💿🐜 Antkeeper source code
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1054 lines
25 KiB
1054 lines
25 KiB
/*
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* Copyright (C) 2023 Christopher J. Howard
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*
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* This file is part of Antkeeper source code.
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*
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* Antkeeper source code is free software: you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation, either version 3 of the License, or
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* (at your option) any later version.
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*
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* Antkeeper source code is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with Antkeeper source code. If not, see <http://www.gnu.org/licenses/>.
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*/
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#ifndef ANTKEEPER_MATH_QUATERNION_HPP
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#define ANTKEEPER_MATH_QUATERNION_HPP
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#include <engine/math/numbers.hpp>
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#include <engine/math/matrix.hpp>
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#include <engine/math/vector.hpp>
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#include <array>
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#include <cmath>
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#include <utility>
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namespace math {
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/**
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* Quaternion composed of a real scalar part and imaginary vector part.
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*
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* @tparam T Scalar type.
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*/
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template <class T>
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struct quaternion
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{
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/// Scalar type.
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using scalar_type = T;
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/// Vector type.
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using vector_type = vec3<T>;
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/// Rotation matrix type.
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using matrix_type = mat3<T>;
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/// Quaternion real part.
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scalar_type r;
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/// Quaternion imaginary part.
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vector_type i;
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/// Returns a reference to the quaternion real part.
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/// @{
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[[nodiscard]] inline constexpr scalar_type& w() noexcept
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{
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return r;
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}
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[[nodiscard]] inline constexpr const scalar_type& w() const noexcept
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{
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return r;
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}
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/// @}
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/// Returns a reference to the first element of the quaternion imaginary part.
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/// @{
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[[nodiscard]] inline constexpr scalar_type& x() noexcept
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{
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return i.x();
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}
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[[nodiscard]] inline constexpr const scalar_type& x() const noexcept
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{
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return i.x();
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}
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/// @}
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/// Returns a reference to the second element of the quaternion imaginary part.
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/// @{
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[[nodiscard]] inline constexpr scalar_type& y() noexcept
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{
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return i.y();
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}
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[[nodiscard]] inline constexpr const scalar_type& y() const noexcept
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{
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return i.y();
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}
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/// @}
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/// Returns a reference to the third element of the quaternion imaginary part.
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/// @{
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[[nodiscard]] inline constexpr scalar_type& z() noexcept
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{
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return i.z();
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}
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[[nodiscard]] inline constexpr const scalar_type& z() const noexcept
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{
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return i.z();
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}
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/// @}
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/**
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* Returns a quaternion representing a rotation about the x-axis.
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*
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* @param angle Angle of rotation, in radians.
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*
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* @return Quaternion representing an x-axis rotation.
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*/
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[[nodiscard]] static quaternion rotate_x(scalar_type angle)
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{
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return {std::cos(angle * T{0.5}), std::sin(angle * T{0.5}), T{0}, T{0}};
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}
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/**
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* Returns a quaternion representing a rotation about the y-axis.
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*
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* @param angle Angle of rotation, in radians.
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*
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* @return Quaternion representing an y-axis rotation.
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*/
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[[nodiscard]] static quaternion rotate_y(scalar_type angle)
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{
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return {std::cos(angle * T{0.5}), T{0}, std::sin(angle * T{0.5}), T{0}};
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}
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/**
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* Returns a quaternion representing a rotation about the z-axis.
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*
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* @param angle Angle of rotation, in radians.
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* @return Quaternion representing an z-axis rotation.
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*/
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[[nodiscard]] static quaternion rotate_z(scalar_type angle)
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{
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return {std::cos(angle * T{0.5}), T{0}, T{0}, std::sin(angle * T{0.5})};
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}
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/**
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* Type-casts the quaternion scalars using `static_cast`.
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*
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* @tparam U Target scalar type.
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*
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* @return Type-casted quaternion.
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*/
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template <class U>
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[[nodiscard]] inline constexpr explicit operator quaternion<U>() const noexcept
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{
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return {static_cast<U>(r), vec3<U>(i)};
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}
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/**
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* Constructs a matrix representing the rotation described by the quaternion.
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*
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* @return Rotation matrix.
