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// Copyright (c) 2014 INRIA Sophia-Antipolis (France). // All rights reserved. // // This file is part of CGAL (www.cgal.org). // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Barycentric_coordinates_2/include/CGAL/Barycentric_coordinates_2/Wachspress_2.h $ // $Id: include/CGAL/Barycentric_coordinates_2/Wachspress_2.h b26b07a1242 $ // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Dmitry Anisimov, David Bommes, Kai Hormann, and Pierre Alliez. #ifndef CGAL_BARYCENTRIC_WACHSPRESS_2_H #define CGAL_BARYCENTRIC_WACHSPRESS_2_H #include #include // CGAL headers. #include #include // Barycentric coordinates headers. #include // Boost headers. #include // CGAL namespace. namespace CGAL { // Barycentric coordinates namespace. namespace Barycentric_coordinates { #if !defined(CGAL_NO_DEPRECATED_CODE) || defined(DOXYGEN_RUNNING) // Examples: see the User Manual here - https://doc.cgal.org/latest/Manual/index.html. // [1] Reference: "M. S. Floater, K. Hormann, and G. Kos. A general construction of barycentric coordinates over convex polygons. Advances in Computational Mathematics, 24(1-4):311-331, 2006.". /*! * \ingroup PkgBarycentricCoordinates2RefDeprecated * The class `Wachspress_2` implements 2D Wachspress coordinates ( \cite cgal:bc:fhk-gcbcocp-06, \cite cgal:bc:mlbd-gbcip-02, \cite cgal:bc:w-rfeb-75 ). * This class is parameterized by a traits class `Traits`, and it is used as a coordinate class to complete the class `Generalized_barycentric_coordinates_2`. * For a polygon with three vertices it is better to use the class `Triangle_coordinates_2`. * Wachspress coordinates can be computed exactly, and they are always positive in the closure of a strictly convex polygon. * \deprecated This part of the package is deprecated since the version 5.4 of \cgal. \tparam Traits must be a model of the concepts `BarycentricTraits_2` and `PolygonTraits_2`. \cgalModels{BarycentricCoordinates_2} \pre The provided polygon is strictly convex. */ template class #ifndef DOXYGEN_RUNNING CGAL_DEPRECATED_MSG("This part of the package is deprecated since the version 5.4 of CGAL!") #endif Wachspress_2 { public: /// \name Types /// @{ /// Number type. typedef typename Traits::FT FT; /// Point type. typedef typename Traits::Point_2 Point_2; /// @} // \name Creation // Creates the class `Wachspress_2` that implements the behavior of Wachspress coordinates for any query point that does not belong to the polygon's boundary. // The polygon is given by a range of vertices of the type `Traits::Point_2` stored in a container of the type `std::vector`. Wachspress_2(const std::vector &vertices, const Traits &b_traits) : vertex(vertices), barycentric_traits(b_traits), number_of_vertices(vertex.size()), area_2(barycentric_traits.compute_area_2_object()), collinear_2(barycentric_traits.collinear_2_object()) { // Resize all the internal containers. A.resize(number_of_vertices); C.resize(number_of_vertices); weight.resize(number_of_vertices); } // Computation of Wachspress Weight Functions // This function computes Wachspress weights (unnormalized coordinates) for a chosen query point. template inline std::optional weights(const Point_2 &query_point, OutputIterator &output) { return weights_2(query_point, output); } // Computation of Wachspress Basis Functions // This function computes Wachspress barycentric coordinates for a chosen query point on the bounded side of a strictly convex polygon. // \pre The provided polygon is strictly convex. template inline std::optional coordinates_on_bounded_side(const Point_2 &query_point, OutputIterator &output, const Type_of_algorithm type_of_algorithm) { switch(type_of_algorithm) { case PRECISE: return