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include/fennec/math/geometric.h
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include/fennec/math/geometric.h
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// =====================================================================================================================
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// fennec, a free and open source game engine
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// Copyright (C) 2025 Medusa Slockbower
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//
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// This program 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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// This program 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 this program. If not, see <https://www.gnu.org/licenses/>.
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// =====================================================================================================================
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///
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/// \file geometric.h
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/// \brief \ref page_fennec_math_geometric
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///
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///
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/// \details
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/// \author Medusa Slockbower
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///
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/// \copyright Copyright © 2025 Medusa Slockbower ([GPLv3](https://www.gnu.org/licenses/gpl-3.0.en.html))
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///
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///
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#ifndef FENNEC_MATH_GEOMETRIC_H
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#define FENNEC_MATH_GEOMETRIC_H
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#include <fennec/math/vector.h>
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#include <fennec/math/exponential.h>
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///
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///
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///
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/// \page page_fennec_math_geometric Geometric
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///
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/// \brief The Geometric Functions defined in the [OpenGL 4.6 Shading Language Specification](https://registry.khronos.org/OpenGL/specs/gl/GLSLangSpec.4.60.pdf).
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///
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///
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/// \section Geometric Functions
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///
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/// <table width="100%" class="fieldtable" id="table_fennec_math_geometric_functions">
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/// <tr><th style="vertical-align: top">Syntax
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/// <th style="vertical-align: top">Description
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///
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/// <tr><td width="50%" style="vertical-align: top"> <br>
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/// \ref fennec::dot(const genFType&, const genFType&) "float dot(genFType x, genFType y)" <br>
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/// \ref fennec::dot(const genDType&, const genDType&) "double dot(genDType x, genDType x)"
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/// <td width="50%" style="vertical-align: top">
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/// \copydoc fennec::dot(const genFType&, const genFType&)
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///
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/// <tr><td width="50%" style="vertical-align: top" class="odd_c"> <br>
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/// \ref fennec::length2(const genFType&) "float length2(genFType x)" <br>
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/// \ref fennec::length2(const genDType&) "double length2(genDType x)"
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/// <td width="50%" style="vertical-align: top" class="odd_c">
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/// \copydoc fennec::length2(const genFType&)
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///
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/// <tr><td width="50%" style="vertical-align: top"> <br>
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/// \ref fennec::length(const genFType&) "float length(genFType x)" <br>
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/// \ref fennec::length(const genDType&) "double length(genDType x)"
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/// <td width="50%" style="vertical-align: top">
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/// \copydoc fennec::length(const genFType&)
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///
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/// <tr><td width="50%" style="vertical-align: top" class="odd_c"> <br>
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/// \ref fennec::distance(const genFType&, const genFType&) "float distance(genFType x, genFType y)" <br>
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/// \ref fennec::distance(const genDType&, const genDType&) "double distance(genDType x, genDType x)"
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/// <td width="50%" style="vertical-align: top" class="odd_c">
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/// \copydoc fennec::distance(const genFType&, const genFType&)
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///
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/// <tr><td width="50%" style="vertical-align: top"> <br>
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/// \ref fennec::normalize(const genFType&) "float normalize(genFType x)" <br>
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/// \ref fennec::normalize(const genDType&) "double normalize(genDType x)"
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/// <td width="50%" style="vertical-align: top">
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/// \copydoc fennec::normalize(const genFType&)
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///
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/// <tr><td width="50%" style="vertical-align: top" class="odd_c"> <br>
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/// \ref fennec::cross(const fennec::vec3&, const fennec::vec3&) "vec3 cross(vec3 x, vec3 y)" <br>
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/// \ref fennec::cross(const fennec::dvec3&, const fennec::dvec3&) "dvec3 cross(dvec3 x, dvec3 x)"
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/// <td width="50%" style="vertical-align: top" class="odd_c">
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/// \copydoc fennec::cross(const fennec::vec3&, const fennec::vec3&)
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///
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/// <tr><td width="50%" style="vertical-align: top"> <br>
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/// \ref fennec::faceforward(const genFType&, const genFType&, const genFType&) "genFType faceforward(genFType N, genFType I, genFType Nref)" <br>
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/// \ref fennec::faceforward(const genDType&, const genDType&, const genDType&) "genDType faceforward(genDType N, genDType I, genDType Nref)"
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/// <td width="50%" style="vertical-align: top">
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/// \copydoc fennec::faceforward(const genFType&, const genFType&, const genFType&)
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///
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/// <tr><td width="50%" style="vertical-align: top" class="odd_c"> <br>
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/// \ref fennec::reflect(const genFType&, const genFType&) "genFType reflect(genFType I, genFType N)" <br>
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/// \ref fennec::reflect(const genDType&, const genDType&) "genDType reflect(genDType I, genDType N)"
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/// <td width="50%" style="vertical-align: top" class="odd_c">
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/// \copydoc fennec::reflect(const genFType&, const genFType&)
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///
