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// Copyright Nick Thompson, 2017 // Use, modification and distribution are subject to the // Boost Software License, Version 1.0. // (See accompanying file LICENSE_1_0.txt // or copy at http://www.boost.org/LICENSE_1_0.txt) // This computes the Catmull-Rom spline from a list of points. #ifndef BOOST_MATH_INTERPOLATORS_CATMULL_ROM #define BOOST_MATH_INTERPOLATORS_CATMULL_ROM #include #include #include #include #include #include namespace std_workaround { #if defined(__cpp_lib_nonmember_container_access) || (defined(_MSC_VER) && (_MSC_VER >= 1900)) using std::size; #else template inline constexpr std::size_t size(const C& c) { return c.size(); } template inline constexpr std::size_t size(const T(&array)[N]) noexcept { return N; } #endif } namespace boost{ namespace math{ namespace detail { template typename Point::value_type alpha_distance(Point const & p1, Point const & p2, typename Point::value_type alpha) { using std::pow; using std_workaround::size; typename Point::value_type dsq = 0; for (size_t i = 0; i < size(p1); ++i) { typename Point::value_type dx = p1[i] - p2[i]; dsq += dx*dx; } return pow(dsq, alpha/2); } } template > class catmull_rom { typedef typename Point::value_type value_type; public: catmull_rom(RandomAccessContainer&& points, bool closed = false, value_type alpha = (value_type) 1/ (value_type) 2); catmull_rom(std::initializer_list l, bool closed = false, value_type alpha = (value_type) 1/ (value_type) 2) : catmull_rom(RandomAccessContainer(l), closed, alpha) {} value_type max_parameter() const { return m_max_s; } value_type parameter_at_point(size_t i) const { return m_s[i+1]; } Point operator()(const value_type s) const; Point prime(const value_type s) const; RandomAccessContainer&& get_points() { return std::move(m_pnts); } private: RandomAccessContainer m_pnts; std::vector m_s; value_type m_max_s; }; template catmull_rom::catmull_rom(RandomAccessContainer&& points, bool closed, typename Point::value_type alpha) : m_pnts(std::move(points)) { std::size_t num_pnts = m_pnts.size(); //std::cout << "Number of points = " << num_pnts << "\n"; if (num_pnts < 4) { throw std::domain_error("The Catmull-Rom curve requires at least 4 points."); } if (alpha < 0 || alpha > 1) { throw std::domain_error("The parametrization alpha must be in the range [0,1]."); } using std::abs; m_s.resize(num_pnts+3); m_pnts.resize(num_pnts+3); //std::cout << "Number of points now = " << m_pnts.size() << "\n"; m_pnts[num_pnts+1] = m_pnts[0]; m_pnts[num_pnts+2] = m_pnts[1]; auto tmp = m_pnts[num_pnts-1]; for (auto i = num_pnts; i > 0; --i) { m_pnts[i] = m_pnts[i - 1]; } m_pnts[0] = tmp; m_s[0] = -detail::alpha_distance(m_pnts[0], m_pnts[1], alpha); if (abs(m_s[0]) < std::numeric_limits::epsilon()) { throw std::domain_error("The first and last point should not be the same.\n"); } m_s[1] = 0; for (size_t i = 2; i < m_s.size(); ++i) { typename Point::value_type d = detail::alpha_distance(m_pnts[i], m_pnts[i-1], alpha); if (abs(d) < std::numeric_limits::epsilon()) { throw std::domain_error("The control points of the Catmull-Rom curve are too close together; this will lead to ill-conditioning.