All numeric-related classes and functions. More...

Namespaces
namespace	ufo
	All vision-related classes and functions.

Concepts
concept	ufo::Matrix
	Concept satisfied by any specialization of `Mat<Rows, Cols, U>`.

Classes
struct	ufo::Mat< Rows, Cols, T >
	A fixed-size matrix with `Rows` rows and `Cols` columns. More...

struct	ufo::Quat< T >
	Unit quaternion representing an orientation or rotation in 3-D space. More...

struct	ufo::Transform< Dim, T >
	Rigid-body transform: a rotation matrix plus a translation vector. More...

struct	ufo::Vec< Dim, T >
	A fixed-size arithmetic vector of up to 4 dimensions. More...

Typedefs
template<class T = float>
using	ufo::Mat2 = Mat< 2, 2, T >

using	ufo::Quatf = Quat< float >

template<std::floating_point T = float>
using	ufo::Transform2 = Transform< 2, T >

using	ufo::Vec1b = Vec< 1, bool >

Functions
constexpr	ufo::Quat< T >::Quat () noexcept=default
	Default constructor; initializes to the identity rotation (1, 0, 0, 0).

constexpr	ufo::Transform< Dim, T >::Transform () noexcept=default
	Default constructor; initializes to the identity transform (zero translation and identity rotation).

template<std::floating_point T>
T	ufo::angle (Quat< T > const &q) noexcept
	Extracts the rotation angle (in radians) from a unit quaternion.

template<class T , std::size_t Dim, class U >
constexpr Vec< Dim, T >	ufo::cast (Vec< Dim, U > const &v) noexcept
	Casts each element of a vector to a new type.

template<SquareMatrix M>
constexpr auto	ufo::determinant (M const &m) noexcept
	Computes the determinant of a square matrix.

template<std::floating_point T>
constexpr T	ufo::dot (Quat< T > const &a, Quat< T > const &b) noexcept
	Computes the four-component dot product `a.wb.w + a.xb.x + a.yb.y + a.zb.z`.

template<std::floating_point T>
constexpr Vec< 3, T >	ufo::eigenValues (Mat< 3, 3, T > const &m) noexcept
	Computes the eigenvalues of a real symmetric 3×3 matrix, sorted ascending.

template<std::size_t I, std::size_t Rows, std::size_t Cols, class T > requires (I < Rows)
constexpr Vec< Cols, T > &	ufo::get (Mat< Rows, Cols, T > &m) noexcept
	Structured-binding accessor — lvalue reference to row `I`.

template<std::size_t I, std::size_t Dim, class T > requires (I < Dim)
constexpr T &	ufo::get (Vec< Dim, T > &v) noexcept
	Returns a reference to the `I`-th element for structured bindings.

template<std::size_t Dim, std::floating_point T>
constexpr Transform< Dim, T >	ufo::inverse (Transform< Dim, T > const &t)
	Returns the inverse of a rigid-body transform.

template<std::floating_point T>
Vec< 4, bool >	ufo::isnan (Quat< T > const &q) noexcept
	Returns a bool vector indicating which components are NaN.

template<std::floating_point T>
Quat< T >	ufo::mix (Quat< T > const &x, Quat< T > const &y, T a)
	Spherical or linear interpolation between two quaternions.

constexpr	ufo::Quat< T >::operator Mat< 3, 3, T > () const noexcept
	Converts to a row-major 3×3 rotation matrix.

template<std::floating_point U>
constexpr	ufo::Transform< Dim, T >::operator Mat< Dim, Dim, U > () const
	Extracts the rotation component as a `Dim × Dim` matrix.

template<std::floating_point U>
constexpr Vec< Dim, U >	ufo::Transform< Dim, T >::operator() (Vec< Dim, U > const &v) const
	Applies the transform to a vector: `result = R * v + t`.

template<std::size_t Rows, std::size_t Cols, class T >
constexpr Vec< Rows, T >	ufo::operator* (Mat< Rows, Cols, T > const &m, Vec< Cols, T > const &v) noexcept
	Multiplies a matrix by a column vector (`M * v`).

template<std::size_t Rows, std::size_t Shared, std::size_t Cols, class T >
constexpr Mat< Rows, Cols, T >	ufo::operator* (Mat< Rows, Shared, T > const &a, Mat< Shared, Cols, T > const &b) noexcept
	Matrix × matrix product.

template<std::size_t Dim, std::floating_point T>
constexpr Transform< Dim, T >	ufo::operator* (Transform< Dim, T > t1, Transform< Dim, T > const &t2)
	Composes two transforms: `(t1 * t2)(v) = t1(t2(v))`.

