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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/Mean_value_2.h $ // $Id: include/CGAL/Barycentric_coordinates_2/Mean_value_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_MEAN_VALUE_2_H #define CGAL_BARYCENTRIC_MEAN_VALUE_2_H #include #include // STL headers. #include // CGAL headers. #include #include #include // Boost headers. #include #include // Barycentric coordinates headers. #include // CGAL namespace. namespace CGAL { // Barycentric coordinates namespace. namespace Barycentric_coordinates { #if !defined(CGAL_NO_DEPRECATED_CODE) || defined(DOXYGEN_RUNNING) /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// // Try to find a square root object in the provided `Traits` class. If not, then use the default square root from CGAL. // Finds a square root of the provided value of the type `Kernel::FT` by first converting it to the double type and then taking the square root using the `CGAL::sqrt()` function. template class Default_sqrt { typedef typename Traits::FT FT; public: FT operator()(const FT &value) const { return FT(CGAL::sqrt(CGAL::to_double(value))); } }; BOOST_MPL_HAS_XXX_TRAIT_NAMED_DEF(Has_nested_type_Sqrt, Sqrt, false) // Case: do_not_use_default = false. template::value> class Get_sqrt { public: typedef Default_sqrt Sqrt; static Sqrt sqrt_object(const Traits&) { return Sqrt(); } }; // Case: do_not_use_default = true. template class Get_sqrt { public: typedef typename Traits::Sqrt Sqrt; static Sqrt sqrt_object(const Traits &traits) { return traits.sqrt_object(); } }; /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// // Examples: see the User Manual here - https://doc.cgal.org/latest/Manual/index.html. // [1] Reference: "K. Hormann and M. Floater. Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics, 25(4):1424-1441, 2006.". // [2] Reference: "M. S. Floater, Wachspress and mean value coordinates, to appear in the Proceedings of the 14th International Conference on Approximation Theory, G. Fasshauer and L. L. Schumaker (eds.)." /*! * \ingroup PkgBarycentricCoordinates2RefDeprecated * The class `Mean_value_2` implements 2D mean value coordinates ( \cite cgal:bc:hf-mvcapp-06, \cite cgal:bc:fhk-gcbcocp-06, \cite cgal:f-mvc-03 ). * 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 (triangle) it is better to use the class `Triangle_coordinates_2`. * Mean value coordinates can be computed only approximately due to an inevitable square root operation, and they are necessarily positive only inside the kernel of a star-shaped polygon and inside any quadrilateral. * \deprecated This part of the package is deprecated since the version 5.4 of \cgal. \tparam Traits must be a model of the concept `BarycentricTraits_2`. \cgalModels{BarycentricCoordinates_2} */ template class #ifndef DOXYGEN_RUNNING CGAL_DEPRECATED_MSG("This part of the package is deprecated since the version 5.4 of CGAL!") #endif Mean_value_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 `Mean_value_2` that implements the behavior of mean value 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`. Mean_value_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()), squared_length_2(barycentric_traits.compute_squared_length_2_object()), scalar_product_2(barycentric_traits.compute_scalar_product_2_object()), sqrt(Get_sqrt::sqrt_object(barycentric_traits)) { // Resize all the internal containers. s.resize(number_of_vertices); r.resize(number_of_vertices); A.resize(number_of_vertices); B.resize(number_of_vertices); D.resize(number_of_vertices); P.resize(number_of_vertices); t.resize(number_of_vertices); weight.resize(number_of_vertices); } // Computation of Mean Value Weight Functions // This function computes mean value 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 Mean Value Basis Functions // This function computes mean value barycentric coordinates for a chosen query point on the bounded side of a simple polygon. 