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/// <tr><td width="50%" style="vertical-align: top"> <br>
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/// \ref fennec::refract(const genFType&, const genFType&, float) "genFType refract(genFType N, genFType I, float eta)" <br>
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/// \ref fennec::refract(const genDType&, const genDType&, double) "genDType refract(genDType N, genDType I, double eta)"
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/// <td width="50%" style="vertical-align: top">
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/// \copydoc fennec::refract(const genFType&, const genFType&, float)
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///
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/// </table>
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///
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///
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///
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namespace fennec
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{
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// dot -----------------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief Returns the dot product of \f$x\f$ and \f$y\f$, i.e., \f$x_0 \cdot y_0 + x_0 \cdot y_0 + \ldots\f$
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///
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/// \returns the dot product of \f$x\f$ and \f$y\f$, i.e., \f$x_0 \cdot y_0 + x_0 \cdot y_0 + \ldots\f$ <br><br>
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/// \details we can represent this in linear algebra as the following, <br><br>
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/// let \f$X=\left[\begin{array}\\ x_0 \\ x_1 \\ \vdots \\ x_N \end{array}\right]\f$ <br>
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/// let \f$Y=\left[\begin{array}\\ y_0 \\ y_1 \\ \vdots \\ y_N \end{array}\right]\f$ <br><br>
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///
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/// then \f$\text{dot}(X, Y)=X \cdot Y^T\f$ <br><br>
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///
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/// \param x first vector
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/// \param y second vector
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template<typename genType, size_t...i>
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constexpr genType dot(const vector<genType, i...>& x, const vector<genType, i...>& y)
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{ return ((x[i] * y[i]) + ...); }
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// length2 -------------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief Returns the squared length of vector \f$x\f$, i.e., \f$x_0^2 + x_1^2 + \ldots\f$
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///
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/// \returns the squared length of vector \f$x\f$, i.e., \f$x_0^2 + x_1^2 + \ldots\f$ <br><br>
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/// \details we can represent this in linear algebra as the following, <br><br>
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/// let \f$X=\left[\begin{array}\\ x_0 \\ x_1 \\ \vdots \\ x_N \end{array}\right]\f$ <br><br>
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///
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/// then \f$\text{length2}(X)=X \cdot X^T\f$ <br><br>
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///
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/// \param x the vector
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template<typename genType, size_t...i>
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constexpr genType length2(const vector<genType, i...>& x)
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{ return fennec::dot(x, x); }
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// length --------------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief Returns the length of vector \f$x\f$, i.e., \f$\sqrt{x_0^2 + x_1^2 + \ldots}\f$
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///
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/// \returns the length of vector \f$x\f$, i.e., \f$\sqrt{x_0^2 + x_1^2 + \ldots}\f$<br><br>
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/// \details we can represent this in linear algebra as the following, <br><br>
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/// let \f$X=\left[\begin{array}\\ x_0 \\ x_1 \\ \vdots \\ x_N \end{array}\right]\f$ <br><br>
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///
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/// then, \f$\text{length}(X)=\left|\left|X\right|\right|\f$ <br><br>
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///
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/// \param x the vector
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template<typename genType, size_t...i>
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constexpr genType length(const vector<genType, i...>& x) { return fennec::sqrt(fennec::length2(x)); }
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// distance ------------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief Returns the length of vector \f$x\f$, i.e., \f$\sqrt{x_0^2 + x_1^2 + \ldots}\f$
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///
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/// \returns the distance between \f$p_0\f$ and \f$p_1\f$, i.e., \f$\left|{p_1-p_0}\right|\f$
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/// \details we can represent this in linear algebra as the following, <br><br>
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/// let \f$X=\left[\begin{array}\\ x_0 \\ x_1 \\ \vdots \\ x_N \end{array}\right]\f$ <br>
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/// let \f$Y=\left[\begin{array}\\ y_0 \\ y_1 \\ \vdots \\ y_N \end{array}\right]\f$ <br><br>
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///
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/// then \f$\text{distance}(X, Y)=\left|\left|Y-X\right|\right|\f$ <br><br>
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///
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/// \param p0 first vector
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/// \param p1 second vector
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template<typename genType, size_t...i>
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constexpr genType distance(const vector<genType, i...>& p0, const vector<genType, i...>& p1)
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{ return fennec::length(p1 - p0); }
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// cross ---------------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief Returns the cross product of \f$x\f$ and \f$y\f$, i.e.,
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/// \f$\left({x_1 \cdot y_2 - y_1 \cdot x_2, x_2 \cdot y_0 - y_2 \cdot x_0, x_0 \cdot y_1 - y_0 \cdot x_1}\right)\f$
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///
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/// \returns the cross product of \f$x\f$ and \f$y\f$, i.e.,
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/// \f$\left({x_1 \cdot y_2 - y_1 \cdot x_2, x_2 \cdot y_0 - y_2 \cdot x_0, x_0 \cdot y_1 - y_0 \cdot x_1}\right)\f$ <br><br>
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/// \details we can represent this in linear algebra as the following, <br><br>
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/// let \f$X=\left[\begin{array}\\ x_0 \\ x_1 \\ \vdots \\ x_N \end{array}\right]\f$ <br>
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/// let \f$Y=\left[\begin{array}\\ y_0 \\ y_1 \\ \vdots \\ y_N \end{array}\right]\f$ <br><br>
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///
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/// then \f$\text{cross}(X, Y)=X \times Y\f$ <br><br>
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///
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/// \param x first vector
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/// \param y second vector
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template<typename genType, size_t...i> requires(sizeof...(i) == 3)
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constexpr vector<genType, i...> cross(const vector<genType, i...>& x, const vector<genType, i...>& y)
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{ return vector<genType, i...>(x[1]*y[2]-y[1]*x[2], x[2]*y[0]-y[2]*x[0], x[0]*y[1]-y[0]*x[1]); }
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// normalize -----------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief Returns a vector in the same direction as \f$x\f$, but with a length of \f$1\f$, i.e.
