle-limited rotation from one direction towards another direction. More...
template<class T >
quaternion< T >	slerp (const quaternion< T > &a, const quaternion< T > &b, T t, T tolerance=T{1e-6})
	Performs spherical linear interpolation between two quaternions. More...
template<class T >
constexpr T	sqr_length (const quaternion< T > &q) noexcept
	Calculates the square length of a quaternion. More...
template<class T >
constexpr quaternion< T >	sub (const quaternion< T > &a, const quaternion< T > &b) noexcept
	Subtracts a quaternion from another quaternion. More...
template<class T >
std::array< quaternion< T >, 2 >	swing_twist (const quaternion< T > &q, const vec3< T > &twist_axis, T tolerance=T{1e-6})
	Decomposes a quaternion into swing and twist rotation components. More...
template<class T >
quaternion< T >	quaternion_cast (const mat3< T > &m)
	Converts a 3x3 rotation matrix to a quaternion. More...
template<typename T = float>
T	random (T start, T end)
	Generates a pseudo-random floating point number on [start, end) using std::rand(). More...
template<class T >
transform< T >	inverse (const transform< T > &t) noexcept
	Calculates the inverse of a transform. More...
template<class T >
transform< T >	mul (const transform< T > &x, const transform< T > &y)
	Combines two transforms. More...
template<class T >
constexpr vector< T, 3 >	mul (const transform< T > &t, const vector< T, 3 > &v) noexcept
	Transforms a vector by a transform. More...
template<class T >
constexpr vector< T, 3 >	mul (const vector< T, 3 > &v, const transform< T > &t) noexcept
	Transforms a vector by the inverse of a transform. More...
template<class T , std::size_t N>
constexpr vector< T, N >	abs (const vector< T, N > &x)
	Returns the absolute values of each element. More...
template<std::size_t N>
constexpr bool	all (const vector< bool, N > &x) noexcept
	Checks if all elements of a boolean vector are true. More...
template<std::floating_point T, std::size_t N>
T	angle (const vector< T, N > &from, const vector< T, N > &to)
	Calculates the angle between two direction vectors. More...
template<std::size_t N>
constexpr bool	any (const vector< bool, N > &x) noexcept
	Checks if any elements of a boolean vector are true. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	ceil (const vector< T, N > &x)
	Performs an element-wise ceil operation. More...
template<std::floating_point T, std::size_t N>
vector< T, N >	clamp_length (const vector< T, N > &x, T max_length)
	Clamps the length of a vector. More...
template<class T >
constexpr vector< T, 3 >	cross (const vector< T, 3 > &x, const vector< T, 3 > &y) noexcept
	Calculates the cross product of two vectors. More...
template<std::floating_point T, std::size_t N>
T	distance (const vector< T, N > &p0, const vector< T, N > &p1)
	Calculates the distance between two points. More...
template<class T , std::size_t N>
constexpr T	dot (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Calculates the dot product of two vectors. More...
template<class T , std::size_t N>
constexpr vector< bool, N >	equal (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Compares two vectors for equality. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	floor (const vector< T, N > &x)
	Performs a element-wise floor operation. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	fract (const vector< T, N > &x)
	Returns a vector containing the fractional part of each element. More...
template<class T , std::size_t N>
constexpr vector< bool, N >	greater_than (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Performs a element-wise greater-than comparison of two vectors. More...
template<class T , std::size_t N>
constexpr vector< bool, N >	greater_than_equal (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Performs a element-wise greater-than or equal-to comparison of two vectors. More...
template<std::floating_point T, std::size_t N>
T	inv_length (const vector< T, N > &x)
	Calculates the inverse length of a vector. More...
template<std::floating_point T, std::size_t N>
T	length (const vector< T, N > &x)
	Calculates the length of a vector. More...
template<class T , std::size_t N>
constexpr vector< bool, N >	less_than (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Performs a element-wise less-than comparison of two vectors. More...
template<class T , std::size_t N>
constexpr vector< bool, N >	less_than_equal (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Performs a element-wise less-than or equal-to comparison of two vectors. More...
template<class T , std::size_t N>
constexpr vector< T, N >	max (const vector< T, N > &x, const vector< T, N > &y)
	Returns a vector containing the maximum elements of two vectors. More...
template<class T , std::size_t N>
constexpr T	max (const vector< T, N > &x)
	Returns the value of the greatest element in a vector. More...
template<class T , std::size_t N>
constexpr vector< T, N >	min (const vector< T, N > &x, const vector< T, N > &y)
	Returns a vector containing the minimum elements of two vectors. More...
template<class T , std::size_t N>
constexpr T	min (const vector< T, N > &x)
	Returns the value of the smallest element in a vector. More...
template<class T , std::size_t N>
constexpr vector< T, N >	negate (const vector< T, N > &x) noexcept
	Negates a vector. More...
template<std::floating_point T, std::size_t N>
vector< T, N >	normalize (const vector< T, N > &x)
	Calculates the unit vector in the same direction as the original vector. More...
template<class T , std::size_t N>
constexpr vector< bool, N >	logical_not (const vector< T, N > &x) noexcept
	Logically inverts a boolean vector. More...
template<class T , std::size_t N>
constexpr vector< bool, N >	not_equal (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Compares two vectors for inequality. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	round (const vector< T, N > &x)
	Performs a element-wise round operation. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	sign (const vector< T, N > &x)
	Returns a vector containing the signs of each element. More...
template<std::floating_point T>
T	signed_angle (const vector< T, 3 > &x, const vector< T, 3 > &y, const vector< T, 3 > &n)
	Calculates the signed angle between two direction vectors about axis. More...
template<class T , std::size_t N>
constexpr T	sqr_distance (const vector< T, N > &p0, const vector< T, N > &p1) noexcept
	Calculates the square distance between two points. More...
template<class T , std::size_t N>
constexpr T	sqr_length (const vector< T, N > &x) noexcept
	Calculates the square length of a vector. More...
template<std::floating_point T, std::size_t N>
vector< T, N >	sqrt (const vector< T, N > &x)
	Takes the square root of each element. More...
template<class T , std::size_t N>
constexpr T	sum (const vector< T, N > &x) noexcept
	Calculates the sum of all elements in a vector. More...
template<std::size_t... Indices, class T , std::size_t N>
constexpr vector< T, sizeof...(Indices)>	swizzle (const vector< T, N > &x) noexcept
	Makes an m-dimensional vector by rearranging and/or duplicating elements of an n-dimensional vector. More...
template<class T >
constexpr T	triple (const vector< T, 3 > &x, const vector< T, 3 > &y, const vector< T, 3 > &z) noexcept
	Calculates the triple product of three vectors. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	trunc (const vector< T, N > &x)
	Performs a element-wise trunc operation. More...
template<std::floating_point T, std::size_t N>
vector< T, N >	sqrt (const vector< T, N > &x, const vector< T, N > &y)
template<rotation_sequence Sequence, class T >
vec3< T >	euler_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_zxz_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_xyx_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_yzy_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_zyz_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_xzx_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_yxy_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_xyz_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_yzx_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_zxy_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_xzy_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_zyx_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_yxz_from_quat (const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<class T >
vec3< T >	euler_from_quat (rotation_sequence sequence, const quat< T > &q, T tolerance=T{1e-6})
	Derives Euler angles from a unit quaternion. More...
template<rotation_sequence Sequence, class T >
quat< T >	euler_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_zxz_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_xyx_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_yzy_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_zyz_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_xzx_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_yxy_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_xyz_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_yzx_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_zxy_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_xzy_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_zyx_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_yxz_to_quat (const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<class T >
quat< T >	euler_to_quat (rotation_sequence sequence, const vec3< T > &angles)
	Constructs a quaternion from Euler angles. More...
template<std::size_t I, class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >::column_vector_type &	get (matrix< T, N, M > &m) noexcept
	Extracts the Ith column from a matrix. More...
template<std::size_t I, class T , std::size_t N, std::size_t M>
constexpr matrix< T, N, M >::column_vector_type &&	get (matrix< T, N, M > &&m) noexcept
	Extracts the Ith column from a matrix. More...
template<std::size_t I, class T , std::size_t N, std::size_t M>
constexpr const matrix< T, N, M >::column_vector_type &	get (const matrix< T, N, M > &m) noexcept
	Extracts the Ith column from a matrix. More...
template<std::size_t I, class T , std::size_t N, std::size_t M>
constexpr const matrix< T, N, M >::column_vector_type &&	get (const matrix< T, N, M > &&m) noexcept
	Extracts the Ith column from a matrix. More...
template<class T >
constexpr quaternion< T >	sub (const quaternion< T > &a, T b) noexcept
	Subtracts a quaternion and a scalar. More...
template<class T >
constexpr quaternion< T >	sub (T a, const quaternion< T > &b) noexcept
	Subtracts a quaternion and a scalar. More...
template<class T , std::size_t N>
constexpr vector< T, N >	add (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Adds two values. More...
template<class T , std::size_t N>
constexpr vector< T, N >	add (const vector< T, N > &x, T y) noexcept
	Adds two values. More...
template<class T , std::size_t N>
constexpr vector< T, N >	clamp (const vector< T, N > &x, const vector< T, N > &min, const vector< T, N > &max)
	Clamps the values of a vector's elements. More...
template<class T , std::size_t N>
constexpr vector< T, N >	clamp (const vector< T, N > &x, T min, T max)
	Clamps the values of a vector's elements. More...
template<class T , std::size_t N>
constexpr vector< T, N >	div (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Divides a vector by a value. More...
template<class T , std::size_t N>
constexpr vector< T, N >	div (const vector< T, N > &x, T y) noexcept
	Divides a vector by a value. More...
template<class T , std::size_t N>
constexpr vector< T, N >	div (T x, const vector< T, N > &y) noexcept
	Divides a vector by a value. More...
template<class T , std::size_t N>
constexpr vector< T, N >	fma (const vector< T, N > &x, const vector< T, N > &y, const vector< T, N > &z)
	Performs a multiply-add operation. More...
template<class T , std::size_t N>
constexpr vector< T, N >	fma (const vector< T, N > &x, T y, T z)
	Performs a multiply-add operation. More...
template<std::size_t I, class T , std::size_t N>
constexpr T &	get (math::vector< T, N > &v) noexcept
	Extracts the Ith element from a vector. More...
template<std::size_t I, class T , std::size_t N>
constexpr T &&	get (math::vector< T, N > &&v) noexcept
	Extracts the Ith element from a vector. More...
template<std::size_t I, class T , std::size_t N>
constexpr const T &	get (const math::vector< T, N > &v) noexcept
	Extracts the Ith element from a vector. More...
template<std::size_t I, class T , std::size_t N>
constexpr const T &&	get (const math::vector< T, N > &&v) noexcept
	Extracts the Ith element from a vector. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	mod (const vector< T, N > &x, const vector< T, N > &y)
	Calculates the element-wise remainder of the division operation x / y. More...
template<std::floating_point T, std::size_t N>
constexpr vector< T, N >	mod (const vector< T, N > &x, T y)
	Calculates the element-wise remainder of the division operation x / y. More...
template<class T , std::size_t N>
constexpr vector< T, N >	mul (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Multiplies two values. More...
template<class T , std::size_t N>
constexpr vector< T, N >	mul (const vector< T, N > &x, T y) noexcept
	Multiplies two values. More...
template<std::floating_point T, std::size_t N>
vector< T, N >	pow (const vector< T, N > &x, const vector< T, N > &y)
	Raises each element to a power. More...
template<std::floating_point T, std::size_t N>
vector< T, N >	pow (const vector< T, N > &x, T y)
	Raises each element to a power. More...
template<class T , std::size_t N>
constexpr vector< T, N >	sub (const vector< T, N > &x, const vector< T, N > &y) noexcept
	Subtracts a value by another value. More...
template<class T , std::size_t N>
constexpr vector< T, N >	sub (const vector< T, N > &x, T y) noexcept
	Subtracts a value by another value. More...
template<class T , std::size_t N>
constexpr vector< T, N >	sub (T x, const vector< T, N > &y) noexcept
	Subtracts a value by another value. More...