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*/
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/// @{
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[[nodiscard]] constexpr explicit operator matrix_type() const noexcept
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{
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const T xx = x() * x();
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const T xy = x() * y();
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const T xz = x() * z();
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const T xw = x() * w();
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const T yy = y() * y();
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const T yz = y() * z();
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const T yw = y() * w();
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const T zz = z() * z();
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const T zw = z() * w();
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return
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{
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T{1} - (yy + zz) * T{2}, (xy + zw) * T{2}, (xz - yw) * T{2},
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(xy - zw) * T{2}, T{1} - (xx + zz) * T{2}, (yz + xw) * T{2},
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(xz + yw) * T{2}, (yz - xw) * T{2}, T{1} - (xx + yy) * T{2}
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};
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}
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[[nodiscard]] inline constexpr matrix_type matrix() const noexcept
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{
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return matrix_type(*this);
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}
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/// @}
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/**
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* Casts the quaternion to a 4-element vector, with the real part as the first element and the imaginary part as the following three elements.
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*
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* @return Vector containing the real and imaginary parts of the quaternion.
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*/
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[[nodiscard]] inline constexpr explicit operator vec4<T>() const noexcept
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{
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return {r, i.x(), i.y(), i.z()};
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}
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/// Returns a zero quaternion, where every scalar is equal to zero.
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[[nodiscard]] static constexpr quaternion zero() noexcept
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{
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return {};
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}
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/// Returns a rotation identity quaternion.
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[[nodiscard]] static constexpr quaternion identity() noexcept
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{
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return {T{1}, vector_type::zero()};
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}
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};
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/// Quaternion types.
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namespace quaternion_types {
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/// @copydoc math::quaternion
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template <class T>
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using quat = quaternion<T>;
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/**
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* Quaternion with single-precision floating-point scalars.
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*
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* @tparam T Scalar type.
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*/
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using fquat = quat<float>;
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/**
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* Quaternion with double-precision floating-point scalars.
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*
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* @tparam T Scalar type.
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*/
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using dquat = quat<double>;
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} // namespace quaternion_types
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// Bring quaternion types into math namespace
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using namespace quaternion_types;
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// Bring quaternion types into math::types namespace
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namespace types { using namespace math::quaternion_types; }
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/**
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* Adds two quaternions.
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*
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* @param a First quaternion.
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* @param b Second quaternion.
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*
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* @return Sum of the two quaternions.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> add(const quaternion<T>& a, const quaternion<T>& b) noexcept;
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/**
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* Adds a quaternion and a scalar.
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*
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* @param a First value.
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* @param b Second second value.
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*
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* @return Sum of the quaternion and scalar.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> add(const quaternion<T>& a, T b) noexcept;
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/**
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* Calculates the conjugate of a quaternion.
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*
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* @param q Quaternion from which the conjugate will be calculated.
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*
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* @return Conjugate of the quaternion.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> conjugate(const quaternion<T>& q) noexcept;
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/**
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* Calculates the dot product of two quaternions.
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*
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* @param a First quaternion.
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* @param b Second quaternion.
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*
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* @return Dot product of the two quaternions.
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*/
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template <class T>
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[[nodiscard]] constexpr T dot(const quaternion<T>& a, const quaternion<T>& b) noexcept;
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/**
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* Divides a quaternion by another quaternion.
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*
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* @param a First value.
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* @param b Second value.
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*
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* @return Result of the division.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> div(const quaternion<T>& a, const quaternion<T>& b) noexcept;
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/**
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* Divides a quaternion by a scalar.
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*
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* @param a Quaternion.
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* @param b Scalar.
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*
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* @return Result of the division.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> div(const quaternion<T>& a, T b) noexcept;
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/**
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* Divides a scalar by a quaternion.
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*
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* @param a Scalar.
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* @param b Quaternion.
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*
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* @return Result of the division.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> div(T a, const quaternion<T>& b) noexcept;
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/**
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* Calculates the inverse length of a quaternion.
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*
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* @param q Quaternion to calculate the inverse length of.
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*
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* @return Inverse length of the quaternion.
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*/
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template <class T>
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[[nodiscard]] T inv_length(const quaternion<T>& q);
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/**
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* Calculates the length of a quaternion.
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*
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* @param q Quaternion to calculate the length of.
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*
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* @return Length of the quaternion.
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*/
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template <class T>
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[[nodiscard]] T length(const quaternion<T>& q);
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/**
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* Performs linear interpolation between two quaternions.
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*
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* @param a First quaternion.