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 mean value barycentric coordinates for a chosen query point on the unbounded side of a simple polygon. template inline std::optional coordinates_on_unbounded_side(const Point_2 &query_point, OutputIterator &output, const Type_of_algorithm type_of_algorithm) { switch(type_of_algorithm) { case PRECISE: return coordinates_on_unbounded_side_precise_2(query_point, output); break; case FAST: return coordinates_on_unbounded_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(); } // Information Functions // This function prints some information about mean value 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; typedef typename Traits::Vector_2 Vector; typedef typename std::vector Vector_vector; // Internal global variables. const Point_vector &vertex; const Traits &barycentric_traits; const size_t number_of_vertices; Vector_vector s; FT_vector r, A, B, D, P, t, weight; FT mv_denominator, inverted_mv_denominator; typename Traits::Compute_area_2 area_2; typename Traits::Compute_squared_length_2 squared_length_2; typename Traits::Compute_scalar_product_2 scalar_product_2; typename Get_sqrt::Sqrt sqrt; // WEIGHTS. // Compute mean value 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 vectors s following the pseudo-code in the Figure 10 from [1]. for(int i = 0; i < n; ++i) s[i] = vertex[i] - query_point; // Compute lengths r, areas A, and dot products D following the pseudo-code in the Figure 10 from [1]. // Split the loop to make this computation faster. r[0] = sqrt(squared_length_2(s[0])); A[0] = area_2(vertex[0], vertex[1], query_point); D[0] = scalar_product_2(s[0], s[1]); for(int i = 1; i < n-1; ++i) { r[i] = sqrt(squared_length_2(s[i])); A[i] = area_2(vertex[i], vertex[i+1], query_point); D[i] = scalar_product_2(s[i], s[i+1]); } r[n-1] = sqrt(squared_length_2(s[n-1])); A[n-1] = area_2(vertex[n-1], vertex[0], query_point); D[n-1] = scalar_product_2(s[n-1], s[0]); // Compute intermediate values t using the formulas from slide 19 here // - https://www.inf.usi.ch/hormann/nsfworkshop/presentations/Hormann.pdf for(int i = 0; i < n-1; ++i) { CGAL_precondition( (r[i]*r[i+1] + D[i]) != FT(0) ); t[i] = A[i] / (r[i]*r[i+1] + D[i]); } CGAL_precondition( (r[n-1]*r[0] + D[n-1]) != FT(0) ); t[n-1] = A[n-1] / (r[n-1]*r[0] + D[n-1]); // Compute mean value weights using the same pseudo-code as before. CGAL_precondition( r[0] != FT(0) ); *output = (t[n-1] + t[0]) / r[0]; ++output; for(int i = 1; i < n-1; ++i) { CGAL_precondition( r[i] != FT(0) ); *output = (t[i-1] + t[i]) / r[i]; ++output; } CGAL_precondition( r[n-1] != FT(0) ); *output = (t[n-2] + t[n-1]) / r[n-1]; ++output; // Return weights. return std::optional(output); } // COORDINATES ON BOUNDED SIDE. // Compute mean value 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) { // Get the number of vertices in the polygon. const int n = int(number_of_vertices); // Compute vectors s and its lengths r following the pseudo-code in the Figure 10 from [1]. s[0] = vertex[0] - query_point; r[0] = sqrt(squared_length_2(s[0])); // Compute areas A and B following the notation from [1] (see Figure 2). Split the loop to make this computation faster. A[0] = area_2(vertex[0] , vertex[1], query_point); B[0] = area_2(vertex[n-1], vertex[1], query_point); for(int i = 1; i < n-1; ++i) { s[i] = vertex[i] - query_point; r[i] = sqrt(squared_length_2(s[i])); A[i] = area_2(vertex[i] , vertex[i+1], query_point); B[i] = area_2(vertex[i-1], vertex[i+1], query_point); } s[n-1] = vertex[n-1] - query_point; r[n-1] = sqrt(squared_length_2(s[n-1])); A[n-1] = area_2(vertex[n-1], vertex[0], query_point); B[n-1] = area_2(vertex[n-2], vertex[0], query_point); // Following section 4.2 from [2] we denote P_j = r_j*r_{j+1} + dot_product(d_j, d_{j+1}). // Vector s_i from [1] corresponds to that one with the name d_i in [2]. for(int j = 0; j < n-1; ++j) P[j] = CGAL::max(r[j]*r[j+1] + scalar_product_2(s[j], s[j+1]), 0); P[n-1] = CGAL::max(r[n-1]*r[0] + scalar_product_2(s[n-1], s[0]), 0); // Compute mean value weights using the formula (16) from [2]. // Since the formula (16) always gives positive values, we have to add a proper sign to all the weight functions. weight[0] = r[n-1]*r[1] - scalar_product_2(s[n-1], s[1]); for(int j = 1; j < n-1; ++j) weight[0] *= P[j]; weight[0] = sign_of_weight(A[n-1], A[0], B[0])*sqrt(weight[0]); for(int i = 1; i < n-1; ++i) { weight[i] = r[i-1]*r[i+1] - scalar_product_2(s[i-1], s[i+1]); for(int j = 0; j < i-1; ++j) weight[i] *= P[j]; for(int j = i+1; j < n; ++j) weight[i] *= P[j]; weight[i] = sign_of_weight(A[i-1], A[i], B[i])*sqrt(weight[i]); } weight[n-1] = r[n-2]*r[0] - scalar_product_2(s[n-2], s[0]); for(int j = 0; j < n-2; ++j) weight[n-1] *= P[j]; weight[n-1] = sign_of_weight(A[n-2], A[n-1], B[n-1])*sqrt(weight[n-1]); // Compute the sum of all weights - denominator of mean value coordinates. mv_denominator = weight[0]; for(int i = 1; i < n; ++i) mv_denominator += weight[i]; // Invert this denominator. CGAL_precondition( mv_denominator != FT(0) ); inverted_mv_denominator = FT(1) / mv_denominator; // Normalize weights and save them as resulting mean value coordinates. for(int i = 0; i < n-1; ++i) { *output = weight[i] * inverted_mv_denominator; ++output; } *output = weight[n-1] * inverted_mv_denominator; ++output; // Return