constexpr Mat	ufo::Mat< Rows, Cols, T >::operator+ () const noexcept
	Unary identity operator.

constexpr Vec	ufo::Vec< Dim, T >::operator+ () const noexcept
	Unary identity operator.

template<std::size_t Rows, std::size_t Cols, class T >
constexpr Mat< Rows, Cols, T >	ufo::operator+ (Mat< Rows, Cols, T > lhs, Mat< Rows, Cols, T > const &rhs) noexcept
	Element-wise addition of two matrices.

template<std::size_t Rows, std::size_t Cols, class T >
constexpr Mat< Rows, Cols, T >	ufo::operator+ (Mat< Rows, Cols, T > lhs, T rhs) noexcept
	Adds scalar `rhs` to every element of `lhs`.

template<std::floating_point T>
constexpr Quat< T >	ufo::operator+ (Quat< T > const &q) noexcept
	Unary plus — returns a copy of `q` unchanged.

template<std::floating_point T>
constexpr Quat< T >	ufo::operator+ (Quat< T > lhs, Quat< T > const &rhs) noexcept
	Component-wise addition of two quaternions.

template<std::size_t Dim, class T >
constexpr Vec< Dim, T >	ufo::operator+ (Vec< Dim, T > lhs, T rhs) noexcept
	Adds scalar `rhs` to every component of `lhs`.

template<std::size_t Dim, class T >
constexpr Vec< Dim, T >	ufo::operator+ (Vec< Dim, T > lhs, Vec< Dim, T > const &rhs) noexcept
	Component-wise addition of two vectors.

constexpr Mat &	ufo::Mat< Rows, Cols, T >::operator+= (Mat const &rhs) noexcept
	Adds `rhs` element-wise to this matrix.

constexpr Quat &	ufo::Quat< T >::operator+= (Quat const &q) noexcept
	Component-wise addition.

constexpr Mat &	ufo::Mat< Rows, Cols, T >::operator+= (T rhs) noexcept
	Adds scalar `rhs` to every element.

constexpr Vec &	ufo::Vec< Dim, T >::operator+= (T rhs) noexcept
	Adds scalar `rhs` to every component.

constexpr Vec &	ufo::Vec< Dim, T >::operator+= (Vec const &rhs) noexcept
	Adds each component of `rhs` to the corresponding component.

template<std::size_t Rows, std::size_t Cols, class T >
std::ostream &	ufo::operator<< (std::ostream &out, Mat< Rows, Cols, T > const &m)
	Writes a matrix to an output stream, one row per line.

template<std::floating_point T>
std::ostream &	ufo::operator<< (std::ostream &out, Quat< T > const &q)
	Writes the quaternion to an output stream as `"qw: W qx: X qy: Y qz: Z"`.

template<std::size_t Dim, class T >
std::ostream &	ufo::operator<< (std::ostream &out, Vec< Dim, T > const &v)
	Writes a human-readable representation of a vector to a stream.

constexpr Quat &	ufo::Quat< T >::operator= (Quat const &) noexcept=default
	Copy assignment operator.

constexpr Transform &	ufo::Transform< Dim, T >::operator= (Transform const &) noexcept=default
	Default copy assignment operator.

constexpr auto &	ufo::Vec< Dim, T >::operator[] (this auto &self, size_type i) noexcept
	Accesses the element at index `i`.

constexpr auto &	ufo::Quat< T >::operator[] (this auto &self, size_type pos) noexcept
	Component access by index: 0->w, 1->x, 2->y, 3->z.

constexpr auto &	ufo::Mat< Rows, Cols, T >::operator[] (this auto &self, size_type r, size_type c) noexcept
	Accesses the element at row `r`, column `c`.

template<std::floating_point T>
Mat< 4, 4, T >	ufo::orthogonal (T left, T right, T bottom, T top)
	Builds a right-handed 2-D orthographic projection matrix (no depth clipping).

template<std::floating_point T>
T	ufo::roll (Quat< T > const &q) noexcept
	Extracts the roll angle (rotation about the X-axis) in radians.

template<std::floating_point T>
Quat< T >	ufo::rotate (Quat< T > const &q, T angle, Vec< 3, T > axis)
	Appends an angle-axis rotation to a quaternion.

static constexpr size_type	ufo::Quat< T >::size () noexcept
	Returns the number of components (always 4: w, x, y, z).