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* @param b Second quaternion.
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* @param t Interpolation factor.
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*
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* @return Interpolated quaternion.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> lerp(const quaternion<T>& a, const quaternion<T>& b, T t) noexcept;
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/**
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* Creates a unit quaternion rotation using forward and up vectors.
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*
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* @param forward Unit forward vector.
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* @param up Unit up vector.
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*
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* @return Unit rotation quaternion.
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*/
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template <class T>
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[[nodiscard]] quaternion<T> look_rotation(const vec3<T>& forward, vec3<T> up);
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/**
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* Multiplies two quaternions.
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*
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* @param a First quaternion.
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* @param b Second quaternion.
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*
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* @return Product of the two quaternions.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> mul(const quaternion<T>& a, const quaternion<T>& b) noexcept;
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/**
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* Multiplies a quaternion by a scalar.
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*
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* @param a First value.
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* @param b Second value.
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*
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* @return Product of the quaternion and scalar.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> mul(const quaternion<T>& a, T b) noexcept;
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/**
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* Rotates a vector by a unit quaternion.
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*
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* @param q Unit quaternion.
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* @param v Vector to rotate.
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*
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* @return Rotated vector.
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*
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* @warning @p q must be a unit quaternion.
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*
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* @see https://fgiesen.wordpress.com/2019/02/09/rotating-a-single-vector-using-a-quaternion/
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*/
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template <class T>
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[[nodiscard]] constexpr vec3<T> mul(const quaternion<T>& q, const vec3<T>& v) noexcept;
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/**
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* Rotates a vector by the inverse of a unit quaternion.
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*
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* @param v Vector to rotate.
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* @param q Unit quaternion.
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*
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* @return Rotated vector.
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*
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* @warning @p q must be a unit quaternion.
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*/
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template <class T>
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[[nodiscard]] constexpr vec3<T> mul(const vec3<T>& v, const quaternion<T>& q) noexcept;
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/**
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* Negates a quaternion.
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*
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* @param q Quaternion to negate.
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*
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* @return Negated quaternion.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> negate(const quaternion<T>& q) noexcept;
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/**
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* Performs normalized linear interpolation between two quaternions.
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*
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* @param a First quaternion.
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* @param b Second quaternion.
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* @param t Interpolation factor.
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*
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* @return Interpolated quaternion.
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*/
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template <class T>
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[[nodiscard]] quaternion<T> nlerp(const quaternion<T>& a, const quaternion<T>& b, T t);
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/**
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* Normalizes a quaternion.
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*
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* @param q Quaternion to normalize.
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*
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* @return Unit quaternion.
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*/
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template <class T>
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[[nodiscard]] quaternion<T> normalize(const quaternion<T>& q);
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/**
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* Creates a rotation from an angle and axis.
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*
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* @param angle Angle of rotation (in radians).
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* @param axis Axis of rotation
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*
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* @return Quaternion representing the rotation.
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*/
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template <class T>
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[[nodiscard]] quaternion<T> angle_axis(T angle, const vec3<T>& axis);
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/**
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* Constructs a quaternion representing the minimum rotation from one direction to another direction.
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*
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* @param from Unit vector pointing in the source direction.
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* @param to Unit vector pointing in the target direction.
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* @param tolerance Floating-point tolerance.
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*
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* @return Unit quaternion representing the minimum rotation from direction @p from to direction @p to.
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*
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* @warning @p from and @p to must be unit vectors.
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*/
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template <class T>
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[[nodiscard]] quaternion<T> rotation(const vec3<T>& from, const vec3<T>& to, T tolerance = T{1e-6});
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/**
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* Constructs a quaternion representing an angle-limited rotation from one direction towards another direction.
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*
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* @param from Unit vector pointing in the source direction.
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* @param to Unit vector pointing in the target direction.
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* @param max_angle Maximum angle of rotation, in radians.
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*
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* @return Unit quaternion representing a rotation from direction @p from towards direction @p to.
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*
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* @warning @p from and @p to must be unit vectors.
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*/
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template <class T>
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[[nodiscard]] quaternion<T> rotate_towards(const vec3<T>& from, const vec3<T>& to, T max_angle);
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/**
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* Performs spherical linear interpolation between two quaternions.
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*
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* @param a First quaternion.
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* @param b Second quaternion.
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* @param t Interpolation factor.
