mirror of https://github.com/CGAL/cgal
removed all notes about flattening from both user and ref manuals
This commit is contained in:
Dmitry Anisimov
parent
c4ab20e8b2
commit
f63e34fca7
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@ -96,7 +96,7 @@ being the power parameter.
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Here, `q` is a query point and the points `p0`, `p1`, and `p2` are ordered.
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Flattening)
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Coplanarity)
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while alternative formulations are explained in \ref Weights_Implementation.
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\cgalFigureBegin{three_point_family_weight, three_point_family.svg}
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@ -133,7 +133,7 @@ with notations shown in the figure below.
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Here, `q` is a query point and the points `p0`, `p1`, and `p2` are ordered.
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Flattening)
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Coplanarity)
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while alternative formulations are explained in \ref Weights_Implementation.
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\cgalFigureBegin{wachspress_weight, wachspress.svg}
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@ -198,7 +198,7 @@ The \f$\pm\f$ sign is a sign of the weight that depends on the configuration.
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Here, `q` is a query point and the points `p0`, `p1`, and `p2` are ordered.
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Flattening)
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Coplanarity)
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while alternative formulations are explained in \ref Weights_Implementation.
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\cgalFigureBegin{mean_value_weight, mean_value.svg}
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@ -262,7 +262,7 @@ with notations shown in the figure below.
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Here, `q` is a query point and the points `p0`, `p1`, and `p2` are ordered.
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Flattening)
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This weight supports only planar configurations (see more in section about \ref Weights_Implementation_Coplanarity)
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while alternative formulations are explained in \ref Weights_Implementation.
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\cgalFigureBegin{discrete_harmonic_weight, discrete_harmonic.svg}
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@ -93,40 +93,25 @@ The parameter `traits` with a traits class can be omitted for all functions and
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if it can be deduced from the input point type.
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Several weights in this package have different implementations. One reason to have it is
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explained in section about \ref Weights_Implementation_Flattening. Another reason is that
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explained in section about \ref Weights_Implementation_Coplanarity. Another reason is that
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in different communities the same weights are named and computed differently. If one searches
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for these weights, one needs to know all their alternative names, which is tiresome. We provide
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the most common names and implementations of these weights.
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\subsection Weights_Implementation_Flattening Coplanarity
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\cgalFigureBegin{flattening, flattening.svg}
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The non-planar configuration (top) is flattened to the planar configuration (bottom).
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\cgalFigureEnd
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\subsection Weights_Implementation_Coplanarity Coplanarity
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When computing weights for a query point \f$q\f$ with respect to its neighbors
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\f$p_0\f$, \f$p_1\f$, and \f$p_2\f$, the local configuration is a quadrilateral
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[\f$p_0\f$, \f$p_1\f$, \f$p_2\f$, \f$q\f$] or two connected triangles [\f$q\f$, \f$p_0\f$, \f$p_1\f$]
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and [\f$q\f$, \f$p_1\f$, \f$p_2\f$]. When working in 3D, these triangles are not
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necessarily coplanar, in other words, they do not belong to the same common plane.
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When they are not coplanar, they can be made coplanar through the process called *flattening* (see the Figure above),
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however the latter introduces a distortion because the weights are computed with respect to the
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flattened configuration rather than to the original non-flat configuration.
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Certain weights in this package support only coplanar configurations, while other weights support both.
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The weights, which support non-coplanar configurations, provide the corresponding overloads with 3D points,
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while other weights accept only 2D point types. For example, \ref PkgWeightsRefCotangentWeights "cotangent weights"
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support both flat and non-flat configurations, while \ref PkgWeightsRefDiscreteHarmonicWeights "discrete harmonic weights"
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support only flat configurations.
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If you still want to compute the coplanar weights, such as discrete harmonic weights, in 3D,
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you can either provide your own way to flatten the 3D configuration or you can use one of the projection
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traits available in CGAL as the last parameter to the weight function (see \ref Weights_Examples_ProjectionTraits
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for example, where the common plane is a horizontal plane). If the two triangles are not exactly coplanar,
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this projection traits will introduce an error, but this error is proportional to how much these triangles
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differ from the common plane. If the non-coplanarity difference between them is due to noise or inexact computations,
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it won't affect much the final computation result.
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support both coplanar and non-coplanar configurations, while \ref PkgWeightsRefDiscreteHarmonicWeights "discrete harmonic weights"
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support only coplanar configurations.
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\subsection Weights_Implementation_Boundaries Boundaries
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@ -179,22 +164,10 @@ The used type of barycentric weights is `Wachspress_weights_2`.
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\cgalExample{Weights/coordinates_multiple_queries.cpp}
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\subsection Weights_Examples_ProjectionTraits Computing 2D Weights in 3D
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If you have a 2D polygon in 3D plane that is not an XY plane, you can still compute
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the 2D weights, however you need to provide a special projection traits class.
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The common plane that is used in this example is projectable to the XY plane. We first
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compute `Mean_value_weights_2` for a 3D polygon in this plane. We then also show how to use
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the projection traits to compute the \ref PkgWeightsRefWachspressWeights "2D Wachspress weight"
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for 3D points, which are not strictly coplanar.
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\cgalExample{Weights/projection_traits.cpp}
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\subsection Weights_Examples_CustomTraits Weights with Custom Traits
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As you could see from the previous example, it is possible to provide your own traits
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class with basic geometric objects, constructions, and predicates required by the algorithm.