Detailed Description

Mathematical functions and data types.

Enumeration Type Documentation

rotation_sequence

enum math::rotation_sequence : std::uint8_t

strong

Rotation sequences of proper Euler and Tait-Bryan angles.

Indices of the first, second, and third rotation axes are encoded in bits 0-1, 2-3, and 4-5, respectively.

Enumerator
zxz	z-x-z rotation sequence (proper Euler angles).
xyx	x-y-x rotation sequence (proper Euler angles).
yzy	y-z-y rotation sequence (proper Euler angles).
zyz	z-y-z rotation sequence (proper Euler angles).
xzx	x-z-x rotation sequence (proper Euler angles).
yxy	y-x-y rotation sequence (proper Euler angles).
xyz	x-y-z rotation sequence (Tait-Bryan angles).
yzx	y-z-x rotation sequence (Tait-Bryan angles).
zxy	z-x-y rotation sequence (Tait-Bryan angles).
xzy	x-z-y rotation sequence (Tait-Bryan angles).
zyx	z-y-x rotation sequence (Tait-Bryan angles).
yxz	y-x-z rotation sequence (Tait-Bryan angles).

Definition at line 38 of file euler-angles.hpp.

Function Documentation

abs()

template<class T , std::size_t N>

constexpr vector< T, N > math::abs ( const vector< T, N > &  x )

inlineconstexpr

Returns the absolute values of each element.

Parameters

x	Input vector.

Returns

Absolute values of input vector elements.

Definition at line 985 of file vector.hpp.

add() [1/6]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::add ( const matrix< T, N, M > &  a, const matrix< T, N, M > &  b  )

constexprnoexcept

Adds two matrices.

Parameters

a	First matrix.
b	Second matrix.

Returns

Sum of the two matrices.

Definition at line 776 of file math/matrix.hpp.

add() [2/6]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::add ( const matrix< T, N, M > &  a, T  b  )

constexprnoexcept

Adds a matrix and a scalar.

Parameters

a	Matrix.
b	scalar.

Returns

Sum of the matrix and scalar.

Definition at line 789 of file math/matrix.hpp.

add() [3/6]

template<class T >

constexpr quaternion< T > math::add ( const quaternion< T > &  a, const quaternion< T > &  b  )

inlineconstexprnoexcept

Adds two quaternions.

Parameters

a	First quaternion.
b	Second quaternion.

Returns

Sum of the two quaternions.

Definition at line 554 of file quaternion.hpp.

add() [4/6]

template<class T >

constexpr quaternion< T > math::add ( const quaternion< T > &  a, T  b  )

inlineconstexprnoexcept

Adds a quaternion and a scalar.

Parameters

a	First value.
b	Second second value.

Returns

Sum of the quaternion and scalar.

Definition at line 560 of file quaternion.hpp.

add() [5/6]

template<class T , std::size_t N>

constexpr vector< T, N > math::add ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Adds two values.

Parameters

x	First value.
y	Second value.

Returns

Sum of the two values.

Definition at line 998 of file vector.hpp.

add() [6/6]

template<class T , std::size_t N>

constexpr vector< T, N > math::add ( const vector< T, N > &  x, T  y  )

inlineconstexprnoexcept

Adds two values.

Parameters

x	First value.
y	Second value.

Returns

Sum of the two values.

Definition at line 1011 of file vector.hpp.

all()

template<std::size_t N>

constexpr bool math::all ( const vector< bool, N > &  x )

inlineconstexprnoexcept

Checks if all elements of a boolean vector are true.

Parameters

x	Vector to be tested for truth.

Returns

true if all elements are true, false otherwise.

Definition at line 1024 of file vector.hpp.

angle()

template<std::floating_point T, std::size_t N>

T math::angle	(	const vector< T, N > &	from,
		const vector< T, N > &	to
	)

Calculates the angle between two direction vectors.

Parameters

from	First direction vector.
to	Second direction vector.

Returns

Angle between the two direction vectors, in radians.

Definition at line 1030 of file vector.hpp.

angle_axis()

template<class T >

quaternion< T > math::angle_axis	(	T	angle,
		const vec3< T > &	axis
	)

Creates a rotation from an angle and axis.

Parameters

angle	Angle of rotation (in radians).
axis	Axis of rotation

Returns

Quaternion representing the rotation.

Definition at line 685 of file quaternion.hpp.

any()

template<std::size_t N>

constexpr bool math::any ( const vector< bool, N > &  x )

inlineconstexprnoexcept

Checks if any elements of a boolean vector are true.

Parameters

x	Vector to be tested for truth.

Returns

true if any elements are true, false otherwise.

Definition at line 1043 of file vector.hpp.

ceil()

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::ceil ( const vector< T, N > &  x )

inlineconstexpr

Performs an element-wise ceil operation.

Parameters

x	Input vector.

Returns

Component-wise ceil of input vector.

Definition at line 1056 of file vector.hpp.

clamp() [1/2]

template<class T , std::size_t N>

constexpr vector< T, N > math::clamp ( const vector< T, N > &  x, const vector< T, N > &  min, const vector< T, N > &  max  )

inlineconstexpr

Clamps the values of a vector's elements.

Parameters

x	Vector to clamp.
min	Minimum value.
max	Maximum value.

Returns

Clamped vector.

Definition at line 1069 of file vector.hpp.

clamp() [2/2]

template<class T , std::size_t N>

constexpr vector< T, N > math::clamp ( const vector< T, N > &  x, T  min, T  max  )

inlineconstexpr

Clamps the values of a vector's elements.