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* @param tolerance Floating-point tolerance.
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*
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* @return Interpolated quaternion.
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*/
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template <class T>
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[[nodiscard]] quaternion<T> slerp(const quaternion<T>& a, const quaternion<T>& b, T t, T tolerance = T{1e-6});
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/**
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* 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.
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*
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* @param q Quaternion to calculate the square length of.
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*
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* @return Square length of the quaternion.
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*/
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template <class T>
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[[nodiscard]] constexpr T sqr_length(const quaternion<T>& q) noexcept;
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/**
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* Subtracts a quaternion from another quaternion.
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*
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* @param a First quaternion.
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* @param b Second quaternion.
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*
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* @return Difference between the quaternions.
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*/
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template <class T>
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[[nodiscard]] constexpr quaternion<T> sub(const quaternion<T>& a, const quaternion<T>& b) noexcept;
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/**
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* Subtracts a quaternion and a scalar.
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*
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* @param a First value.
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* @param b Second second.
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*
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* @return Difference between the quaternion and scalar.
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*/
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/// @{
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template <class T>
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[[nodiscard]] constexpr quaternion<T> sub(const quaternion<T>& a, T b) noexcept;
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template <class T>
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[[nodiscard]] constexpr quaternion<T> sub(T a, const quaternion<T>& b) noexcept;
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/// @}
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/**
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* Decomposes a quaternion into swing and twist rotation components.
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*
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* @param[in] q Unit quaternion to decompose.
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* @param[in] twist_axis Axis of twist rotation.
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* @param[out] swing Swing rotation component.
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* @param[out] twist Twist rotation component.
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* @param[in] tolerance Floating-point tolerance.
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*
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* @return Array containing the swing rotation component, followed by the twist rotation component.
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*
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* @warning @p q must be a unit quaternion.
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* @warning @p twist_axis must be a unit vector.
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*