void	ufo::Quat< T >::swap (Quat &other) noexcept
	Swaps all components with `other`.

template<std::floating_point T>
void	ufo::swap (Quat< T > &lhs, Quat< T > &rhs) noexcept
	Non-member swap — exchanges all components of `lhs` and `rhs`.

constexpr T	ufo::Mat< Rows, Cols, T >::trace () const noexcept
	Computes the trace (sum of diagonal elements) of a square matrix.

template<class T >
constexpr int	ufo::sign (T val) noexcept
	Returns the sign of a value.

template<std::floating_point T>
constexpr T	ufo::radians (T deg) noexcept
	Converts degrees to radians.

template<std::floating_point T>
constexpr T	ufo::degrees (T rad) noexcept
	Converts radians to degrees.

template<class T > requires std::is_arithmetic_v<T>
constexpr T	ufo::ipow (T base, int exp) noexcept
	Computes integer power of a base.

template<std::floating_point T>
constexpr T	ufo::probabilityToLogit (T probability)
	Converts probability to logit value.

template<std::floating_point T>
constexpr T	ufo::logitToProbability (T logit)
	Converts logit value to probability.

Detailed Description

All numeric-related classes and functions.

Author: Daniel Duberg (danie.nosp@m.ldub.nosp@m.erg@g.nosp@m.mail.nosp@m..com)

See also: https://github.com/UnknownFreeOccupied/ufo

Version: 1.0

Date: 2026-02-22

Copyright: BSD 3-Clause License

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Typedef Documentation

◆ Mat2

template<class T = float>

using ufo::Mat2 = typedef Mat<2, 2, T>

Definition at line 751 of file mat.hpp.

◆ Quatf

using ufo::Quatf = typedef Quat<float>

Definition at line 427 of file quat.hpp.

◆ Transform2

template<std::floating_point T = float>

using ufo::Transform2 = typedef Transform<2, T>

Definition at line 382 of file transform.hpp.

◆ Vec1b

using ufo::Vec1b = typedef Vec<1, bool>

Definition at line 502 of file vec.hpp.

Function Documentation

◆ angle()

template<std::floating_point T>

T ufo::angle ( Quat< T > const & q )

noexcept

Extracts the rotation angle (in radians) from a unit quaternion.

Template Parameters

T	Floating-point scalar type.

Parameters

[in] q Unit quaternion.

Returns: Rotation angle in [0, 2π).

Definition at line 852 of file quat.hpp.

◆ cast()

template<class T , std::size_t Dim, class U >

constexpr Vec< Dim, T > ufo::cast ( Vec< Dim, U > const & v )

constexprnoexcept

Casts each element of a vector to a new type.

Template Parameters

T	Target element type.
Dim	Number of dimensions.
U	Source element type.

Parameters

[in] v The source vector.

Returns: A new vector with elements cast to type T.

Definition at line 1042 of file vec.hpp.

◆ degrees()

template<std::floating_point T>

constexpr T ufo::degrees ( T rad )

constexprnoexcept

Converts radians to degrees.

Template Parameters

T	Floating point type (e.g., float, double).

Parameters

[in] rad The angle in radians.

Returns: The angle in degrees.

Definition at line 98 of file math.hpp.

◆ determinant()

template<SquareMatrix M>

constexpr auto ufo::determinant ( M const & m )

constexprnoexcept

Computes the determinant of a square matrix.

Closed-form implementations are provided for 2×2, 3×3, and 4×4 matrices. Requesting any other size triggers a static_assert at compile time.

Template Parameters

M	A square matrix type satisfying `SquareMatrix`.

Parameters

[in] m The matrix.

Returns: The determinant of m as M::value_type.

Definition at line 1056 of file mat.hpp.

◆ dot()

template<std::floating_point T>

constexpr T ufo::dot	(	Quat< T > const &	a,
		Quat< T > const &	b
	)

constexprnoexcept

Computes the four-component dot product a.w*b.w + a.x*b.x + a.y*b.y + a.z*b.z.

Template Parameters

T	Floating-point scalar type.

Parameters

[in]	a	First quaternion.
[in]	b	Second quaternion.

Returns: Scalar dot product.

Definition at line 643 of file quat.hpp.

◆ eigenValues()

template<std::floating_point T>

constexpr Vec< 3, T > ufo::eigenValues ( Mat< 3, 3, T > const & m )

constexprnoexcept

Computes the eigenvalues of a real symmetric 3×3 matrix, sorted ascending.