Parameters

x	Vector to clamp.
min	Minimum value.
max	Maximum value.

Returns

Clamped vector.

Definition at line 1082 of file vector.hpp.

clamp_length()

template<std::floating_point T, std::size_t N>

vector< T, N > math::clamp_length	(	const vector< T, N > &	x,
		T	max_length
	)

Clamps the length of a vector.

Parameters

x	Vector to clamp.
max_length	Maximum length.

Returns

Length-clamped vector.

Definition at line 1088 of file vector.hpp.

componentwise_mul()

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::componentwise_mul ( const matrix< T, N, M > &  a, const matrix< T, N, M > &  b  )

constexprnoexcept

Performs a component-wise multiplication of two matrices.

Parameters

x	First matrix multiplicand.
y	Second matrix multiplicand.

Returns

Product of the component-wise multiplcation.

Definition at line 843 of file math/matrix.hpp.

conjugate()

template<class T >

constexpr quaternion< T > math::conjugate ( const quaternion< T > &  q )

inlineconstexprnoexcept

Calculates the conjugate of a quaternion.

Parameters

q	Quaternion from which the conjugate will be calculated.

Returns

Conjugate of the quaternion.

Definition at line 566 of file quaternion.hpp.

cross()

template<class T >

constexpr vector< T, 3 > math::cross ( const vector< T, 3 > &  x, const vector< T, 3 > &  y  )

inlineconstexprnoexcept

Calculates the cross product of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Cross product of the two vectors.

Definition at line 1095 of file vector.hpp.

degrees()

template<class T >

constexpr T math::degrees ( T  radians )

inlineconstexprnoexcept

Converts an angle from radians to degrees.

Parameters

radians

Angle in radians.

Returns

Angle in degrees.

Definition at line 36 of file angles.hpp.

determinant()

template<class T , std::size_t N>

constexpr T math::determinant ( const matrix< T, N, N > &  m )

constexprnoexcept

Calculates the determinant of a square matrix.

Parameters

m	Matrix of which to take the determinant.

Returns

Determinant of m.

Warning

Currently only implemented for 2x2, 3x3, and 4x4 matrices.

distance()

template<std::floating_point T, std::size_t N>

T math::distance ( const vector< T, N > &  p0, const vector< T, N > &  p1  )

inline

Calculates the distance between two points.

Parameters

p0	First of two points.
p1	Second of two points.

Returns

Distance between the two points.

Definition at line 1106 of file vector.hpp.

div() [1/9]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::div ( const matrix< T, N, M > &  a, const matrix< T, N, M > &  b  )

constexprnoexcept

Divides a matrix by a matrix.

Parameters

a	First matrix.
b	Second matrix.

Returns

Result of the division.

Definition at line 856 of file math/matrix.hpp.

div() [2/9]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::div ( const matrix< T, N, M > &  a, T  b  )

constexprnoexcept

Divides a matrix by a scalar.

Parameters

a	Matrix.
b	Scalar.

Returns

Result of the division.

Definition at line 869 of file math/matrix.hpp.

div() [3/9]

template<class T >

constexpr quaternion< T > math::div ( const quaternion< T > &  a, const quaternion< T > &  b  )

inlineconstexprnoexcept

Divides a quaternion by another quaternion.

Parameters

a	First value.
b	Second value.

Returns

Result of the division.

Definition at line 578 of file quaternion.hpp.

div() [4/9]

template<class T >

constexpr quaternion< T > math::div ( const quaternion< T > &  a, T  b  )

inlineconstexprnoexcept

Divides a quaternion by a scalar.

Parameters

a	Quaternion.
b	Scalar.

Returns

Result of the division.

Definition at line 584 of file quaternion.hpp.

div() [5/9]

template<class T , std::size_t N>

constexpr vector< T, N > math::div ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Divides a vector by a value.

Parameters

x	First value.
y	Second value.

Returns

Result of the division.

Definition at line 1119 of file vector.hpp.

div() [6/9]

template<class T , std::size_t N>

constexpr vector< T, N > math::div ( const vector< T, N > &  x, T  y  )

inlineconstexprnoexcept

Divides a vector by a value.

Parameters

x	First value.
y	Second value.

Returns

Result of the division.

Definition at line 1132 of file vector.hpp.

div() [7/9]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::div ( T  a, const matrix< T, N, M > &  b  )

constexprnoexcept

Divides a scalar by a matrix.

Parameters

a	Scalar.
b	Matrix.

Returns

Result of the division.

Definition at line 882 of file math/matrix.hpp.

div() [8/9]

template<class T >

constexpr quaternion< T > math::div ( T  a, const quaternion< T > &  b  )

inlineconstexprnoexcept

Divides a scalar by a quaternion.

Parameters

a	Scalar.
b	Quaternion.

Returns

Result of the division.

Definition at line 590 of file quaternion.hpp.

div() [9/9]

template<class T , std::size_t N>

constexpr vector< T, N > math::div ( T  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Divides a vector by a value.

Parameters

x	First value.
y	Second value.

Returns

Result of the division.

Definition at line 1145 of file vector.hpp.

dot() [1/2]

template<class T >

constexpr T math::dot ( const quaternion< T > &  a, const quaternion< T > &  b  )

inlineconstexprnoexcept

Calculates the dot product of two quaternions.

Parameters

a	First quaternion.
b	Second quaternion.

Returns

Dot product of the two quaternions.

Definition at line 572 of file quaternion.hpp.

dot() [2/2]

template<class T , std::size_t N>

constexpr T math::dot ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Calculates the dot product of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Dot product of the two vectors.

Definition at line 1158 of file vector.hpp.

equal()

template<class T , std::size_t N>

constexpr vector< bool, N > math::equal ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Compares two vectors for equality.

Parameters

x	First vector.
y	Second vector.

Returns

Boolean vector containing the result of the element comparisons.

Definition at line 1171 of file vector.hpp.

euler_from_quat() [1/2]

template<rotation_sequence Sequence, class T >

vec3<T> math::euler_from_quat	(	const quat< T > &	q,
		T	tolerance = T{1e-6}
	)

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 109 of file euler-angles.hpp.

euler_from_quat() [2/2]

template<class T >

vec3<T> math::euler_from_quat ( rotation_sequence  sequence, const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 235 of file euler-angles.hpp.

euler_to_quat() [1/2]

template<rotation_sequence Sequence, class T >

quat<T> math::euler_to_quat

(

const vec3< T > &

angles

)

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 283 of file euler-angles.hpp.

euler_to_quat() [2/2]

template<class T >

quat<T> math::euler_to_quat ( rotation_sequence  sequence, const vec3< T > &  angles  )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 488 of file euler-angles.hpp.

euler_xyx_from_quat()

template<class T >

vec3<T> math::euler_xyx_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 169 of file euler-angles.hpp.

euler_xyx_to_quat()

template<class T >

quat<T> math::euler_xyx_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 422 of file euler-angles.hpp.

euler_xyz_from_quat()

template<class T >

vec3<T> math::euler_xyz_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 199 of file euler-angles.hpp.

euler_xyz_to_quat()

template<class T >

quat<T> math::euler_xyz_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 452 of file euler-angles.hpp.

euler_xzx_from_quat()

template<class T >

vec3<T> math::euler_xzx_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 187 of file euler-angles.hpp.

euler_xzx_to_quat()

template<class T >

quat<T> math::euler_xzx_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 440 of file euler-angles.hpp.

euler_xzy_from_quat()

template<class T >

vec3<T> math::euler_xzy_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 217 of file euler-angles.hpp.

euler_xzy_to_quat()

template<class T >

quat<T> math::euler_xzy_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 470 of file euler-angles.hpp.

euler_yxy_from_quat()

template<class T >

vec3<T> math::euler_yxy_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 193 of file euler-angles.hpp.

euler_yxy_to_quat()

template<class T >

quat<T> math::euler_yxy_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 446 of file euler-angles.hpp.

euler_yxz_from_quat()

template<class T >

vec3<T> math::euler_yxz_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 229 of file euler-angles.hpp.

euler_yxz_to_quat()

template<class T >

quat<T> math::euler_yxz_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 482 of file euler-angles.hpp.

euler_yzx_from_quat()

template<class T >

vec3<T> math::euler_yzx_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 205 of file euler-angles.hpp.

euler_yzx_to_quat()

template<class T >

quat<T> math::euler_yzx_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 458 of file euler-angles.hpp.

euler_yzy_from_quat()

template<class T >

vec3<T> math::euler_yzy_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 175 of file euler-angles.hpp.

euler_yzy_to_quat()

template<class T >

quat<T> math::euler_yzy_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 428 of file euler-angles.hpp.

euler_zxy_from_quat()

template<class T >

vec3<T> math::euler_zxy_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 211 of file euler-angles.hpp.

euler_zxy_to_quat()

template<class T >

quat<T> math::euler_zxy_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 464 of file euler-angles.hpp.

euler_zxz_from_quat()

template<class T >

vec3<T> math::euler_zxz_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 163 of file euler-angles.hpp.

euler_zxz_to_quat()

template<class T >

quat<T> math::euler_zxz_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 416 of file euler-angles.hpp.

euler_zyx_from_quat()

template<class T >

vec3<T> math::euler_zyx_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 223 of file euler-angles.hpp.

euler_zyx_to_quat()

template<class T >

quat<T> math::euler_zyx_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 476 of file euler-angles.hpp.

euler_zyz_from_quat()

template<class T >

vec3<T> math::euler_zyz_from_quat ( const quat< T > &  q, T  tolerance = T{1e-6}  )

inline

Derives Euler angles from a unit quaternion.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

sequence	Rotation sequence of the Euler angles.
q	Quaternion to convert.
tolerance	Floating-point tolerance, in radians, for handling singularities.