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* @see https://www.euclideanspace.com/maths/geometry/rotations/for/decomposition/
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*/
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template <class T>
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[[nodiscard]] std::array<quaternion<T>, 2> swing_twist(const quaternion<T>& q, const vec3<T>& twist_axis, T tolerance = T{1e-6});
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/**
|
|
* Converts a 3x3 rotation matrix to a quaternion.
|
|
*
|
|
* @param m Rotation matrix.
|
|
*
|
|
* @return Unit quaternion representing the rotation described by @p m.
|
|
*/
|
|
template <class T>
|
|
[[nodiscard]] quaternion<T> quaternion_cast(const mat3<T>& m);
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> add(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return {a.r + b.r, a.i + b.i};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> add(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return {a.r + b, a.i + b};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> conjugate(const quaternion<T>& q) noexcept
|
|
{
|
|
return {q.r, -q.i};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr T dot(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return a.r * b.r + dot(a.i, b.i);
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> div(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return {a.r / b.r, a.i / b.i};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> div(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return {a.r / b, a.i / b};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> div(T a, const quaternion<T>& b) noexcept
|
|
{
|
|
return {a / b.r, a / b.i};
|
|
}
|
|
|
|
template <class T>
|
|
inline T inv_length(const quaternion<T>& q)
|
|
{
|
|
return T{1} / length(q);
|
|
}
|
|
|
|
template <class T>
|
|
inline T length(const quaternion<T>& q)
|
|
{
|
|
return std::sqrt(sqr_length(q));
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> lerp(const quaternion<T>& a, const quaternion<T>& b, T t) noexcept
|
|
{
|
|
return
|
|
{
|
|
(b.r - a.r) * t + a.r,
|
|
(b.i - a.i) * t + a.i
|
|
};
|
|
}
|
|
|
|
template <class T>
|
|
quaternion<T> look_rotation(const vec3<T>& forward, vec3<T> up)
|
|
{
|
|
const vec3<T> right = normalize(cross(forward, up));
|
|
up = cross(right, forward);
|
|
|
|
const mat3<T> m =
|
|
{
|
|
right,
|
|
up,
|
|
-forward
|
|
};
|
|
|
|
// Convert to quaternion
|
|
return normalize(quaternion_cast(m));
|
|
}
|
|
|
|
template <class T>
|
|
constexpr quaternion<T> mul(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return
|
|
{
|
|
a.w() * b.w() - a.x() * b.x() - a.y() * b.y() - a.z() * b.z(),
|
|
a.w() * b.x() + a.x() * b.w() + a.y() * b.z() - a.z() * b.y(),
|
|
a.w() * b.y() - a.x() * b.z() + a.y() * b.w() + a.z() * b.x(),
|
|
a.w() * b.z() + a.x() * b.y() - a.y() * b.x() + a.z() * b.w()
|
|
};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> mul(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return {a.r * b, a.i * b};
|
|
}
|
|
|
|
template <class T>
|
|
constexpr vec3<T> mul(const quaternion<T>& q, const vec3<T>& v) noexcept
|
|
{
|
|
const auto t = cross(q.i, v) * T{2};
|
|
return v + q.r * t + cross(q.i, t);
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr vec3<T> mul(const vec3<T>& v, const quaternion<T>& q) noexcept
|
|
{
|
|
return mul(conjugate(q), v);
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> negate(const quaternion<T>& q) noexcept
|
|
{
|
|
return {-q.r, -q.i};
|
|
}
|
|
|
|
template <class T>
|
|
quaternion<T> nlerp(const quaternion<T>& a, const quaternion<T>& b, T t)
|
|
{
|
|
return normalize(add(mul(a, T{1} - t), mul(b, std::copysign(t, dot(a, b)))));
|
|
}
|
|
|
|
template <class T>
|
|
inline quaternion<T> normalize(const quaternion<T>& q)
|
|
{
|
|
return mul(q, inv_length(q));
|
|
}
|
|
|
|
template <class T>
|
|
quaternion<T> angle_axis(T angle, const vec3<T>& axis)
|
|
{
|
|
return {std::cos(angle * T{0.5}), axis * std::sin(angle * T{0.5})};
|
|
}
|
|
|
|
template <class T>
|
|
quaternion<T> rotation(const vec3<T>& from, const vec3<T>& to, T tolerance)
|
|
{
|
|
const auto cos_theta = dot(from, to);
|
|
|
|
if (cos_theta <= T{-1} + tolerance)
|
|
{
|
|
// Direction vectors are opposing, return 180 degree rotation about arbitrary axis
|
|
return quaternion<T>{T{0}, {T{1}, T{0}, T{0}}};
|
|
}
|
|
else if (cos_theta >= T{1} - tolerance)
|
|
{
|
|
// Direction vectors are parallel, return identity quaternion
|
|