Uses the closed-form trigonometric solution (Cardano's method adapted for symmetric 3×3 matrices). Only the upper triangle of m is read; the lower triangle is assumed to equal the upper triangle.

Template Parameters

T	Floating-point scalar type.

Parameters

[in] m A real symmetric 3×3 matrix.

Returns: Vec<3, T> with eigenvalues e[0] ≤ e[1] ≤ e[2].

Definition at line 1252 of file mat.hpp.

◆ get() [1/2]

template<std::size_t I, std::size_t Rows, std::size_t Cols, class T >
requires (I < Rows)

constexpr Vec< Cols, T > & ufo::get ( Mat< Rows, Cols, T > & m )

constexprnoexcept

Structured-binding accessor — lvalue reference to row I.

Template Parameters

I	Row index (must satisfy `I < Rows`).
Rows	Number of rows in the matrix.
Cols	Number of columns in the matrix.
T	Element type.

Parameters

[in] m The matrix.

Returns: A mutable reference to row I as Vec<Cols, T>&.

Definition at line 1586 of file mat.hpp.

◆ get() [2/2]

template<std::size_t I, std::size_t Dim, class T >
requires (I < Dim)

constexpr T & ufo::get ( Vec< Dim, T > & v )

constexprnoexcept

Returns a reference to the I-th element for structured bindings.

Template Parameters

I	Zero-based compile-time index (must be less than `Dim`).
Dim	Number of dimensions.
T	Element type.

Parameters

[in] v The vector.

Returns: Lvalue reference to v[I].

Definition at line 1729 of file vec.hpp.

◆ inverse()

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

constexpr Transform< Dim, T > ufo::inverse ( Transform< Dim, T > const & t )

constexpr

Returns the inverse of a rigid-body transform.

For an orthogonal rotation matrix R, the inverse is R^T. The inverse translation is -(R^T * t), so inverse(t)(v) = R^T * (v - t).

Template Parameters

Dim	Spatial dimension.
T	Floating-point scalar type.

Parameters

[in] t The transform to invert (rotation must be orthogonal).

Returns: The inverse transform such that inverse(t) * t = identity.

Definition at line 478 of file transform.hpp.

◆ ipow()

template<class T >
requires std::is_arithmetic_v<T>

constexpr T ufo::ipow	(	T	base,
		int	exp
	)

constexprnoexcept

Computes integer power of a base.

Template Parameters

T	Numeric type (e.g., int, float).

Parameters

[in]	base	The base value.
[in]	exp	The exponent (can be negative).

Returns: The result of raising the base to the exponent.

Definition at line 112 of file math.hpp.

◆ isnan()

template<std::floating_point T>

Vec< 4, bool > ufo::isnan ( Quat< T > const & q )

noexcept

Returns a bool vector indicating which components are NaN.

Template Parameters

T	Floating-point scalar type.

Parameters

[in] q Input quaternion.

Returns: Vec<4, bool> where element i is true iff component i is NaN.

Definition at line 1010 of file quat.hpp.

◆ logitToProbability()

template<std::floating_point T>

constexpr T ufo::logitToProbability ( T logit )

constexpr

Converts logit value to probability.

Template Parameters

T	Floating point type (e.g., float, double).

Parameters

[in] logit The logit value.

Returns: The corresponding probability value in the range [0, 1].

Definition at line 148 of file math.hpp.

◆ mix()

template<std::floating_point T>

Quat< T > ufo::mix	(	Quat< T > const &	x,
		Quat< T > const &	y,
		T	a
	)

Spherical or linear interpolation between two quaternions.

Falls back to lerp when the quaternions are nearly parallel (cos θ > 1 − ε) to avoid division by near-zero sine.

Template Parameters

T	Floating-point scalar type.

Parameters

[in]	x	Start quaternion.
[in]	y	End quaternion.
[in]	a	Interpolation factor in `[0, 1]`.

Returns: Interpolated quaternion (not necessarily unit-length if inputs are not).

Definition at line 746 of file quat.hpp.

◆ operator Mat< 3, 3, T >()

template<std::floating_point T = float>

constexpr ufo::Quat< T >::operator Mat< 3, 3, T > ( ) const

inlineexplicitconstexprnoexcept

Converts to a row-major 3×3 rotation matrix.

Returns: Mat<3, 3, T> where m[row][col] = R_{row,col}.

Assumes the quaternion is unit-length.

Definition at line 255 of file quat.hpp.