Returns

Euler angles representing the rotation described by the quaternion, in radians.

See also

Bernardes, Evandro & Viollet, Stéphane. (2022). Quaternion to Euler angles conversion: A direct, general and computationally efficient method. PLoS ONE. 17. 10.1371/journal.pone.0276302.

Definition at line 181 of file euler-angles.hpp.

euler_zyz_to_quat()

template<class T >

quat<T> math::euler_zyz_to_quat ( const vec3< T > &  angles )

inline

Constructs a quaternion from Euler angles.

Template Parameters

Sequence	Rotation sequence of the Euler angles.
T	Scalar type.

Parameters

angles

Vector of Euler angles.

Returns

Quaternion.

See also

Diebel, J. (2006). Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix, 58(15-16), 1-35.

Definition at line 434 of file euler-angles.hpp.

floor()

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::floor ( const vector< T, N > &  x )

inlineconstexpr

Performs a element-wise floor operation.

Parameters

x	Input vector.

Returns

Component-wise floor of input vector.

Definition at line 1184 of file vector.hpp.

fma() [1/2]

template<class T , std::size_t N>

constexpr vector< T, N > math::fma ( const vector< T, N > &  x, const vector< T, N > &  y, const vector< T, N > &  z  )

inlineconstexpr

Performs a multiply-add operation.

Parameters

x	Input vector
y	Value to multiply.
z	Value to add.

Returns

Vector containing the multiply-add results.

Definition at line 1197 of file vector.hpp.

fma() [2/2]

template<class T , std::size_t N>

constexpr vector< T, N > math::fma ( const vector< T, N > &  x, T  y, T  z  )

inlineconstexpr

Performs a multiply-add operation.

Parameters

x	Input vector
y	Value to multiply.
z	Value to add.

Returns

Vector containing the multiply-add results.

Definition at line 1210 of file vector.hpp.

fract() [1/2]

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::fract ( const vector< T, N > &  x )

inlineconstexpr

Returns a vector containing the fractional part of each element.

Parameters

x	Input vector.

Returns

Fractional parts of input vector.

Definition at line 1223 of file vector.hpp.

fract() [2/2]

template<std::floating_point T>

T math::fract ( T  x )

inline

Returns the fractional part of a floating-point number.

Template Parameters

T	Floating-point type.

Parameters

x	Floating-point number.

Returns

Fractional part of x.

Definition at line 38 of file fract.hpp.

get() [1/8]

template<std::size_t I, class T , std::size_t N>

constexpr const T && math::get ( const math::vector< T, N > &&  v )

inlineconstexprnoexcept

Extracts the Ith element from a vector.

Template Parameters

I	Index of an element.
T	Element type.
N	Number of elements.

Parameters

v	Vector from which to extract an element.

Returns

Reference to the Ith element of v.

Definition at line 1250 of file vector.hpp.

get() [2/8]

template<std::size_t I, class T , std::size_t N>

constexpr const T & math::get ( const math::vector< T, N > &  v )

inlineconstexprnoexcept

Extracts the Ith element from a vector.

Template Parameters

I	Index of an element.
T	Element type.
N	Number of elements.

Parameters

v	Vector from which to extract an element.

Returns

Reference to the Ith element of v.

Definition at line 1243 of file vector.hpp.

get() [3/8]

template<std::size_t I, class T , std::size_t N, std::size_t M>

constexpr const matrix< T, N, M >::column_vector_type && math::get ( const matrix< T, N, M > &&  m )

inlineconstexprnoexcept

Extracts the Ith column from a matrix.

Template Parameters

I	Index of a column.
T	Element type.
N	Number of columns.
M	Number of rows.

Parameters

m	Matrix from which to extract a column.

Returns

Reference to the Ith column of m.

Definition at line 909 of file math/matrix.hpp.

get() [4/8]

template<std::size_t I, class T , std::size_t N, std::size_t M>

constexpr const matrix< T, N, M >::column_vector_type & math::get ( const matrix< T, N, M > &  m )

inlineconstexprnoexcept

Extracts the Ith column from a matrix.

Template Parameters

I	Index of a column.
T	Element type.
N	Number of columns.
M	Number of rows.

Parameters

m	Matrix from which to extract a column.

Returns

Reference to the Ith column of m.

Definition at line 902 of file math/matrix.hpp.

get() [5/8]

template<std::size_t I, class T , std::size_t N>

constexpr T && math::get ( math::vector< T, N > &&  v )

inlineconstexprnoexcept

Extracts the Ith element from a vector.

Template Parameters

I	Index of an element.
T	Element type.
N	Number of elements.

Parameters

v	Vector from which to extract an element.

Returns

Reference to the Ith element of v.

Definition at line 1236 of file vector.hpp.

get() [6/8]

template<std::size_t I, class T , std::size_t N>

constexpr T & math::get ( math::vector< T, N > &  v )

inlineconstexprnoexcept

Extracts the Ith element from a vector.

Template Parameters

I	Index of an element.
T	Element type.
N	Number of elements.

Parameters

v	Vector from which to extract an element.

Returns

Reference to the Ith element of v.

Definition at line 1229 of file vector.hpp.

get() [7/8]

template<std::size_t I, class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M >::column_vector_type && math::get ( matrix< T, N, M > &&  m )

inlineconstexprnoexcept

Extracts the Ith column from a matrix.

Template Parameters

I	Index of a column.
T	Element type.
N	Number of columns.
M	Number of rows.

Parameters

m	Matrix from which to extract a column.

Returns

Reference to the Ith column of m.

Definition at line 895 of file math/matrix.hpp.

get() [8/8]

template<std::size_t I, class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M >::column_vector_type & math::get ( matrix< T, N, M > &  m )

inlineconstexprnoexcept

Extracts the Ith column from a matrix.

Template Parameters

I	Index of a column.
T	Element type.
N	Number of columns.
M	Number of rows.

Parameters

m	Matrix from which to extract a column.

Returns

Reference to the Ith column of m.

Definition at line 888 of file math/matrix.hpp.

greater_than()

template<class T , std::size_t N>

constexpr vector< bool, N > math::greater_than ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Performs a element-wise greater-than comparison of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Boolean vector containing the result of the element comparisons.

Definition at line 1264 of file vector.hpp.

greater_than_equal()

template<class T , std::size_t N>

constexpr vector< bool, N > math::greater_than_equal ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Performs a element-wise greater-than or equal-to comparison of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Boolean vector containing the result of the element comparisons.

Definition at line 1277 of file vector.hpp.

horizontal_fov()

template<class T >

T math::horizontal_fov	(	T	v,
		T	r
	)

Calculates a horizontal FoV given a vertical FoV and aspect ratio.

Parameters

v	Vertical FoV, in radians.
r	Ratio of width to height.

Returns

Horizontal FoV, in radians.

See also

https://en.wikipedia.org/wiki/Field_of_view_in_video_games

Definition at line 40 of file math/projection.hpp.

inf_perspective_half_z_reverse()

template<class T >

mat4<T> math::inf_perspective_half_z_reverse	(	T	vertical_fov,
		T	aspect_ratio,
		T	near
	)

Constructs a perspective projection matrix, with an infinite far plane, which will transform the near and far clipping planes to [1, 0], respectively.

Parameters

vertical_fov	Vertical field of view angle, in radians.
aspect_ratio	Aspect ratio which determines the horizontal field of view.
near	Distance to the near clipping plane.

Returns

Perspective projection matrix.

Definition at line 319 of file math/projection.hpp.

inf_perspective_half_z_reverse_inv()

template<class T >

std::tuple<mat4<T>, mat4<T> > math::inf_perspective_half_z_reverse_inv	(	T	vertical_fov,
		T	aspect_ratio,
		T	near
	)

Constructs a perspective projection matrix, with an infinite far plane, which will transform the near and far clipping planes to [1, 0], respectively, along with its inverse.