coordinates. return std::optional(output); } // Compute mean value 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) { // Get the number of vertices in the polygon. const int n = int(number_of_vertices); // Compute vectors s following the pseudo-code in the Figure 10 from [1]. for(int i = 0; i < n; ++i) s[i] = vertex[i] - query_point; // Compute lengths r, areas A, and dot products D following the pseudo-code in the Figure 10 from [1]. // Split the loop to make this computation faster. r[0] = sqrt(squared_length_2(s[0])); A[0] = area_2(vertex[0], vertex[1], query_point); D[0] = scalar_product_2(s[0], s[1]); for(int i = 1; i < n-1; ++i) { r[i] = sqrt(squared_length_2(s[i])); A[i] = area_2(vertex[i], vertex[i+1], query_point); D[i] = scalar_product_2(s[i], s[i+1]); } r[n-1] = sqrt(squared_length_2(s[n-1])); A[n-1] = area_2(vertex[n-1], vertex[0], query_point); D[n-1] = scalar_product_2(s[n-1], s[0]); // Compute intermediate values t using the formulas from slide 19 here // - https://www.inf.usi.ch/hormann/nsfworkshop/presentations/Hormann.pdf for(int i = 0; i < n-1; ++i) { CGAL_precondition( (r[i]*r[i+1] + D[i]) != FT(0) ); t[i] = A[i] / (r[i]*r[i+1] + D[i]); } CGAL_precondition( (r[n-1]*r[0] + D[n-1]) != FT(0) ); t[n-1] = A[n-1] / (r[n-1]*r[0] + D[n-1]); // Compute mean value weights using the same pseudo-code as before. CGAL_precondition( r[0] != FT(0) ); weight[0] = (t[n-1] + t[0]) / r[0]; for(int i = 1; i < n-1; ++i) { CGAL_precondition( r[i] != FT(0) ); weight[i] = (t[i-1] + t[i]) / r[i]; } CGAL_precondition( r[n-1] != FT(0) ); weight[n-1] = (t[n-2] + t[n-1]) / r[n-1]; // Compute the sum of all weights - denominator of mean value coordinates. mv_denominator = weight[0]; for(int i = 1; i < n; ++i) mv_denominator += weight[i]; // Invert this denominator. CGAL_precondition( mv_denominator != FT(0) ); inverted_mv_denominator = FT(1) / mv_denominator; // Normalize weights and save them as resulting mean value coordinates. for(int i = 0; i < n-1; ++i) { *output = weight[i] * inverted_mv_denominator; ++output; } *output = weight[n-1] * inverted_mv_denominator; ++output; // Return coordinates. return std::optional(output); } // COORDINATES ON UNBOUNDED SIDE. // Compute mean value 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) { // 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 mean value 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) { // 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. // Return the sign of a mean value weight function. // We can have 3 different values: 0 if the weight = 0, -1 if the weight is negative, and +1 if the weight is positive. inline FT sign_of_weight(const FT &A_prev, const FT &A, const FT &B) const { if(A_prev > FT(0) && A > FT(0) && B <= FT(0)) return FT(1); if(A_prev < FT(0) && A < FT(0) && B >= FT(0)) return FT(-1); if(B > FT(0)) return FT(1); if(B < FT(0)) return FT(-1); return FT(0); } // Print some information about mean value coordinates. void print_coordinates_information_2(std::ostream &output_stream) const { output_stream << std::endl << "TYPE OF COORDINATES: " << std::endl << std::endl; output_stream << "The coordinate functions to be computed are mean value coordinates." << std::endl; output_stream << std::endl << "INFORMATION ABOUT COORDINATES: " << std::endl << std::endl; output_stream << "Mean value coordinates are well-defined for an arbitrary simple polygon and can be computed only approximately due to the involved computation of the square root." << 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. Linearity along edges;" << std::endl; output_stream << "5. Smoothness;" << std::endl; output_stream << "6. Similarity invariance;" << std::endl; output_stream << "7. Linear independence;" << std::endl; output_stream << "8. Refinability." << std::endl; output_stream << std::endl; output_stream << "Mean value coordinates satisfy the non-negativity and boundedness between 0 and 1 properties inside the kernel of an arbitrary star-shaped polygon, too." << std::endl; output_stream << std::endl << "REFERENCES: " << std::endl << std::endl; output_stream << "K. Hormann and M. Floater. Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics, 25(4):1424-1441, 2006." << std::endl; output_stream << "M. S. Floater, Wachspress and mean value coordinates, to appear in the Proceedings of the 14th International Conference on Approximation Theory, G. Fasshauer and L. L. Schumaker (eds.)." << std::endl; } }; #endif // CGAL_NO_DEPRECATED_CODE } // namespace Barycentric_coordinates } // namespace CGAL #include #endif // CGAL_BARYCENTRIC_MEAN_VALUE_2_H