138 lines
4.2 KiB
C++
138 lines
4.2 KiB
C++
/**
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BSD 3-Clause License
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This file is part of the Basalt project.
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https://gitlab.com/VladyslavUsenko/basalt-headers.git
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Copyright (c) 2019, Vladyslav Usenko and Nikolaus Demmel.
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All rights reserved.
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions are met:
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* Redistributions of source code must retain the above copyright notice, this
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list of conditions and the following disclaimer.
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* Redistributions in binary form must reproduce the above copyright notice,
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this list of conditions and the following disclaimer in the documentation
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and/or other materials provided with the distribution.
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* Neither the name of the copyright holder nor the names of its
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contributors may be used to endorse or promote products derived from
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this software without specific prior written permission.
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THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
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DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
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FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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@file
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@brief Common functions for B-spline evaluation
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*/
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#pragma once
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#include <Eigen/Dense>
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#include <cstdint>
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namespace basalt {
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/// @brief Compute binomial coefficient.
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///
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/// Computes number of combinations that include k objects out of n.
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/// @param[in] n
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/// @param[in] k
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/// @return binomial coefficient
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constexpr inline uint64_t binomialCoefficient(uint64_t n, uint64_t k) {
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if (k > n) {
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return 0;
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}
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uint64_t r = 1;
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for (uint64_t d = 1; d <= k; ++d) {
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r *= n--;
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r /= d;
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}
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return r;
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}
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/// @brief Compute blending matrix for uniform B-spline evaluation.
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///
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/// @param _N order of the spline
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/// @param _Scalar scalar type to use
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/// @param _Cumulative if the spline should be cumulative
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template <int _N, typename _Scalar = double, bool _Cumulative = false>
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Eigen::Matrix<_Scalar, _N, _N> computeBlendingMatrix() {
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Eigen::Matrix<double, _N, _N> m;
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m.setZero();
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for (int i = 0; i < _N; ++i) {
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for (int j = 0; j < _N; ++j) {
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double sum = 0;
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for (int s = j; s < _N; ++s) {
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sum += std::pow(-1.0, s - j) * binomialCoefficient(_N, s - j) *
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std::pow(_N - s - 1.0, _N - 1.0 - i);
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}
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m(j, i) = binomialCoefficient(_N - 1, _N - 1 - i) * sum;
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}
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}
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if (_Cumulative) {
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for (int i = 0; i < _N; i++) {
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for (int j = i + 1; j < _N; j++) {
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m.row(i) += m.row(j);
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}
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}
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}
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uint64_t factorial = 1;
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for (int i = 2; i < _N; ++i) {
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factorial *= i;
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}
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return (m / factorial).template cast<_Scalar>();
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}
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/// @brief Compute base coefficient matrix for polynomials of size N.
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///
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/// In each row starting from 0 contains the derivative coefficients of the
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/// polynomial. For _N=5 we get the following matrix: \f[ \begin{bmatrix}
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/// 1 & 1 & 1 & 1 & 1
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/// \\0 & 1 & 2 & 3 & 4
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/// \\0 & 0 & 2 & 6 & 12
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/// \\0 & 0 & 0 & 6 & 24
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/// \\0 & 0 & 0 & 0 & 24
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/// \\ \end{bmatrix}
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/// \f]
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/// Functions \ref RdSpline::baseCoeffsWithTime and \ref
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/// So3Spline::baseCoeffsWithTime use this matrix to compute derivatives of the
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/// time polynomial.
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///
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/// @param _N order of the polynomial
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/// @param _Scalar scalar type to use
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template <int _N, typename _Scalar = double>
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Eigen::Matrix<_Scalar, _N, _N> computeBaseCoefficients() {
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Eigen::Matrix<double, _N, _N> base_coefficients;
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base_coefficients.setZero();
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base_coefficients.row(0).setOnes();
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constexpr int DEG = _N - 1;
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int order = DEG;
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for (int n = 1; n < _N; n++) {
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for (int i = DEG - order; i < _N; i++) {
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base_coefficients(n, i) = (order - DEG + i) * base_coefficients(n - 1, i);
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}
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order--;
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}
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return base_coefficients.template cast<_Scalar>();
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}
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} // namespace basalt
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