Parameters

vertical_fov	Vertical field of view angle, in radians.
aspect_ratio	Aspect ratio which determines the horizontal field of view.
near	Distance to the near clipping plane.

Returns

Tuple containing the perspective projection matrix, followed by its inverse.

Note

Constructing the inverse perspective projection matrix from projection parameters is faster and more precise than inverting matrix.

Definition at line 345 of file math/projection.hpp.

inv_length() [1/2]

template<class T >

T math::inv_length ( const quaternion< T > &  q )

inline

Calculates the inverse length of a quaternion.

Parameters

q	Quaternion to calculate the inverse length of.

Returns

Inverse length of the quaternion.

Definition at line 596 of file quaternion.hpp.

inv_length() [2/2]

template<std::floating_point T, std::size_t N>

T math::inv_length ( const vector< T, N > &  x )

inline

Calculates the inverse length of a vector.

Parameters

x	Vector of which to calculate the inverse length.

Returns

Inverse length of the vector.

Definition at line 1283 of file vector.hpp.

inverse() [1/2]

template<class T , std::size_t N>

constexpr matrix<T, N, N> math::inverse ( const matrix< T, N, N > &  m )

constexprnoexcept

Calculates the inverse of a square matrix.

Parameters

m	Square matrix.

Returns

Inverse of matrix m.

Warning

Currently only implemented for 2x2, 3x3, and 4x4 matrices.

inverse() [2/2]

template<class T >

transform< T > math::inverse ( const transform< T > &  t )

noexcept

Calculates the inverse of a transform.

Parameters

t	Transform of which to take the inverse.

Definition at line 155 of file transform.hpp.

length() [1/2]

template<class T >

T math::length ( const quaternion< T > &  q )

inline

Calculates the length of a quaternion.

Parameters

q	Quaternion to calculate the length of.

Returns

Length of the quaternion.

Definition at line 602 of file quaternion.hpp.

length() [2/2]

template<std::floating_point T, std::size_t N>

T math::length ( const vector< T, N > &  x )

inline

Calculates the length of a vector.

Parameters

x	Vector of which to calculate the length.

Returns

Length of the vector.

Definition at line 1289 of file vector.hpp.

lerp() [1/2]

template<class T >

constexpr quaternion< T > math::lerp ( const quaternion< T > &  a, const quaternion< T > &  b, T  t  )

inlineconstexprnoexcept

Performs linear interpolation between two quaternions.

Parameters

a	First quaternion.
b	Second quaternion.
t	Interpolation factor.

Returns

Interpolated quaternion.

Definition at line 608 of file quaternion.hpp.

lerp() [2/2]

template<typename T , typename S = float>

constexpr T math::lerp ( const T &  x, const T &  y, S  a  )

constexprnoexcept

Linearly interpolates between x and y.

Requires the following operators to be defined:

T operator+(const T&, const T&);
T operator-(const T&, const T&);
T operator*(const T&, S);

Template Parameters

T	Value type.
S	Scalar type.

Definition at line 41 of file interpolation.hpp.

lerp_angle()

template<typename T >

constexpr T math::lerp_angle ( T  x, T  y, T  a  )

constexpr

Linearly interpolates between two angles, x and y.

Template Parameters

T	Value type.

Parameters

x	Start angle, in radians.
y	End angle, in radians.
a	Interpolation ratio.

Returns

Interpolated angle, in radians.

Definition at line 56 of file interpolation.hpp.

less_than()

template<class T , std::size_t N>

constexpr vector< bool, N > math::less_than ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Performs a element-wise less-than comparison of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Boolean vector containing the result of the element comparisons.

Definition at line 1302 of file vector.hpp.

less_than_equal()

template<class T , std::size_t N>

constexpr vector< bool, N > math::less_than_equal ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Performs a element-wise less-than or equal-to comparison of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Boolean vector containing the result of the element comparisons.

Definition at line 1315 of file vector.hpp.

log_lerp()

template<typename T , typename S = float>

T math::log_lerp	(	const T &	x,
		const T &	y,
		S	a
	)

Logarithmically interpolates between x and y.

Warning

Undefined behavior when x is zero.

Template Parameters

T	Value type.
S	Scalar type.

Definition at line 70 of file interpolation.hpp.

logical_not()

template<class T , std::size_t N>

constexpr vector< bool, N > math::logical_not ( const vector< T, N > &  x )

inlineconstexprnoexcept

Logically inverts a boolean vector.

Parameters

x	Vector to be inverted.

Returns

Logically inverted vector.

Definition at line 1437 of file vector.hpp.

look_at_rh()

template<class T >

constexpr mat4< T > math::look_at_rh ( const vec3< T > &  position, const vec3< T > &  target, const vec3< T > &  up  )

constexpr

Constructs a right-handed viewing transformation matrix.

Parameters

position	Position of the view point.
target	Position of the target.
up	Unit vector pointing in the up direction.

Returns

Viewing transformation matrix.

Definition at line 983 of file math/matrix.hpp.

look_at_rh_inv()

template<class T >

constexpr std::tuple< mat4< T >, mat4< T > > math::look_at_rh_inv ( const vec3< T > &  position, const vec3< T > &  target, const vec3< T > &  up  )

constexpr

Constructs a right-handed viewing transformation matrix and its inverse.

Parameters

position	Position of the view point.
target	Position of the target.
up	Unit vector pointing in the up direction.

Returns

Tuple containing the viewing transformation matrix, followed by its inverse.

Note

Constructing the inverse viewing transformation matrix from viewing parameters is faster and more precise than inverting matrix.

Definition at line 1000 of file math/matrix.hpp.

look_rotation()

template<class T >

quaternion< T > math::look_rotation	(	const vec3< T > &	forward,
		vec3< T >	up
	)

Creates a unit quaternion rotation using forward and up vectors.

Parameters

forward	Unit forward vector.
up	Unit up vector.

Returns

Unit rotation quaternion.

Definition at line 618 of file quaternion.hpp.

map()

template<class T >

constexpr T math::map ( T  x, T  from_min, T  from_max, T  to_min, T  to_max  )

constexprnoexcept

Remaps a number from one range to another.

Parameters

x	Value to remap.
from_min	Start of the first range.
from_max	End of the first range.
to_min	Start of the second range.
to_max	End of the second range.

Returns

Unclamped remapped value.

Definition at line 37 of file map.hpp.

max() [1/2]

template<class T , std::size_t N>

constexpr T math::max ( const vector< T, N > &  x )

inlineconstexpr

Returns the value of the greatest element in a vector.

Parameters

x	Input vector.

Returns

Value of the greatest element in the input vector.

Definition at line 1334 of file vector.hpp.

max() [2/2]

template<class T , std::size_t N>

constexpr vector< T, N > math::max ( const vector< T, N > &  x, const vector< T, N > &  y  )

constexpr

Returns a vector containing the maximum elements of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Maximum elements of the two vectors.

Definition at line 1328 of file vector.hpp.

min() [1/2]

template<class T , std::size_t N>

constexpr T math::min ( const vector< T, N > &  x )

inlineconstexpr

Returns the value of the smallest element in a vector.

Parameters

x	Input vector.

Returns

Value of the smallest element in the input vector.

Definition at line 1353 of file vector.hpp.

min() [2/2]

template<class T , std::size_t N>

constexpr vector< T, N > math::min ( const vector< T, N > &  x, const vector< T, N > &  y  )

constexpr

Returns a vector containing the minimum elements of two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Minimum elements of the two vectors.

Definition at line 1347 of file vector.hpp.

mod() [1/2]

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::mod ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexpr

Calculates the element-wise remainder of the division operation x / y.

Parameters

x	First vector.
y	Second vector.

Returns

Remainders of x / y.

Definition at line 1366 of file vector.hpp.

mod() [2/2]

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::mod ( const vector< T, N > &  x, T  y  )

inlineconstexpr

Calculates the element-wise remainder of the division operation x / y.

Parameters

x	First vector.
y	Second vector.

Returns

Remainders of x / y.

Definition at line 1379 of file vector.hpp.

mul() [1/13]

template<class T , std::size_t N, std::size_t M, std::size_t P>

constexpr matrix< T, P, M > math::mul ( const matrix< T, N, M > &  a, const matrix< T, P, N > &  b  )

constexprnoexcept

Multiplies two matrices.

Template Parameters

T	Matrix element type.
N	Number of columns in matrix a, and rows in matrix b.
M	Number of rows in matrix a.
P	Number of columns in matrix b.

Parameters

a	First matrix.
b	Second matrix.

Returns

Product of a * b.

Definition at line 1027 of file math/matrix.hpp.

mul() [2/13]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M >::column_vector_type math::mul ( const matrix< T, N, M > &  a, const typename matrix< T, N, M >::row_vector_type &  b  )

constexprnoexcept

Calculates the product of a matrix and a row vector.