coordinates_on_bounded_side_precise_2(query_point, output); break; case FAST: return coordinates_on_bounded_side_fast_2(query_point, output); break; } // Pointer cannot be here. Something went wrong. const bool type_of_algorithm_failure = true; CGAL_postcondition( !type_of_algorithm_failure ); if(!type_of_algorithm_failure) return std::optional(output); else return std::optional(); } // This function computes Wachspress barycentric coordinates for a chosen query point on the unbounded side of a strictly convex polygon. // \pre The provided polygon is strictly convex. template inline std::optional coordinates_on_unbounded_side(const Point_2 &query_point, OutputIterator &output, const Type_of_algorithm type_of_algorithm, const bool warning_tag = true) { switch(type_of_algorithm) { case PRECISE: return coordinates_on_unbounded_side_precise_2(query_point, output, warning_tag); break; case FAST: return coordinates_on_unbounded_side_fast_2(query_point, output, warning_tag); break; } // Pointer cannot be here. Something went wrong. const bool type_of_algorithm_failure = true; CGAL_postcondition( !type_of_algorithm_failure ); if(!type_of_algorithm_failure) return std::optional(output); else return std::optional(); } // Information Functions // This function prints some information about Wachspress coordinates. void print_coordinates_information(std::ostream &output_stream) const { return print_coordinates_information_2(output_stream); } private: // Some convenient typedefs. typedef typename std::vector FT_vector; typedef typename std::vector Point_vector; // Internal global variables. const Point_vector &vertex; const Traits &barycentric_traits; const size_t number_of_vertices; FT_vector A, C, weight; FT wp_denominator, inverted_wp_denominator; typename Traits::Compute_area_2 area_2; typename Traits::Collinear_2 collinear_2; // WEIGHTS. // Compute Wachspress weights without normalization. template std::optional weights_2(const Point_2 &query_point, OutputIterator &output) { // Get the number of vertices in the polygon. const int n = int(number_of_vertices); // Compute areas A and C following the area notation from [1]. Split the loop to make this computation faster. A[0] = area_2(vertex[0] , vertex[1], query_point); C[0] = area_2(vertex[n-1], vertex[0], vertex[1] ); for(int i = 1; i < n-1; ++i) { A[i] = area_2(vertex[i] , vertex[i+1], query_point); C[i] = area_2(vertex[i-1], vertex[i] , vertex[i+1]); } A[n-1] = area_2(vertex[n-1], vertex[0] , query_point); C[n-1] = area_2(vertex[n-2], vertex[n-1], vertex[0] ); // Compute unnormalized weights following the formula (28) from [1]. CGAL_precondition( A[n-1] != FT(0) && A[0] != FT(0) ); *output = C[0] / (A[n-1] * A[0]); ++output; for(int i = 1; i < n-1; ++i) { CGAL_precondition( A[i-1] != FT(0) && A[i] != FT(0) ); *output = C[i] / (A[i-1] * A[i]); ++output; } CGAL_precondition( A[n-2] != FT(0) && A[n-1] != FT(0) ); *output = C[n-1] / (A[n-2] * A[n-1]); ++output; // Return weights. return std::optional(output); } // COORDINATES ON BOUNDED SIDE. // Compute Wachspress coordinates on the bounded side of the polygon with the slow O(n^2) but precise algorithm. // Here, n - is the number of the polygon's vertices. template std::optional coordinates_on_bounded_side_precise_2(const Point_2 &query_point, OutputIterator &output) { CGAL_precondition( type_of_polygon() == STRICTLY_CONVEX ); // Get the number of vertices in the polygon. const int n = int(number_of_vertices); // Compute areas A following the area notation from [1]. Split the loop to make this computation faster. A[0] = area_2(vertex[0], vertex[1], query_point); for(int i = 1; i < n-1; ++i) A[i] = area_2(vertex[i], vertex[i+1], query_point); A[n-1] = area_2(vertex[n-1], vertex[0], query_point); // Initialize