◆ operator Mat< Dim, Dim, U >()

template<std::size_t Dim, std::floating_point T = float>

template<std::floating_point U>

constexpr ufo::Transform< Dim, T >::operator Mat< Dim, Dim, U > ( ) const

inlineexplicitconstexpr

Extracts the rotation component as a Dim × Dim matrix.

Template Parameters

U	Target scalar type.

Returns: A copy of the rotation matrix converted to U.

Definition at line 275 of file transform.hpp.

◆ operator()()

template<std::size_t Dim, std::floating_point T = float>

template<std::floating_point U>

constexpr Vec< Dim, U > ufo::Transform< Dim, T >::operator() ( Vec< Dim, U > const & v ) const

inlineconstexpr

Applies the transform to a vector: result = R * v + t.

Template Parameters

U	Scalar type of the vector (may differ from `T`).

Parameters

[in] v Input vector.

Returns: Transformed vector.

Definition at line 323 of file transform.hpp.

◆ operator*() [1/3]

template<std::size_t Rows, std::size_t Cols, class T >

constexpr Vec< Rows, T > ufo::operator*	(	Mat< Rows, Cols, T > const &	m,
		Vec< Cols, T > const &	v
	)

constexprnoexcept

Multiplies a matrix by a column vector (M * v).

Template Parameters

Rows	Number of rows in `m`.
Cols	Number of columns in `m` (equals the dimension of `v`).
T	Element type.

Parameters

[in]	m	Left-hand side matrix.
[in]	v	Column vector.

Returns: A Vec<Rows, T> where element r is dot(m.row(r), v).

Definition at line 975 of file mat.hpp.

◆ operator*() [2/3]

template<std::size_t Rows, std::size_t Shared, std::size_t Cols, class T >

constexpr Mat< Rows, Cols, T > ufo::operator*	(	Mat< Rows, Shared, T > const &	a,
		Mat< Shared, Cols, T > const &	b
	)

constexprnoexcept

Matrix × matrix product.

Computes the standard matrix product a * b via the triple-loop formula. The number of columns in a must equal the number of rows in b (Shared).

Template Parameters

Rows	Number of rows in `a` (and in the result).
Shared	Inner dimension: columns of `a` and rows of `b`.
Cols	Number of columns in `b` (and in the result).
T	Element type.

Parameters

[in]	a	Left-hand side matrix of shape `Rows × Shared`.
[in]	b	Right-hand side matrix of shape `Shared × Cols`.

Returns: A Mat<Rows, Cols, T> where result[r][c] = sum_k a[r][k] * b[k][c].

Definition at line 1027 of file mat.hpp.

◆ operator*() [3/3]

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

constexpr Transform< Dim, T > ufo::operator*	(	Transform< Dim, T >	t1,
		Transform< Dim, T > const &	t2
	)

constexpr

Composes two transforms: (t1 * t2)(v) = t1(t2(v)).

Template Parameters

Dim	Spatial dimension (2 or 3).
T	Floating-point scalar type.

Parameters

[in]	t1	Outer transform.
[in]	t2	Inner transform.

Returns: The composed transform.

Definition at line 406 of file transform.hpp.

◆ operator+() [1/8]

template<std::size_t Rows, std::size_t Cols, class T >

constexpr Mat ufo::Mat< Rows, Cols, T >::operator+ ( ) const

inlineconstexprnoexcept

Unary identity operator.

Returns: A copy of this matrix, unchanged.

Definition at line 639 of file mat.hpp.

◆ operator+() [2/8]

template<std::size_t Dim, class T >

constexpr Vec ufo::Vec< Dim, T >::operator+ ( ) const

inlineconstexprnoexcept

Unary identity operator.

Returns: A copy of this vector, unchanged.

Definition at line 219 of file vec.hpp.

◆ operator+() [3/8]

template<std::size_t Rows, std::size_t Cols, class T >

constexpr Mat< Rows, Cols, T > ufo::operator+	(	Mat< Rows, Cols, T >	lhs,
		Mat< Rows, Cols, T > const &	rhs
	)

constexprnoexcept

Element-wise addition of two matrices.

Template Parameters

Rows	Number of rows.
Cols	Number of columns.
T	Element type.

Parameters

[in]	lhs	Left-hand side matrix.
[in]	rhs	Right-hand side matrix.

Returns: A matrix whose element [r][c] is lhs[r][c] + rhs[r][c].

Definition at line 789 of file mat.hpp.

◆ operator+() [4/8]

template<std::size_t Rows, std::size_t Cols, class T >

constexpr Mat< Rows, Cols, T > ufo::operator+	(	Mat< Rows, Cols, T >	lhs,
		T	rhs
	)

constexprnoexcept

Adds scalar rhs to every element of lhs.