Parameters

a	Matrix.
b	Row vector

Returns

Product of the matrix and the row vector.

Definition at line 1066 of file math/matrix.hpp.

mul() [3/13]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::mul ( const matrix< T, N, M > &  a, T  b  )

constexprnoexcept

Multiplies a matrix by a scalar.

Parameters

a	Matrix.
b	Scalar.

Returns

Product of the matrix and the scalar.

Definition at line 1053 of file math/matrix.hpp.

mul() [4/13]

template<class T >

constexpr quaternion< T > math::mul ( const quaternion< T > &  a, const quaternion< T > &  b  )

constexprnoexcept

Multiplies two quaternions.

Parameters

a	First quaternion.
b	Second quaternion.

Returns

Product of the two quaternions.

Definition at line 635 of file quaternion.hpp.

mul() [5/13]

template<class T >

constexpr quaternion< T > math::mul ( const quaternion< T > &  a, T  b  )

inlineconstexprnoexcept

Multiplies a quaternion by a scalar.

Parameters

a	First value.
b	Second value.

Returns

Product of the quaternion and scalar.

Definition at line 647 of file quaternion.hpp.

mul() [6/13]

template<class T >

constexpr vec3< T > math::mul ( const quaternion< T > &  q, const vec3< T > &  v  )

constexprnoexcept

Rotates a vector by a unit quaternion.

Parameters

q	Unit quaternion.
v	Vector to rotate.

Returns

Rotated vector.

Warning

q must be a unit quaternion.

See also

https://fgiesen.wordpress.com/2019/02/09/rotating-a-single-vector-using-a-quaternion/

Definition at line 653 of file quaternion.hpp.

mul() [7/13]

template<class T >

constexpr vector< T, 3 > math::mul ( const transform< T > &  t, const vector< T, 3 > &  v  )

constexprnoexcept

Transforms a vector by a transform.

Parameters

t	Transform.
v	Vector.

Returns

Transformed vector.

Definition at line 172 of file transform.hpp.

mul() [8/13]

template<class T >

transform< T > math::mul	(	const transform< T > &	x,
		const transform< T > &	y
	)

Combines two transforms.

Parameters

x	First transform.
y	Second transform.

Returns

Product of the two transforms.

Definition at line 161 of file transform.hpp.

mul() [9/13]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M >::row_vector_type math::mul ( const typename matrix< T, N, M >::column_vector_type &  a, const matrix< T, N, M > &  b  )

constexprnoexcept

Calculates the product of a column vector and a matrix.

Parameters

a	Column vector.
b	Matrix.

Returns

Product of the column vector and the matrix.

Definition at line 1079 of file math/matrix.hpp.

mul() [10/13]

template<class T >

constexpr vec3< T > math::mul ( const vec3< T > &  v, const quaternion< T > &  q  )

inlineconstexprnoexcept

Rotates a vector by the inverse of a unit quaternion.

Parameters

v	Vector to rotate.
q	Unit quaternion.

Returns

Rotated vector.

Warning

q must be a unit quaternion.

Definition at line 660 of file quaternion.hpp.

mul() [11/13]

template<class T >

constexpr vector< T, 3 > math::mul ( const vector< T, 3 > &  v, const transform< T > &  t  )

inlineconstexprnoexcept

Transforms a vector by the inverse of a transform.

Parameters

v	Vector.
t	Transform.

Returns

Transformed vector.

Definition at line 178 of file transform.hpp.

mul() [12/13]

template<class T , std::size_t N>

constexpr vector< T, N > math::mul ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Multiplies two values.

Parameters

x	First value.
y	Second value.

Returns

Product of the two values.

Definition at line 1392 of file vector.hpp.

mul() [13/13]

template<class T , std::size_t N>

constexpr vector< T, N > math::mul ( const vector< T, N > &  x, T  y  )

inlineconstexprnoexcept

Multiplies two values.

Parameters

x	First value.
y	Second value.

Returns

Product of the two values.

Definition at line 1405 of file vector.hpp.

negate() [1/2]

template<class T >

constexpr quaternion< T > math::negate ( const quaternion< T > &  q )

inlineconstexprnoexcept

Negates a quaternion.

Parameters

q	Quaternion to negate.

Returns

Negated quaternion.

Definition at line 667 of file quaternion.hpp.

negate() [2/2]

template<class T , std::size_t N>

constexpr vector< T, N > math::negate ( const vector< T, N > &  x )

inlineconstexprnoexcept

Negates a vector.

Parameters

x	Vector to negate.

Returns

Negated vector.

Definition at line 1418 of file vector.hpp.

nlerp()

template<class T >

quaternion< T > math::nlerp	(	const quaternion< T > &	a,
		const quaternion< T > &	b,
		T	t
	)

Performs normalized linear interpolation between two quaternions.

Parameters

a	First quaternion.
b	Second quaternion.
t	Interpolation factor.

Returns

Interpolated quaternion.

Definition at line 673 of file quaternion.hpp.

normalize() [1/2]

template<class T >

quaternion< T > math::normalize ( const quaternion< T > &  q )

inline

Normalizes a quaternion.

Parameters

q	Quaternion to normalize.

Returns

Unit quaternion.

Definition at line 679 of file quaternion.hpp.

normalize() [2/2]

template<std::floating_point T, std::size_t N>

vector< T, N > math::normalize ( const vector< T, N > &  x )

inline

Calculates the unit vector in the same direction as the original vector.

Parameters

x	Vector to normalize.

Returns

Unit vector.

Definition at line 1424 of file vector.hpp.

not_equal()

template<class T , std::size_t N>

constexpr vector< bool, N > math::not_equal ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Compares two vectors for inequality.

Parameters

x	First vector.
y	Second vector.

Returns

Boolean vector containing the result of the element comparisons.

Definition at line 1450 of file vector.hpp.

ortho()

template<class T >

constexpr mat4<T> math::ortho ( T  left, T  right, T  bottom, T  top, T  near, T  far  )

constexprnoexcept

Constructs an orthographic projection matrix which will transform the near and far clipping planes to [-1, 1], respectively.

Parameters

left	Signed distance to the left clipping plane.
right	Signed distance to the right clipping plane.
bottom	Signed distance to the bottom clipping plane.
top	Signed distance to the top clipping plane.
near	Signed distance to the near clipping plane.
far	Signed distance to the far clipping plane.

Returns

Orthographic projection matrix.

Definition at line 74 of file math/projection.hpp.

ortho_half_z()

template<class T >

constexpr mat4<T> math::ortho_half_z ( T  left, T  right, T  bottom, T  top, T  near, T  far  )

constexprnoexcept

Constructs an orthographic projection matrix which will transform the near and far clipping planes to [0, 1], respectively.

Parameters

left	Signed distance to the left clipping plane.
right	Signed distance to the right clipping plane.
bottom	Signed distance to the bottom clipping plane.
top	Signed distance to the top clipping plane.
near	Signed distance to the near clipping plane.
far	Signed distance to the far clipping plane.

Returns

Orthographic projection matrix.

Definition at line 135 of file math/projection.hpp.

ortho_half_z_inv()

template<class T >

constexpr std::tuple<mat4<T>, mat4<T> > math::ortho_half_z_inv ( T  left, T  right, T  bottom, T  top, T  near, T  far  )

constexprnoexcept

Constructs an orthographic projection matrix which will transform the near and far clipping planes to [0, 1], respectively, along with its inverse.

Parameters

left	Signed distance to the left clipping plane.
right	Signed distance to the right clipping plane.
bottom	Signed distance to the bottom clipping plane.
top	Signed distance to the top clipping plane.
near	Signed distance to the near clipping plane.
far	Signed distance to the far clipping plane.

Returns

Tuple containing the orthographic projection matrix, followed by its inverse.

Note

Constructing the inverse orthographic projection matrix from projection parameters is faster and more precise than inverting matrix.

Definition at line 161 of file math/projection.hpp.

ortho_inv()

template<class T >

constexpr std::tuple<mat4<T>, mat4<T> > math::ortho_inv ( T  left, T  right, T  bottom, T  top, T  near, T  far  )

constexprnoexcept

Constructs an orthographic projection matrix which will transform the near and far clipping planes to [-1, 1], respectively, along with its inverse.

Parameters

left	Signed distance to the left clipping plane.
right	Signed distance to the right clipping plane.
bottom	Signed distance to the bottom clipping plane.
top	Signed distance to the top clipping plane.
near	Signed distance to the near clipping plane.
far	Signed distance to the far clipping plane.

Returns

Tuple containing the orthographic projection matrix, followed by its inverse.

Note

Constructing the inverse orthographic projection matrix from projection parameters is faster and more precise than inverting matrix.