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As you could see from the reference manual, it is possible to provide your own traits class
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with basic geometric objects, constructions, and predicates to the weight functions.
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All weights in this \cgal component are models of the `AnalyticWeightTraits_2` and `AnalyticWeightTraits_3`
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concepts. However, many weights do not require all objects from these concepts. This example shows that
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the inverse distance weight, for instance, requires only the squared distance object which is specified
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@ -3,7 +3,6 @@
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\example Weights/coordinates_one_query.cpp
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\example Weights/coordinates_multiple_queries.cpp
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\example Weights/weighted_laplacian.cpp
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\example Weights/projection_traits.cpp
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\example Weights/custom_traits.cpp
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\example Weights/convergence.cpp
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*/
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@ -432,6 +432,61 @@ namespace internal {
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return Point_2(x, y);
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}
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// Flattening.
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// \cgalFigureBegin{flattening, flattening.svg}
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// The non-planar configuration (top) is flattened to the planar configuration (bottom).
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// \cgalFigureEnd
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// When computing weights for a query point \f$q\f$ with respect to its neighbors
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// \f$p_0\f$, \f$p_1\f$, and \f$p_2\f$, the local configuration is a quadrilateral
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// [\f$p_0\f$, \f$p_1\f$, \f$p_2\f$, \f$q\f$] or two connected triangles [\f$q\f$, \f$p_0\f$, \f$p_1\f$]
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// and [\f$q\f$, \f$p_1\f$, \f$p_2\f$]. When working in 3D, these triangles are not
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// necessarily coplanar, in other words, they do not belong to the same common plane.
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// When they are not coplanar, they can be made coplanar through the process called *flattening* (see the Figure above),
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// however the latter introduces a distortion because the weights are computed with respect to the
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// flattened configuration rather than to the original non-flat configuration.
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// \subsection Weights_Examples_ProjectionTraits Computing 2D Weights in 3D
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// If you have a 2D polygon in 3D plane that is not an XY plane, you can still compute
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// the 2D weights, however you need to provide a special projection traits class.
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// The common plane that is used in this example is projectable to the XY plane. We first
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// compute `Mean_value_weights_2` for a 3D polygon in this plane. We then also show how to use
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// the projection traits to compute the \ref PkgWeightsRefWachspressWeights "2D Wachspress weight"
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// for 3D points, which are not strictly coplanar.
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// \cgalExample{Weights/projection_traits.cpp}
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// Example of flattening:
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// 3D configuration.
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// const Point_3 p0(0, 1, 1);
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// const Point_3 p1(2, 0, 1);
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// const Point_3 p2(7, 1, 1);
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// const Point_3 q0(3, 1, 1);
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// Choose a type of the weight:
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// e.g. 0 - Wachspress (WP) weight.
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// const FT wp = FT(0);
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// Compute WP weights for q1, which is not on the plane [p0, p1, p2].
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// Point_3 q1(3, 1, 2);
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// std::cout << "3D wachspress (WP, q1): ";
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// std::cout << CGAL::Weights::three_point_family_weight(p0, p1, p2, q1, wp) << std::endl;
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// Converge q1 towards q0 that is we flatten the configuration.
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// We also compare the result with the authalic weight.
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// std::cout << "Converge q1 to q0: " << std::endl;
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// for (FT x = FT(0); x <= FT(1); x += step) {
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// std::cout << "3D wachspress/authalic: ";
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// q1 = Point_3(3, 1, FT(2) - x);
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// std::cout << CGAL::Weights::three_point_family_weight(p0, p1, p2, q1, wp) << "/";
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// std::cout << CGAL::Weights::authalic_weight(p0, p1, p2, q1) << std::endl;
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// }
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// Flattens an arbitrary quad into a planar quad.
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template<typename GeomTraits>
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void flatten(
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@ -130,35 +130,6 @@ namespace Weights {
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namespace internal {
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// Example of flattening:
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// 3D configuration.
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// const Point_3 p0(0, 1, 1);
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// const Point_3 p1(2, 0, 1);
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// const Point_3 p2(7, 1, 1);
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// const Point_3 q0(3, 1, 1);
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// Choose a type of the weight:
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// e.g. 0 - Wachspress (WP) weight.
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// const FT wp = FT(0);
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// Compute WP weights for q1, which is not on the plane [p0, p1, p2].
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// Point_3 q1(3, 1, 2);
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// std::cout << "3D wachspress (WP, q1): ";
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// std::cout << CGAL::Weights::three_point_family_weight(p0, p1, p2, q1, wp) << std::endl;
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// Converge q1 towards q0 that is we flatten the configuration.
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// We also compare the result with the authalic weight.
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// std::cout << "Converge q1 to q0: " << std::endl;
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// for (FT x = FT(0); x <= FT(1); x += step) {
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// std::cout << "3D wachspress/authalic: ";
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// q1 = Point_3(3, 1, FT(2) - x);
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// std::cout << CGAL::Weights::three_point_family_weight(p0, p1, p2, q1, wp) << "/";
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// std::cout << CGAL::Weights::authalic_weight(p0, p1, p2, q1) << std::endl;
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// }
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template<typename GeomTraits>
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typename GeomTraits::FT three_point_family_weight(
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const typename GeomTraits::Point_3& t,
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