weights with areas C following the area notation from [1]. // Then we multiply them by areas A as in the formula (5) from [1]. We also split the loop. weight[0] = area_2(vertex[n-1], vertex[0], vertex[1]); for(int j = 1; j < n-1; ++j) weight[0] *= A[j]; for(int i = 1; i < n-1; ++i) { weight[i] = area_2(vertex[i-1], vertex[i], vertex[i+1]); for(int j = 0; j < i-1; ++j) weight[i] *= A[j]; for(int j = i+1; j < n; ++j) weight[i] *= A[j]; } weight[n-1] = area_2(vertex[n-2], vertex[n-1], vertex[0]); for(int j = 0; j < n-2; ++j) weight[n-1] *= A[j]; // Compute the sum of all weights - denominator of Wachspress coordinates. wp_denominator = weight[0]; for(int i = 1; i < n; ++i) wp_denominator += weight[i]; // Invert this denominator. CGAL_precondition( wp_denominator != FT(0) ); inverted_wp_denominator = FT(1) / wp_denominator; // Normalize weights and save them as resulting Wachspress coordinates. for(int i = 0; i < n-1; ++i) { *output = weight[i] * inverted_wp_denominator; ++output; } *output = weight[n-1] * inverted_wp_denominator; ++output; // Return coordinates. return std::optional(output); } // Compute Wachspress coordinates on the bounded side of the polygon with the fast O(n) but less precise algorithm. // Here, n - is the number of the polygon's vertices. Precision is lost near the boundary (~ 1.0e-10 and closer). template std::optional coordinates_on_bounded_side_fast_2(const Point_2 &query_point, OutputIterator &output) { CGAL_precondition( type_of_polygon() == STRICTLY_CONVEX ); // Get the number of vertices in the polygon. const int n = int(number_of_vertices); // Compute areas A and C following the area notation from [1]. Split the loop to make this computation faster. A[0] = area_2(vertex[0] , vertex[1], query_point); C[0] = area_2(vertex[n-1], vertex[0], vertex[1] ); for(int i = 1; i < n-1; ++i) { A[i] = area_2(vertex[i] , vertex[i+1], query_point); C[i] = area_2(vertex[i-1], vertex[i] , vertex[i+1]); } A[n-1] = area_2(vertex[n-1], vertex[0] , query_point); C[n-1] = area_2(vertex[n-2], vertex[n-1], vertex[0] ); // Compute the unnormalized weights following the formula (28) from [1]. CGAL_precondition( A[n-1] != FT(0) && A[0] != FT(0) ); weight[0] = C[0] / (A[n-1] * A[0]); for(int i = 1; i < n-1; ++i) { CGAL_precondition( A[i-1] != FT(0) && A[i] != FT(0) ); weight[i] = C[i] / (A[i-1] * A[i]); } CGAL_precondition( A[n-2] != FT(0) && A[n-1] != FT(0) ); weight[n-1] = C[n-1] / (A[n-2] * A[n-1]); // Compute the sum of all weights - denominator of Wachspress coordinates. wp_denominator = weight[0]; for(int i = 1; i < n; ++i) wp_denominator += weight[i]; // Invert this denominator. CGAL_precondition( wp_denominator != FT(0) ); inverted_wp_denominator = FT(1) / wp_denominator; // Normalize weights and save them as resulting Wachspress coordinates. for(int i = 0; i < n-1; ++i) { *output = weight[i] * inverted_wp_denominator; ++output; } *output = weight[n-1] * inverted_wp_denominator; ++output; // Return coordinates. return std::optional(output); } // COORDINATES ON UNBOUNDED SIDE. // Compute Wachspress coordinates on the unbounded side of the polygon with the slow O(n^2) but precise algorithm. // Here, n - is the number of the polygon's vertices. template std::optional coordinates_on_unbounded_side_precise_2(const Point_2 &query_point, OutputIterator &output, bool warning_tag) { if(warning_tag) std::cout << std::endl << "ATTENTION: Wachspress coordinates might be not well-defined outside the polygon!" << std::endl; // Use the same formulas as for the bounded side since they are also valid on the unbounded side. return coordinates_on_bounded_side_precise_2(query_point, output); } // Compute Wachspress coordinates on