Template Parameters

Rows	Number of rows.
Cols	Number of columns.
T	Element type.

Parameters

[in]	lhs	Left-hand side matrix.
[in]	rhs	Scalar value.

Returns: A matrix whose element [r][c] is lhs[r][c] + rhs.

Definition at line 829 of file mat.hpp.

◆ operator+() [5/8]

template<std::floating_point T>

constexpr Quat< T > ufo::operator+ ( Quat< T > const & q )

constexprnoexcept

Unary plus — returns a copy of q unchanged.

Template Parameters

T	Floating-point scalar type.

Parameters

[in] q Input quaternion.

Returns: q.

Definition at line 443 of file quat.hpp.

◆ operator+() [6/8]

template<std::floating_point T>

constexpr Quat< T > ufo::operator+	(	Quat< T >	lhs,
		Quat< T > const &	rhs
	)

constexprnoexcept

Component-wise addition of two quaternions.

Template Parameters

T	Floating-point scalar type.

Parameters

[in]	lhs	Left-hand side.
[in]	rhs	Right-hand side.

Returns: Component-wise sum.

Definition at line 474 of file quat.hpp.

◆ operator+() [7/8]

template<std::size_t Dim, class T >

constexpr Vec< Dim, T > ufo::operator+	(	Vec< Dim, T >	lhs,
		T	rhs
	)

constexprnoexcept

Adds scalar rhs to every component of lhs.

Template Parameters

Dim	Number of dimensions.
T	Element type.

Parameters

[in]	lhs	Left-hand side vector.
[in]	rhs	Scalar value.

Returns: A vector whose element i is lhs[i] + rhs.

Definition at line 734 of file vec.hpp.

◆ operator+() [8/8]

template<std::size_t Dim, class T >

constexpr Vec< Dim, T > ufo::operator+	(	Vec< Dim, T >	lhs,
		Vec< Dim, T > const &	rhs
	)

constexprnoexcept

Component-wise addition of two vectors.

Template Parameters

Dim	Number of dimensions.
T	Element type.

Parameters

[in]	lhs	Left-hand side vector.
[in]	rhs	Right-hand side vector.

Returns: A vector whose element i is lhs[i] + rhs[i].

Definition at line 538 of file vec.hpp.

◆ operator+=() [1/5]

template<std::size_t Rows, std::size_t Cols, class T >

constexpr Mat & ufo::Mat< Rows, Cols, T >::operator+= ( Mat< Rows, Cols, T > const & rhs )

inlineconstexprnoexcept

Adds rhs element-wise to this matrix.

Parameters

[in] rhs Right-hand side matrix.

Returns: Reference to *this.

Definition at line 663 of file mat.hpp.

◆ operator+=() [2/5]

template<std::floating_point T = float>

constexpr Quat & ufo::Quat< T >::operator+= ( Quat< T > const & q )

inlineconstexprnoexcept

Component-wise addition.

Parameters

[in] q Quaternion to add.

Returns: Reference to *this.

Definition at line 314 of file quat.hpp.

◆ operator+=() [3/5]

template<std::size_t Rows, std::size_t Cols, class T >

constexpr Mat & ufo::Mat< Rows, Cols, T >::operator+= ( T rhs )

inlineconstexprnoexcept

Adds scalar rhs to every element.

Parameters

[in] rhs Scalar value.

Returns: Reference to *this.

Definition at line 703 of file mat.hpp.

◆ operator+=() [4/5]

template<std::size_t Dim, class T >

constexpr Vec & ufo::Vec< Dim, T >::operator+= ( T rhs )

inlineconstexprnoexcept

Adds scalar rhs to every component.

Parameters

[in] rhs Scalar value.

Returns: Reference to *this.

Definition at line 383 of file vec.hpp.

◆ operator+=() [5/5]

template<std::size_t Dim, class T >

constexpr Vec & ufo::Vec< Dim, T >::operator+= ( Vec< Dim, T > const & rhs )

inlineconstexprnoexcept

Adds each component of rhs to the corresponding component.

Parameters

[in] rhs Right-hand side vector.

Returns: Reference to *this.

Definition at line 255 of file vec.hpp.

◆ operator<<() [1/3]

template<std::size_t Rows, std::size_t Cols, class T >

std::ostream & ufo::operator<<	(	std::ostream &	out,
		Mat< Rows, Cols, T > const &	m
	)

Writes a matrix to an output stream, one row per line.