Definition at line 100 of file math/projection.hpp.

perspective()

template<class T >

mat4<T> math::perspective	(	T	vertical_fov,
		T	aspect_ratio,
		T	near,
		T	far
	)

Constructs a perspective projection matrix which will transform the near and far clipping planes to [-1, 1], respectively.

Parameters

vertical_fov	Vertical field of view angle, in radians.
aspect_ratio	Aspect ratio which determines the horizontal field of view.
near	Distance to the near clipping plane.
far	Distance to the far clipping plane.

Returns

Perspective projection matrix.

Definition at line 194 of file math/projection.hpp.

perspective_half_z()

template<class T >

mat4<T> math::perspective_half_z	(	T	vertical_fov,
		T	aspect_ratio,
		T	near,
		T	far
	)

Constructs a perspective projection matrix which will transform the near and far clipping planes to [0, 1], respectively.

Parameters

vertical_fov	Vertical field of view angle, in radians.
aspect_ratio	Aspect ratio which determines the horizontal field of view.
near	Distance to the near clipping plane.
far	Distance to the far clipping plane.

Returns

Perspective projection matrix.

Definition at line 257 of file math/projection.hpp.

perspective_half_z_inv()

template<class T >

std::tuple<mat4<T>, mat4<T> > math::perspective_half_z_inv	(	T	vertical_fov,
		T	aspect_ratio,
		T	near,
		T	far
	)

Constructs a perspective projection matrix which will transform the near and far clipping planes to [0, 1], respectively, along with its inverse.

Parameters

vertical_fov	Vertical field of view angle, in radians.
aspect_ratio	Aspect ratio which determines the horizontal field of view.
near	Distance to the near clipping plane.
far	Distance to the far clipping plane.

Returns

Tuple containing the perspective projection matrix, followed by its inverse.

Note

Constructing the inverse perspective projection matrix from projection parameters is faster and more precise than inverting matrix.

Definition at line 284 of file math/projection.hpp.

perspective_inv()

template<class T >

std::tuple<mat4<T>, mat4<T> > math::perspective_inv	(	T	vertical_fov,
		T	aspect_ratio,
		T	near,
		T	far
	)

Constructs a perspective projection matrix which will transform the near and far clipping planes to [-1, 1], respectively, along with its inverse.

Parameters

vertical_fov	Vertical field of view angle, in radians.
aspect_ratio	Aspect ratio which determines the horizontal field of view.
near	Distance to the near clipping plane.
far	Distance to the far clipping plane.

Returns

Arraay containing the perspective projection matrix, followed by its inverse.

Note

Constructing the inverse perspective projection matrix from projection parameters is faster and more precise than inverting matrix.

Definition at line 221 of file math/projection.hpp.

pow() [1/2]

template<std::floating_point T, std::size_t N>

vector< T, N > math::pow ( const vector< T, N > &  x, const vector< T, N > &  y  )

inline

Raises each element to a power.

Parameters

x	Input vector
y	Exponent.

Returns

Vector with its elements raised to the given power.

Definition at line 1463 of file vector.hpp.

pow() [2/2]

template<std::floating_point T, std::size_t N>

vector< T, N > math::pow ( const vector< T, N > &  x, T  y  )

inline

Raises each element to a power.

Parameters

x	Input vector
y	Exponent.

Returns

Vector with its elements raised to the given power.

Definition at line 1476 of file vector.hpp.

quaternion_cast()

template<class T >

quaternion< T > math::quaternion_cast

(

const mat3< T > &

m

)

Converts a 3x3 rotation matrix to a quaternion.

Parameters

m	Rotation matrix.

Returns

Unit quaternion representing the rotation described by m.

Definition at line 803 of file quaternion.hpp.

radians()

template<class T >

constexpr T math::radians ( T  degrees )

inlineconstexprnoexcept

Converts an angle given in degrees to radians.

Parameters

radians

Angle in radians.

Returns

Angle in degrees.

Definition at line 48 of file angles.hpp.

random()

template<typename T = float>

T math::random	(	T	start,
		T	end
	)

Generates a pseudo-random floating point number on [start, end) using std::rand().

Warning

Don't forget to seed with std::srand() before using!

Parameters

start	Start of the range (inclusive).
end	End of the range (exclusive).

Returns

Pseudo-random floating point number.

Definition at line 38 of file random.hpp.

rotate()

template<class T >

mat3< T > math::rotate	(	T	angle,
		const vector< T, 3 > &	axis
	)

Constructs a rotation matrix.

Parameters

angle	Angle of rotation, in radians.
axis	Axis of rotation.

Returns

Rotation matrix.

Definition at line 1085 of file math/matrix.hpp.

rotate_towards()

template<class T >

quaternion< T > math::rotate_towards	(	const vec3< T > &	from,
		const vec3< T > &	to,
		T	max_angle
	)

Constructs a quaternion representing an angle-limited rotation from one direction towards another direction.

Parameters

from	Unit vector pointing in the source direction.
to	Unit vector pointing in the target direction.
max_angle	Maximum angle of rotation, in radians.

Returns

Unit quaternion representing a rotation from direction from towards direction to.

Warning

from and to must be unit vectors.

Definition at line 716 of file quaternion.hpp.

rotate_x()

template<class T >

mat3< T > math::rotate_x

(

T

angle

)

Produces a matrix which rotates Cartesian coordinates about the x-axis by a given angle.

Parameters

angle

Angle of rotation, in radians.

Returns

Rotation matrix.

Definition at line 1106 of file math/matrix.hpp.

rotate_y()

template<class T >

mat3< T > math::rotate_y

(

T

angle

)

Produces a matrix which rotates Cartesian coordinates about the y-axis by a given angle.

Parameters

angle

Angle of rotation, in radians.

Returns

Rotation matrix.

Definition at line 1120 of file math/matrix.hpp.

rotate_z()

template<class T >

mat3< T > math::rotate_z

(

T

angle

)

Produces a matrix which rotates Cartesian coordinates about the z-axis by a given angle.

Parameters

angle

Angle of rotation, in radians.

Returns

Rotation matrix.

Definition at line 1134 of file math/matrix.hpp.

rotation()

template<class T >

quaternion< T > math::rotation	(	const vec3< T > &	from,
		const vec3< T > &	to,
		T	tolerance = T{1e-6}
	)

Constructs a quaternion representing the minimum rotation from one direction to another direction.

Parameters

from	Unit vector pointing in the source direction.
to	Unit vector pointing in the target direction.
tolerance	Floating-point tolerance.

Returns

Unit quaternion representing the minimum rotation from direction from to direction to.

Warning

from and to must be unit vectors.

Definition at line 691 of file quaternion.hpp.

rotation_axes()

template<std::integral T>

constexpr vec3<T> math::rotation_axes ( rotation_sequence  sequence )

inlineconstexprnoexcept

Returns the indices of the axes of a rotation sequence.

Template Parameters

T	Index type.

Parameters

sequence

Rotation sequence.

Returns

Vector containing the indices of the axes of sequence.

Definition at line 87 of file euler-angles.hpp.

round()

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::round ( const vector< T, N > &  x )

inlineconstexpr

Performs a element-wise round operation.

Parameters

x	Input vector

Returns

Component-wise round of input vector.

Definition at line 1489 of file vector.hpp.

scale()

template<class T >

constexpr mat4< T > math::scale ( const vec3< T > &  v )

constexpr

Constructs a scale matrix.

Parameters

v	Scale vector.

Returns

Scale matrix.

Definition at line 1148 of file math/matrix.hpp.

sign()

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::sign ( const vector< T, N > &  x )

inlineconstexpr

Returns a vector containing the signs of each element.

Parameters

x	Input vector

Returns

Signs of input vector elements.

Definition at line 1502 of file vector.hpp.

signed_angle()

template<std::floating_point T>

T math::signed_angle	(	const vector< T, 3 > &	x,
		const vector< T, 3 > &	y,
		const vector< T, 3 > &	n
	)

Calculates the signed angle between two direction vectors about axis.

Parameters

from	First direction vector.
to	Second direction vector.
axis	Axis direction vector.

Returns

Signed angle between from and to about axis, in radians.

Definition at line 1508 of file vector.hpp.

slerp()

template<class T >

quaternion< T > math::slerp	(	const quaternion< T > &	a,
		const quaternion< T > &	b,
		T	t,
		T	tolerance = T{1e-6}
	)

Performs spherical linear interpolation between two quaternions.

Parameters

a	First quaternion.
b	Second quaternion.
t	Interpolation factor.
tolerance	Floating-point tolerance.

Returns

Interpolated quaternion.

Definition at line 724 of file quaternion.hpp.

sqr_distance()

template<class T , std::size_t N>

constexpr T math::sqr_distance ( const vector< T, N > &  p0, const vector< T, N > &  p1  )

inlineconstexprnoexcept

Calculates the square distance between two points.