\n"); } m_s[i] = m_s[i-1] + d; } if(closed) { m_max_s = m_s[num_pnts+1]; } else { m_max_s = m_s[num_pnts]; } } template Point catmull_rom::operator()(const typename Point::value_type s) const { using std_workaround::size; if (s < 0 || s > m_max_s) { throw std::domain_error("Parameter outside bounds."); } auto it = std::upper_bound(m_s.begin(), m_s.end(), s); //Now *it >= s. We want the index such that m_s[i] <= s < m_s[i+1]: size_t i = std::distance(m_s.begin(), it - 1); // Only denom21 is used twice: typename Point::value_type denom21 = 1/(m_s[i+1] - m_s[i]); typename Point::value_type s0s = m_s[i-1] - s; typename Point::value_type s1s = m_s[i] - s; typename Point::value_type s2s = m_s[i+1] - s; size_t ip2 = i + 2; // When the curve is closed and we evaluate at the end, the endpoint is in fact the startpoint. if (ip2 == m_s.size()) { ip2 = 0; } typename Point::value_type s3s = m_s[ip2] - s; Point A1_or_A3; typename Point::value_type denom = 1/(m_s[i] - m_s[i-1]); for(size_t j = 0; j < size(m_pnts[0]); ++j) { A1_or_A3[j] = denom*(s1s*m_pnts[i-1][j] - s0s*m_pnts[i][j]); } Point A2_or_B2; for(size_t j = 0; j < size(m_pnts[0]); ++j) { A2_or_B2[j] = denom21*(s2s*m_pnts[i][j] - s1s*m_pnts[i+1][j]); } Point B1_or_C; denom = 1/(m_s[i+1] - m_s[i-1]); for(size_t j = 0; j < size(m_pnts[0]); ++j) { B1_or_C[j] = denom*(s2s*A1_or_A3[j] - s0s*A2_or_B2[j]); } denom = 1/(m_s[ip2] - m_s[i+1]); for(size_t j = 0; j < size(m_pnts[0]); ++j) { A1_or_A3[j] = denom*(s3s*m_pnts[i+1][j] - s2s*m_pnts[ip2][j]); } Point B2; denom = 1/(m_s[ip2] - m_s[i]); for(size_t j = 0; j < size(m_pnts[0]); ++j) { B2[j] = denom*(s3s*A2_or_B2[j] - s1s*A1_or_A3[j]); } for(size_t j = 0; j < size(m_pnts[0]); ++j) { B1_or_C[j] = denom21*(s2s*B1_or_C[j] - s1s*B2[j]); } return B1_or_C; } template Point catmull_rom::prime(const typename Point::value_type s) const { using std_workaround::size; // https://math.stackexchange.com/questions/843595/how-can-i-calculate-the-derivative-of-a-catmull-rom-spline-with-nonuniform-param // http://denkovacs.com/2016/02/catmull-rom-spline-derivatives/ if (s < 0 || s > m_max_s) { throw std::domain_error("Parameter outside bounds.\n"); } auto it = std::upper_bound(m_s.begin(), m_s.end(), s); //Now *it >= s. We want the index such that m_s[i] <= s < m_s[i+1]: size_t i = std::distance(m_s.begin(), it - 1); Point A1; typename Point::value_type denom = 1/(m_s[i] - m_s[i-1]); typename Point::value_type k1 = (m_s[i]-s)*denom; typename Point::value_type k2 = (s - m_s[i-1])*denom; for (size_t j = 0; j < size(m_pnts[0]); ++j) { A1[j] = k1*m_pnts[i-1][j] + k2*m_pnts[i][j]; } Point A1p; for (size_t j = 0; j < size(m_pnts[0]); ++j) { A1p[j] = denom*(m_pnts[i][j] - m_pnts[i-1][j]); } Point A2; denom = 1/(m_s[i+1] - m_s[i]); k1 = (m_s[i+1]-s)*denom; k2 = (s - m_s[i])*denom; for (size_t j = 0; j < size(m_pnts[0]); ++j) { A2[j] = k1*m_pnts[i][j] + k2*m_pnts[i+1][j]; } Point A2p; for (size_t j = 0; j < size(m_pnts[0]); ++j) { A2p[j] = denom*(m_pnts[i+1][j] - m_pnts[i][j]); } Point B1; for (size_t j = 0; j < size(m_pnts[0]); ++j) { B1[j] = k1*A1[j] + k2*A2[j]; } Point A3; denom = 1/(m_s[i+2] - m_s[i+1]); k1 = (m_s[i+2]-s)*denom; k2 = (s - m_s[i+1])*denom; for (size_t j = 0; j < size(m_pnts[0]); ++j) { A3[j] = k1*m_pnts[i+1][j] + k2*m_pnts[i+2][j]; } Point A3p; for (size_t j = 0; j < size(m_pnts[0]); ++j) { A3p[j] = denom*(m_pnts[i+2][j] - m_pnts[i+1][j]); } Point B2; denom = 1/(m_s[i+2] - m_s[i]); k1 = (m_s[i+2]-s)*denom; k2 = (s - m_s[i])*denom; for (size_t j = 0; j < size(m_pnts[0]); ++j) { B2[j] = k1*A2[j] + k2*A3[j]; } Point B1p; denom = 1/(m_s[i+1] - m_s[i-1]); for (size_t j = 0; j < size(m_pnts[0]); ++j) { B1p[j] = denom*(A2[j] - A1[j] + (m_s[i+1]- s)*A1p[j] + (s-m_s[i-1])*A2p[j]); } Point B2p; denom = 1/(m_s[i+2] - m_s[i]); for (size_t j = 0; j < size(m_pnts[0]); ++j) { B2p[j] = denom*(A3[j] - A2[j] + (m_s[i+2] - s)*A2p[j] + (s - m_s[i])*A3p[j]); } Point Cp; denom = 1/(m_s[i+1] - m_s[i]); for (size_t j = 0; j < size(m_pnts[0]); ++j) { Cp[j] = denom*(B2[j] - B1[j] + (m_s[i+1] - s)*B1p[j] + (s - m_s[i])*B2p[j]); } return Cp; } }} #endif