Each row is formatted using Vec's operator<<, separated by newlines.

Template Parameters

Rows	Number of rows.
Cols	Number of columns.
T	Element type.

Parameters

[in]	out	Output stream.
[in]	m	Matrix to print.

Returns: Reference to out.

Definition at line 1640 of file mat.hpp.

◆ operator<<() [2/3]

template<std::floating_point T>

std::ostream & ufo::operator<<	(	std::ostream &	out,
		Quat< T > const &	q
	)

Writes the quaternion to an output stream as "qw: W qx: X qy: Y qz: Z".

Template Parameters

T	Floating-point scalar type.

Parameters

[in]	out	Output stream.
[in]	q	Quaternion to print.

Returns: Reference to out.

Definition at line 1041 of file quat.hpp.

◆ operator<<() [3/3]

template<std::size_t Dim, class T >

std::ostream & ufo::operator<<	(	std::ostream &	out,
		Vec< Dim, T > const &	v
	)

Writes a human-readable representation of a vector to a stream.

Template Parameters

Dim	Number of dimensions.
T	Element type.

Parameters

[in,out]	out	The output stream.
[in]	v	The vector to print.

Returns: Reference to out.

Outputs each component labeled by its name (x, y, z, w), separated by spaces. Example output for a Vec3f{1, 2, 3}: x: 1 y: 2 z: 3.

Definition at line 1783 of file vec.hpp.

◆ operator=() [1/2]

template<std::floating_point T = float>

constexpr Quat & ufo::Quat< T >::operator= ( Quat< T > const & )

constexprdefaultnoexcept

Copy assignment operator.

Returns: Reference to *this.

◆ operator=() [2/2]

template<std::size_t Dim, std::floating_point T = float>

constexpr Transform & ufo::Transform< Dim, T >::operator= ( Transform< Dim, T > const & )

constexprdefaultnoexcept

Default copy assignment operator.

Returns: Reference to *this.

◆ operator[]() [1/3]

template<std::size_t Dim, class T >

constexpr auto & ufo::Vec< Dim, T >::operator[]	(	this auto &	self,
		size_type	i
	)

inlineconstexprnoexcept

Accesses the element at index i.

Parameters

[in] i Zero-based element index.

Returns: Reference to the element (const or non-const, deduced from this).

Definition at line 127 of file vec.hpp.

◆ operator[]() [2/3]

template<std::floating_point T = float>

constexpr auto & ufo::Quat< T >::operator[]	(	this auto &	self,
		size_type	pos
	)

inlineconstexprnoexcept

Component access by index: 0->w, 1->x, 2->y, 3->z.

Parameters

[in] pos Component index (must be < 4; asserted in debug builds).

Returns: Reference to the selected component (const or non-const, deduced from this).

Definition at line 292 of file quat.hpp.

◆ operator[]() [3/3]

template<std::size_t Rows, std::size_t Cols, class T >

constexpr auto & ufo::Mat< Rows, Cols, T >::operator[]	(	this auto &	self,
		size_type	r,
		size_type	c
	)

inlineconstexprnoexcept

Accesses the element at row r, column c.

Parameters

[in]	r	Row index.
[in]	c	Column index.

Returns: Reference to the element (const or non-const, deduced from this).

Definition at line 232 of file mat.hpp.

◆ orthogonal()

template<std::floating_point T>

Mat< 4, 4, T > ufo::orthogonal	(	T	left,
		T	right,
		T	bottom,
		T	top
	)

Builds a right-handed 2-D orthographic projection matrix (no depth clipping).

Maps the rectangle [left, right] × [bottom, top] to the NDC square [-1, 1]². The resulting matrix has m[2][2] = -1 and m[3][3] = 1.

Template Parameters

T	Floating-point element type.

Parameters

[in]	left	Left clipping plane.
[in]	right	Right clipping plane.
[in]	bottom	Bottom clipping plane.
[in]	top	Top clipping plane.

Returns: A Mat<4, 4, T> representing the orthographic projection.

Definition at line 1359 of file mat.hpp.

◆ probabilityToLogit()

template<std::floating_point T>

constexpr T ufo::probabilityToLogit ( T probability )

constexpr

Converts probability to logit value.

Template Parameters

T	Floating point type (e.g., float, double).

Parameters

[in] probability The probability value in the range [0, 1].

Returns: The corresponding logit value.

Handles edge cases for probability == 0 or probability == 1.

Definition at line 129 of file math.hpp.