The square distance can be calculated faster than the distance because a call to std::sqrt is saved.

Parameters

p0	First of two points.
p1	Second of two points.

Returns

Square distance between the two points.

Definition at line 1514 of file vector.hpp.

sqr_length() [1/2]

template<class T >

constexpr T math::sqr_length ( const quaternion< T > &  q )

inlineconstexprnoexcept

Calculates the square length of a quaternion.

The square length can be calculated faster than the length because a call to std::sqrt is saved.

Parameters

q	Quaternion to calculate the square length of.

Returns

Square length of the quaternion.

Definition at line 744 of file quaternion.hpp.

sqr_length() [2/2]

template<class T , std::size_t N>

constexpr T math::sqr_length ( const vector< T, N > &  x )

inlineconstexprnoexcept

Calculates the square length of a vector.

The square length can be calculated faster than the length because a call to std::sqrt is saved.

Parameters

x	Vector of which to calculate the square length.

Returns

Square length of the vector.

Definition at line 1520 of file vector.hpp.

sqrt() [1/2]

template<std::floating_point T, std::size_t N>

vector<T, N> math::sqrt

(

const vector< T, N > &

x

)

Takes the square root of each element.

Parameters

x	Input vector

Returns

Square roots of the input vector elements.

sqrt() [2/2]

template<std::floating_point T, std::size_t N>

vector<T, N> math::sqrt ( const vector< T, N > &  x, const vector< T, N > &  y  )

inline

Definition at line 1533 of file vector.hpp.

sub() [1/9]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::sub ( const matrix< T, N, M > &  a, const matrix< T, N, M > &  b  )

constexprnoexcept

Subtracts a matrix from another matrix.

Parameters

a	First matrix.
b	Second matrix.

Returns

Difference between the two matrices.

Definition at line 1167 of file math/matrix.hpp.

sub() [2/9]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::sub ( const matrix< T, N, M > &  a, T  b  )

constexprnoexcept

Subtracts a scalar from matrix.

Parameters

a	Matrix.
b	Scalar.

Returns

Difference between the matrix and scalar.

Definition at line 1180 of file math/matrix.hpp.

sub() [3/9]

template<class T >

constexpr quaternion< T > math::sub ( const quaternion< T > &  a, const quaternion< T > &  b  )

inlineconstexprnoexcept

Subtracts a quaternion from another quaternion.

Parameters

a	First quaternion.
b	Second quaternion.

Returns

Difference between the quaternions.

Definition at line 750 of file quaternion.hpp.

sub() [4/9]

template<class T >

constexpr quaternion< T > math::sub ( const quaternion< T > &  a, T  b  )

inlineconstexprnoexcept

Subtracts a quaternion and a scalar.

Parameters

a	First value.
b	Second second.

Returns

Difference between the quaternion and scalar.

Definition at line 756 of file quaternion.hpp.

sub() [5/9]

template<class T , std::size_t N>

constexpr vector< T, N > math::sub ( const vector< T, N > &  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Subtracts a value by another value.

Parameters

x	First value.
y	Second value.

Returns

Difference between the two values.

Definition at line 1546 of file vector.hpp.

sub() [6/9]

template<class T , std::size_t N>

constexpr vector< T, N > math::sub ( const vector< T, N > &  x, T  y  )

inlineconstexprnoexcept

Subtracts a value by another value.

Parameters

x	First value.
y	Second value.

Returns

Difference between the two values.

Definition at line 1559 of file vector.hpp.

sub() [7/9]

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, N, M > math::sub ( T  a, const matrix< T, N, M > &  b  )

constexprnoexcept

Subtracts a matrix from a scalar.

Parameters

a	Scalar.
b	Matrix.

Returns

Difference between the scalar and matrix.

Definition at line 1193 of file math/matrix.hpp.

sub() [8/9]

template<class T >

constexpr quaternion< T > math::sub ( T  a, const quaternion< T > &  b  )

inlineconstexprnoexcept

Subtracts a quaternion and a scalar.

Parameters

a	First value.
b	Second second.

Returns

Difference between the quaternion and scalar.

Definition at line 762 of file quaternion.hpp.

sub() [9/9]

template<class T , std::size_t N>

constexpr vector< T, N > math::sub ( T  x, const vector< T, N > &  y  )

inlineconstexprnoexcept

Subtracts a value by another value.

Parameters

x	First value.
y	Second value.

Returns

Difference between the two values.

Definition at line 1572 of file vector.hpp.

sum()

template<class T , std::size_t N>

constexpr T math::sum ( const vector< T, N > &  x )

inlineconstexprnoexcept

Calculates the sum of all elements in a vector.

Parameters

x	Vector to sum.

Returns

Sum of the vector's elements.

Definition at line 1585 of file vector.hpp.

swing_twist()

template<class T >

std::array< quaternion< T >, 2 > math::swing_twist	(	const quaternion< T > &	q,
		const vec3< T > &	twist_axis,
		T	tolerance = T{1e-6}
	)

Decomposes a quaternion into swing and twist rotation components.

Parameters

[in]	q	Unit quaternion to decompose.
[in]	twist_axis	Axis of twist rotation.
[out]	swing	Swing rotation component.
[out]	twist	Twist rotation component.
[in]	tolerance	Floating-point tolerance.

Returns

Array containing the swing rotation component, followed by the twist rotation component.

Warning

q must be a unit quaternion.

twist_axis must be a unit vector.

See also

https://www.euclideanspace.com/maths/geometry/rotations/for/decomposition/

Definition at line 768 of file quaternion.hpp.

swizzle()

template<std::size_t... Indices, class T , std::size_t N>

constexpr vector< T, sizeof...(Indices)> math::swizzle ( const vector< T, N > &  x )

inlineconstexprnoexcept

Makes an m-dimensional vector by rearranging and/or duplicating elements of an n-dimensional vector.

Template Parameters

Indices	List of indices of elements in x.
T	Vector element type.
N	Number of dimensions in x.

Parameters

x	Vector to swizzle.

Returns

Vector containing elements from x in the order specified by Indices. The size of the returned vector is equivalent to the number of indices in Indices.

Definition at line 1591 of file vector.hpp.

trace()

template<class T , std::size_t N>

constexpr T math::trace ( const matrix< T, N, N > &  m )

constexprnoexcept

Calculates the trace of a square matrix.

Parameters

m	Square matrix.

Returns

Sum of elements on the main diagonal.

Definition at line 1206 of file math/matrix.hpp.

translate()

template<class T >

constexpr mat4< T > math::translate ( const vec3< T > &  v )

constexpr

Constructs a translation matrix.

Parameters

v	Translation vector.

Returns

Translation matrix.

Definition at line 1212 of file math/matrix.hpp.

transpose()

template<class T , std::size_t N, std::size_t M>

constexpr matrix< T, M, N > math::transpose ( const matrix< T, N, M > &  m )

constexprnoexcept

Calculates the transpose of a matrix.

Parameters

m	Matrix to transpose.

Returns

Transposed matrix.

Definition at line 1238 of file math/matrix.hpp.

triple()

template<class T >

constexpr T math::triple ( const vector< T, 3 > &  x, const vector< T, 3 > &  y, const vector< T, 3 > &  z  )

inlineconstexprnoexcept

Calculates the triple product of three vectors.

Parameters

x	First vector.
y	Second vector.
z	Third vector.

Returns

Triple product of the three vectors (dot(x, cross(y, z)).

Definition at line 1597 of file vector.hpp.

trunc()

template<std::floating_point T, std::size_t N>

constexpr vector< T, N > math::trunc ( const vector< T, N > &  x )

inlineconstexpr

Performs a element-wise trunc operation.

Parameters

x	Input vector

Returns

Component-wise trunc of input vector.

Definition at line 1610 of file vector.hpp.

vertical_fov()

template<class T >

T math::vertical_fov	(	T	h,
		T	r
	)

Calculates a vertical FoV given a horizontal FoV and aspect ratio.

Parameters

h	Horizontal FoV, in radians.
r	Ratio of width to height.

Returns

Vertical FoV, in radians.

See also

https://en.wikipedia.org/wiki/Field_of_view_in_video_games

Definition at line 56 of file math/projection.hpp.

wrap_degrees()

template<class T >

constexpr T math::wrap_degrees ( T  degrees )

inlineconstexpr

Wraps an angle to [-180, 180].

Parameters

degrees

Angle in degrees.

Returns

Wrapped angle.

Definition at line 60 of file angles.hpp.

wrap_radians()

template<class T >

constexpr T math::wrap_radians ( T  radians )

inlineconstexpr

Wraps an angle to [-pi, pi].

Parameters

radians

Angle in radians.

Returns

Wrapped angle.

Definition at line 72 of file angles.hpp.