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///
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/// \returns a vector in the same direction as \f$x\f$, but with a length of \f$1\f$, i.e.\f$\frac{x}{||x||}<br><br>
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/// \details we can represent this in linear algebra as the following, <br><br>
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/// let \f$X=\left[\begin{array}\\ x_0 \\ x_1 \\ \vdots \\ x_N \end{array}\right]\f$ <br><br>
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///
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/// then, \f$\text{length}(X)=\frac{X}{\left|\left|X\right|\right|}\f$ <br><br>
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///
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/// \param x
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template<typename genType, size_t...i>
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constexpr vector<genType, i...> normalize(const vector<genType, i...>& x)
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{ return x / fennec::length(x); }
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// faceforward ---------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief If \f$\text{dot}(Nref, I)<0\f$ return \f$N\f$, otherwise return \f$-N\f$.
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///
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/// \returns \f$N\f$ if \f$\text{dot}(Nref,I)<0\f$, otherwise, returns \f$-N\f$.<br><br>
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///
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/// \param N the vector
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/// \param I the incident
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/// \param Nref the reference
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template<typename genType, size_t...i>
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constexpr vector<genType, i...> faceforward(const vector<genType, i...>& N, const vector<genType, i...>& I, const vector<genType, i...>& Nref)
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{ return fennec::dot(Nref, I) < 0 ? N : -N; }
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// reflect -------------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief For the incident vector \f$I\f$ and surface orientation \f$N\f$, returns the reflection direction.
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///
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/// \returns The reflection direction, given the incident vector \f$I\f$ and surface orientation \f$N\f$ <br><br>
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/// \details We can express this as, <br><br>
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/// \f$\text{reflect}(I, N) = I - 2 N \cdot \text{dot}(N, I)\f$ <br><br>
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///
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/// \param I the incident
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/// \param N the surface orientation
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template<typename genType, size_t...i>
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constexpr vector<genType, i...> reflect(const vector<genType, i...>& I, const vector<genType, i...>& N)
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{ return I - 2.0 * fennec::dot(N, I) * N; }
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// refract -------------------------------------------------------------------------------------------------------------
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///
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///
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/// \brief or the incident vector \f$I\f$ and surface normal \f$N\f$, and the ratio of indices of refraction \f$eta\f$,
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/// return the refraction vector.
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///
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/// \returns The refraction vector, given the incident vector \f$I\f$, surface normal \f$N\f$, and ratio \f$eta\f$.<br><br>
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/// \details The result is computed by the refraction equation, <br><br>
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/// let \f$k=1.0-eta^2 \cdot (1.0 - \text{dot}(N, I)^2)\f$ <br>
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/// then, \f$\text{refract}(I, N, eta)=\begin{cases} 0.0 & k<0.0, \\ eta \cdot I - N \cdot (eta \cdot \text{dot}(N, I) + \sqrt{k}) \end{cases}\f$ <br><br>
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///
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/// \param I the incident
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/// \param N the surface normal
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/// \param eta the ratio of indices of refraction
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template<typename genType, size_t...i>
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constexpr vector<genType, i...> refract(const vector<genType, i...>& I, const vector<genType, i...>& N, genType eta)
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{ genType ndi = fennec::dot(N, I);
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genType k = 1.0 - eta * eta * (1.0 - ndi * ndi);
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return (k < 0.0) ? 0.0 : eta * I - N * (eta * ndi + fennec::sqrt(k)); }
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}
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#endif // FENNEC_MATH_GEOMETRIC_H
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