◆ radians()

template<std::floating_point T>

constexpr T ufo::radians ( T deg )

constexprnoexcept

Converts degrees to radians.

Template Parameters

T	Floating point type (e.g., float, double).

Parameters

[in] deg The angle in degrees.

Returns: The angle in radians.

Definition at line 86 of file math.hpp.

◆ roll()

template<std::floating_point T>

T ufo::roll ( Quat< T > const & q )

noexcept

Extracts the roll angle (rotation about the X-axis) in radians.

Template Parameters

T	Floating-point scalar type.

Parameters

[in] q Unit quaternion.

Returns: Roll angle in [-π, π].

Definition at line 905 of file quat.hpp.

◆ rotate()

template<std::floating_point T>

Quat< T > ufo::rotate	(	Quat< T > const &	q,
		T	angle,
		Vec< 3, T >	axis
	)

Appends an angle-axis rotation to a quaternion.

Computes q * angleAxis(angle, axis), normalising axis only if it deviates from unit length by more than 0.001.

Template Parameters

T	Floating-point scalar type.

Parameters

[in]	q	Base unit quaternion.
[in]	angle	Additional rotation angle in radians.
[in]	axis	Rotation axis (normalised internally if needed).

Returns: Resulting unit quaternion.

Definition at line 965 of file quat.hpp.

◆ sign()

template<class T >

constexpr int ufo::sign ( T val )

constexprnoexcept

Returns the sign of a value.

Template Parameters

T	Numeric type (e.g., int, float).

Parameters

[in] val The value to check.

Return values

-1	`val < 0`
0	`val == 0`
1	`val > 0`

Definition at line 70 of file math.hpp.

◆ size()

template<std::floating_point T = float>

static constexpr size_type ufo::Quat< T >::size ( )

inlinestaticconstexprnoexcept

Returns the number of components (always 4: w, x, y, z).

Returns: 4.

Definition at line 390 of file quat.hpp.

◆ swap() [1/2]

template<std::floating_point T = float>

void ufo::Quat< T >::swap ( Quat< T > & other )

inlinenoexcept

Swaps all components with other.

Parameters

[in] other Quaternion to swap with.

Definition at line 402 of file quat.hpp.

◆ swap() [2/2]

template<std::floating_point T>

void ufo::swap	(	Quat< T > &	lhs,
		Quat< T > &	rhs
	)

noexcept

Non-member swap — exchanges all components of lhs and rhs.

Template Parameters

T	Floating-point scalar type.

Parameters

[in]	lhs	First quaternion.
[in]	rhs	Second quaternion.

Definition at line 624 of file quat.hpp.

◆ trace()

template<std::size_t Rows, std::size_t Cols, class T >

constexpr T ufo::Mat< Rows, Cols, T >::trace ( ) const

inlineconstexprnoexcept

Computes the trace (sum of diagonal elements) of a square matrix.

Returns: fields[0][0] + fields[1][1] + ... + fields[N-1][N-1].

Definition at line 345 of file mat.hpp.

Namespaces

Concepts

Classes

Typedefs

Functions

Detailed Description

Typedef Documentation

◆ Mat2

◆ Quatf

◆ Transform2

◆ Vec1b

Function Documentation

◆ angle()

◆ cast()

◆ degrees()

◆ determinant()

◆ dot()

◆ eigenValues()

◆ get() [1/2]

◆ get() [2/2]

◆ inverse()

◆ ipow()

◆ isnan()

◆ logitToProbability()

◆ mix()

◆ operator Mat< 3, 3, T >()

◆ operator Mat< Dim, Dim, U >()

◆ operator()()

◆ operator*() [1/3]

◆ operator*() [2/3]

◆ operator*() [3/3]

◆ operator+() [1/8]

◆ operator+() [2/8]

◆ operator+() [3/8]

◆ operator+() [4/8]

◆ operator+() [5/8]

◆ operator+() [6/8]

◆ operator+() [7/8]

◆ operator+() [8/8]

◆ operator+=() [1/5]

◆ operator+=() [2/5]

◆ operator+=() [3/5]

◆ operator+=() [4/5]

◆ operator+=() [5/5]

◆ operator<<() [1/3]

◆ operator<<() [2/3]

◆ operator<<() [3/3]

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator[]() [1/3]

◆ operator[]() [2/3]

◆ operator[]() [3/3]

◆ orthogonal()

◆ probabilityToLogit()

◆ radians()

◆ roll()

◆ rotate()

◆ sign()

◆ size()

◆ swap() [1/2]

◆ swap() [2/2]

◆ trace()