the unbounded side of the polygon with the fast O(n) but less precise algorithm. // Here, n - is the number of the polygon's vertices. Precision is lost near the boundary (~ 1.0e-10 and closer). template std::optional coordinates_on_unbounded_side_fast_2(const Point_2 &query_point, OutputIterator &output, bool warning_tag) { if(warning_tag) std::cout << std::endl << "ATTENTION: Wachspress coordinates might be not well-defined outside the polygon!" << std::endl; // Use the same formulas as for the bounded side since they are also valid on the unbounded side. return coordinates_on_bounded_side_fast_2(query_point, output); } // OTHER FUNCTIONS. // Print some information about Wachspress coordinates. void print_coordinates_information_2(std::ostream &output_stream) const { output_stream << std::endl << "CONVEXITY: " << std::endl << std::endl; if(type_of_polygon() == STRICTLY_CONVEX) { output_stream << "This polygon is strictly convex." << std::endl; } else if(type_of_polygon() == WEAKLY_CONVEX) { output_stream << "This polygon is weakly convex. The correct computation is not expected!" << std::endl; } else if(type_of_polygon() == CONCAVE) { output_stream << "This polygon polygon is not convex. The correct computation is not expected!" << std::endl; } output_stream << std::endl << "TYPE OF COORDINATES: " << std::endl << std::endl; output_stream << "The coordinate functions to be computed are Wachspress coordinates." << std::endl; output_stream << std::endl << "INFORMATION ABOUT COORDINATES: " << std::endl << std::endl; output_stream << "Wachspress coordinates are well-defined in the closure of an arbitrary strictly convex polygon and can be computed exactly." << std::endl; output_stream << std::endl; output_stream << "They satisfy the following properties: " << std::endl; output_stream << "1. Partition of unity or constant precision;" << std::endl; output_stream << "2. Homogeneity or linear precision;" << std::endl; output_stream << "3. Lagrange property;" << std::endl; output_stream << "4. Non-negativity;" << std::endl; output_stream << "5. Boundedness between 0 and 1;" << std::endl; output_stream << "6. Linearity along edges;" << std::endl; output_stream << "7. Smoothness;" << std::endl; output_stream << "8. Similarity invariance." << std::endl; output_stream << std::endl; output_stream << "For polygons, whose vertices lie on a common circle, they coincide with discrete harmonic coordinates." << std::endl; output_stream << std::endl << "REFERENCE: " << std::endl << std::endl; output_stream << "M. S. Floater, K. Hormann, and G. Kos. A general construction of barycentric coordinates over convex polygons. Advances in Computational Mathematics, 24(1-4):311-331, 2006." << std::endl; } // Check the type of the provided polygon - CONVEX, STRICTLY_CONVEX, or CONCAVE. Type_of_polygon type_of_polygon() const { // First, test the polygon on convexity. if(CGAL::is_convex_2(vertex.begin(), vertex.end(), barycentric_traits)) { // Index of the last polygon's vertex. const int last = int(number_of_vertices) - 1; // Test all the consequent triplets of the polygon's vertices on collinearity. // In case we find at least one, return WEAKLY_CONVEX polygon. if(collinear_2(vertex[last], vertex[0], vertex[1])) return WEAKLY_CONVEX; for(int i = 1; i < last; ++i) { if(collinear_2(vertex[i-1], vertex[i], vertex[i+1])) return WEAKLY_CONVEX; } if(collinear_2(vertex[last-1], vertex[last], vertex[0])) return WEAKLY_CONVEX; // Otherwise, return STRICTLY_CONVEX polygon. return STRICTLY_CONVEX; } // Otherwise, return CONCAVE polygon. return CONCAVE; } }; #endif // CGAL_NO_DEPRECATED_CODE } // namespace Barycentric_coordinates } // namespace CGAL #include #endif // CGAL_BARYCENTRIC_WACHSPRESS_2_H