return quaternion<T>::identity();
|
|
}
|
|
else
|
|
{
|
|
const auto r = cos_theta + T{1};
|
|
const auto i = cross(from, to);
|
|
const auto inv_length = T{1.0} / std::sqrt(r + r);
|
|
|
|
return quaternion<T>{r * inv_length, i * inv_length};
|
|
}
|
|
}
|
|
|
|
template <class T>
|
|
quaternion<T> rotate_towards(const vec3<T>& from, const vec3<T>& to, T max_angle)
|
|
{
|
|
const auto angle = std::acos(dot(from, to));
|
|
const auto axis = cross(from, to);
|
|
return angle_axis(std::min(max_angle, angle), axis);
|
|
}
|
|
|
|
template <class T>
|
|
quaternion<T> slerp(const quaternion<T>& a, const quaternion<T>& b, T t, T tolerance)
|
|
{
|
|
T cos_theta = dot(a, b);
|
|
if (cos_theta >= T{1} - tolerance)
|
|
{
|
|
// Use linear interpolation if quaternions are nearly aligned
|
|
return normalize(lerp(a, b, t));
|
|
}
|
|
|
|
// Clamp to acos domain
|
|
cos_theta = std::min<T>(std::max<T>(cos_theta, T{-1}), T{1});
|
|
|
|
const T theta = std::acos(cos_theta) * t;
|
|
|
|
const quaternion<T> c = normalize(sub(b, mul(a, cos_theta)));
|
|
|
|
return add(mul(a, std::cos(theta)), mul(c, std::sin(theta)));
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr T sqr_length(const quaternion<T>& q) noexcept
|
|
{
|
|
return q.r * q.r + sqr_length(q.i);
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> sub(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return {a.r - b.r, a.i - b.i};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> sub(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return {a.r - b, a.i - b};
|
|
}
|
|
|
|
template <class T>
|
|
inline constexpr quaternion<T> sub(T a, const quaternion<T>& b) noexcept
|
|
{
|
|
return {a - b.r, a - b.i};
|
|
}
|
|
|
|
template <class T>
|
|
std::array<quaternion<T>, 2> swing_twist(const quaternion<T>& q, const vec3<T>& twist_axis, T tolerance)
|
|
{
|
|
quaternion<T> swing;
|
|
quaternion<T> twist;
|
|
|
|
if (sqr_length(q.i) <= tolerance)
|
|
{
|
|
// Singularity, rotate 180 degrees about twist axis
|
|
twist = angle_axis(pi<T>, twist_axis);
|
|
|
|
const auto rotated_twist_axis = mul(q, twist_axis);
|
|
const auto swing_axis = cross(twist_axis, rotated_twist_axis);
|
|
const auto swing_axis_sqr_length = sqr_length(swing_axis);
|
|
|
|
if (swing_axis_sqr_length <= tolerance)
|
|
{
|
|
// Swing axis and twist axis are parallel, no swing
|
|
swing = quaternion<T>::identity();
|
|
}
|
|
else
|
|
{
|
|
const auto cos_swing_angle = std::min<T>(std::max<T>(dot(twist_axis, rotated_twist_axis), T{-1}), T{1});
|
|
swing = angle_axis(std::acos(cos_swing_angle), swing_axis * (T{1} / std::sqrt(swing_axis_sqr_length)));
|
|
}
|
|
}
|
|
else
|
|
{
|
|
twist = normalize(quaternion<T>{q.r, twist_axis * dot(twist_axis, q.i)});
|
|
swing = mul(q, conjugate(twist));
|
|
}
|
|
|
|
return {std::move(swing), std::move(twist)};
|
|
}
|
|
|
|
template <class T>
|
|
quaternion<T> quaternion_cast(const mat3<T>& m)
|
|
{
|
|
const T t = trace(m);
|
|
|
|
if (t > T{0})
|
|
{
|
|
const T s = T{0.5} / std::sqrt(t + T{1});
|
|
return
|
|
{
|
|
T{0.25} / s,
|
|
(m[1][2] - m[2][1]) * s,
|
|
(m[2][0] - m[0][2]) * s,
|
|
(m[0][1] - m[1][0]) * s
|
|
};
|
|
}
|
|
else
|
|
{
|
|
if (m[0][0] > m[1][1] && m[0][0] > m[2][2])
|
|
{
|
|
const T s = T{2} * std::sqrt(T{1} + m[0][0] - m[1][1] - m[2][2]);
|
|
|
|
return
|
|
{
|
|
(m[1][2] - m[2][1]) / s,
|
|
T{0.25} * s,
|
|
(m[1][0] + m[0][1]) / s,
|
|
(m[2][0] + m[0][2]) / s
|
|
};
|
|
}
|
|
else if (m[1][1] > m[2][2])
|
|
{
|
|
const T s = T{2} * std::sqrt(T{1} + m[1][1] - m[0][0] - m[2][2]);
|
|
return
|
|
{
|
|
(m[2][0] - m[0][2]) / s,
|
|
(m[1][0] + m[0][1]) / s,
|
|
T{0.25} * s,
|
|
(m[2][1] + m[1][2]) / s
|
|
};
|
|
}
|
|
else
|
|
{
|
|
const T s = T{2} * std::sqrt(T{1} + m[2][2] - m[0][0] - m[1][1]);
|
|
return
|
|
{
|
|
(m[0][1] - m[1][0]) / s,
|
|
(m[2][0] + m[0][2]) / s,
|
|
(m[2][1] + m[1][2]) / s,
|
|
T{0.25} * s
|
|
};
|
|
}
|
|
}
|
|
}
|
|
|
|
namespace operators {
|
|
|
|
/// @copydoc add(const quaternion<T>&, const quaternion<T>&)
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator+(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return add(a, b);
|
|
}
|
|
|
|
/// @copydoc add(const quaternion<T>&, T)
|
|
/// @{
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator+(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return add(a, b);
|
|
}
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator+(T a, const quaternion<T>& b) noexcept
|
|
{
|
|
return add(b, a);
|
|
}
|
|
/// @}
|
|
|
|
