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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 <cmath>
#include <vector>
#include <algorithm>
#include <iterator>
namespace boost{ namespace math{
namespace detail
{
template<class Point>
typename Point::value_type alpha_distance(Point const & p1, Point const & p2, typename Point::value_type alpha)
{
using std::pow;
using std::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 Point>
class catmull_rom
{
public:
catmull_rom(std::vector<Point>&& points, bool closed = false, typename Point::value_type alpha = (typename Point::value_type) 1/ (typename Point::value_type) 2);
catmull_rom(std::initializer_list<Point> l, bool closed = false, typename Point::value_type alpha = (typename Point::value_type) 1/ (typename Point::value_type) 2) : catmull_rom(std::vector<Point>(l), closed, alpha) {}
typename Point::value_type max_parameter() const
{
return m_max_s;
}
typename Point::value_type parameter_at_point(size_t i) const
{
return m_s[i+1];
}
Point operator()(const typename Point::value_type s) const;
Point prime(const typename Point::value_type s) const;
std::vector<Point>&& get_points()
{
return std::move(m_pnts);
}
private:
std::vector<Point> m_pnts;
std::vector<typename Point::value_type> m_s;
typename Point::value_type m_max_s;
};
template<class Point>
catmull_rom<Point>::catmull_rom(std::vector<Point>&& points, bool closed, typename Point::value_type alpha) : m_pnts(std::move(points))
{
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 (int64_t i = num_pnts-1; i >= 0; --i)
{
m_pnts[i+1] = m_pnts[i];
}
m_pnts[0] = tmp;
m_s[0] = -detail::alpha_distance<Point>(m_pnts[0], m_pnts[1], alpha);
if (abs(m_s[0]) < std::numeric_limits<typename Point::value_type>::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<Point>(m_pnts[i], m_pnts[i-1], alpha);
if (abs(d) < std::numeric_limits<typename Point::value_type>::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<class Point>
Point catmull_rom<Point>::operator()(const typename Point::value_type s) const
{
using std::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;
typename Point::value_type s3s = m_s[i+2] - 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[i+2] - 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[i+2][j]);
}
Point B2;
denom = 1/(m_s[i+2] - 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<class Point>
Point catmull_rom<Point>::prime(const typename Point::value_type s) const
{
using std::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
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