/// @copydoc div(const quaternion<T>&, const quaternion<T>&)
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator/(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return div(a, b);
|
|
}
|
|
|
|
/// @copydoc div(const quaternion<T>&, T)
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator/(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return div(a, b);
|
|
}
|
|
|
|
/// @copydoc div(T, const quaternion<T>&)
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator/(T a, const quaternion<T>& b) noexcept
|
|
{
|
|
return div(a, b);
|
|
}
|
|
|
|
/// @copydoc mul(const quaternion<T>&, const quaternion<T>&)
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator*(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return mul(a, b);
|
|
}
|
|
|
|
/// @copydoc mul(const quaternion<T>&, T)
|
|
/// @{
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator*(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return mul(a, b);
|
|
}
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator*(T a, const quaternion<T>& b) noexcept
|
|
{
|
|
return mul(b, a);
|
|
}
|
|
/// @}
|
|
|
|
/// @copydoc mul(const quaternion<T>&, const vec3<T>&)
|
|
template <class T>
|
|
inline constexpr vec3<T> operator*(const quaternion<T>& q, const vec3<T>& v) noexcept
|
|
{
|
|
return mul(q, v);
|
|
}
|
|
|
|
/// @copydoc mul(const vec3<T>&, const quaternion<T>&)
|
|
template <class T>
|
|
inline constexpr vec3<T> operator*(const vec3<T>& v, const quaternion<T>& q) noexcept
|
|
{
|
|
return mul(v, q);
|
|
}
|
|
|
|
/// @copydoc sub(const quaternion<T>&, const quaternion<T>&)
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator-(const quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return sub(a, b);
|
|
}
|
|
|
|
/// @copydoc sub(const quaternion<T>&, T)
|
|
/// @{
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator-(const quaternion<T>& a, T b) noexcept
|
|
{
|
|
return sub(a, b);
|
|
}
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator-(T a, const quaternion<T>& b) noexcept
|
|
{
|
|
return sub(a, b);
|
|
}
|
|
/// @}
|
|
|
|
/// @copydoc negate(const quaternion<T>&)
|
|
template <class T>
|
|
inline constexpr quaternion<T> operator-(const quaternion<T>& q) noexcept
|
|
{
|
|
return negate(q);
|
|
}
|
|
|
|
/**
|
|
* Adds two values and stores the result in the first value.
|
|
*
|
|
* @param a First value.
|
|
* @param b Second value.
|
|
*
|
|
* @return Reference to the first value.
|
|
*/
|
|
/// @{
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator+=(quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return (a = a + b);
|
|
}
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator+=(quaternion<T>& a, T b) noexcept
|
|
{
|
|
return (a = a + b);
|
|
}
|
|
/// @}
|
|
|
|
/**
|
|
* Subtracts the first value by the second value and stores the result in the first value.
|
|
*
|
|
* @param a First value.
|
|
* @param b Second value.
|
|
*
|
|
* @return Reference to the first value.
|
|
*/
|
|
/// @{
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator-=(quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return (a = a - b);
|
|
}
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator-=(quaternion<T>& a, T b) noexcept
|
|
{
|
|
return (a = a - b);
|
|
}
|
|
/// @}
|
|
|
|
/**
|
|
* Multiplies two values and stores the result in the first value.
|
|
*
|
|
* @param a First value.
|
|
* @param b Second value.
|
|
*
|
|
* @return Reference to the first value.
|
|
*/
|
|
/// @{
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator*=(quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return (a = a * b);
|
|
}
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator*=(quaternion<T>& a, T b) noexcept
|
|
{
|
|
return (a = a * b);
|
|
}
|
|
/// @}
|
|
|
|
/**
|
|
* Divides the first value by the second value and stores the result in the first value.
|
|
*
|
|
* @param a First value.
|
|
* @param b Second value.
|
|
*
|
|
* @return Reference to the first value.
|
|
*/
|
|
/// @{
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator/=(quaternion<T>& a, const quaternion<T>& b) noexcept
|
|
{
|
|
return (a = a / b);
|
|
}
|
|
template <class T>
|
|
inline constexpr quaternion<T>& operator/=(quaternion<T>& a, T b) noexcept
|
|
{
|
|
return (a = a / b);
|
|
}
|
|
/// @}
|
|
|
|
} // namespace operators
|
|
|
|
} // namespace math
|
|
|
|
// Bring quaternion operators into global namespace
|
|
using namespace math::operators;
|
|
|
|
#endif // ANTKEEPER_MATH_QUATERNION_HPP
|