mirror of https://github.com/CGAL/cgal
102 lines
4.4 KiB
Plaintext
102 lines
4.4 KiB
Plaintext
/// \defgroup PkgInterpolation2Ref 2D and Surface Function Interpolation Reference
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/// \defgroup PkgInterpolation2Concepts Concepts
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/// \ingroup PkgInterpolation2Ref
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/// \defgroup PkgInterpolation2Interpolation Interpolation Functions
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/// \ingroup PkgInterpolation2Ref
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/// \defgroup PkgInterpolation2NatNeighbor Natural Neighbor Coordinate Computation
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/// \ingroup PkgInterpolation2Ref
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/// \defgroup PkgInterpolation2SurfaceNeighbor Surface Neighbor and Surface Neighbor Coordinate Computation
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/// \ingroup PkgInterpolation2Ref
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/*!
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\addtogroup PkgInterpolation2Ref
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\cgalPkgDescriptionBegin{2D and Surface Function Interpolation,PkgInterpolation2}
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\cgalPkgPicture{interpolation.png}
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\cgalPkgSummaryBegin
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\cgalPkgAuthor{Julia Flötotto}
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\cgalPkgDesc{This package implements different methods for scattered data interpolation:
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Given measures of a function on a set of discrete data points, the task is to interpolate
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this function on an arbitrary query point. The package further offers functions for natural neighbor interpolation.}
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\cgalPkgManuals{Chapter_2D_and_Surface_Function_Interpolation,PkgInterpolation2Ref}
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\cgalPkgSummaryEnd
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\cgalPkgShortInfoBegin
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\cgalPkgSince{3.1}
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\cgalPkgBib{cgal:f-i}
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\cgalPkgLicense{\ref licensesGPL "GPL"}
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\cgalPkgShortInfoEnd
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\cgalPkgDescriptionEnd
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Scattered data interpolation solves the following problem: given
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measures of a function on a set of discrete data points, the task is
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to interpolate this function on an arbitrary query point.
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If the function is a linear function and given barycentric coordinates
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that allow to express the query point as the convex combination of
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some data points, the function can be exactly interpolated. If the
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function gradients are known, we can exactly interpolate quadratic
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functions given barycentric coordinates. Any further properties of
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these interpolation functions depend on the properties of the
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barycentric coordinates. They are provided in this package under the
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names `linear_interpolation()` and `quadratic_interpolation()`.
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\cgalCRPSection{Natural Neighbor Interpolation}
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Natural neighbor coordinates are defined by Sibson in 1980 and are based on the Voronoi
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diagram of the data points. Interpolation methods based on natural
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neighbor coordinates are particularly interesting because they adapt
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easily to non-uniform and highly anisotropic data.
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This package contains Sibson's \f$ C^1\f$ continuous interpolation method which
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interpolates exactly spherical quadrics (of the form
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\f$ \Phi(\mathbf{x})= a + \mathbf{b}^t \mathbf{x} +\gamma\ \mathbf{x}^t\mathbf{x}\f$)
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and Farin's \f$ C^1\f$ continuous interpolation method based on
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Bernstein-Bézier techniques and interpolating exactly quadratic
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functions - assuming that the function gradient is known.
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In addition, Sibson defines a method to approximate the function
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gradients for data points that are in the interior of the convex hull.
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This method is exact for spherical quadrics.
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This \cgal package provides functions to compute the natural neighbor
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coordinates with respect to a two-dimensional Voronoi diagram (i. e.,
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from the Delaunay triangulation of the data points) and to a
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two-dimensional power diagram for weighted points (i. e., from their
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regular triangulation). Natural neighbor coordinates on closed and
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well-sampled surfaces can also be computed if the normal to the
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surface at the query point is known. The latter coordinates are only
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approximately barycentric, see \cgalCite{bf-lcss-02}.
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Furthermore, it implements Sibson's and Farin's interpolation
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functions as well as Sibson's function gradient fitting method.
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For a more thorough introduction, see the \link Chapter_2D_and_Surface_Function_Interpolation
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User Manual \endlink.
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\cgalClassifedRefPages
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\cgalCRPSection{Concepts}
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- `InterpolationTraits`
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- `GradientFittingTraits`
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\cgalCRPSection{Interpolation Functions}
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- `CGAL::linear_interpolation()`
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- `CGAL::sibson_c1_interpolation()`
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- `CGAL::farin_c1_interpolation()`
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- `CGAL::quadratic_interpolation()`
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- `CGAL::sibson_gradient_fitting()`
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- `CGAL::Interpolation_traits_2<K>`
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- `CGAL::Interpolation_gradient_fitting_traits_2<K>`
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\cgalCRPSection{Natural Neighbor Coordinate Computation}
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- `CGAL::natural_neighbor_coordinates_2()`
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- `CGAL::regular_neighbor_coordinates_2()`
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\cgalCRPSection{Surface Neighbor and Surface Neighbor Coordinate Computation}
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- `CGAL::Voronoi_intersection_2_traits_3<K>`
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- `CGAL::surface_neighbor_coordinates_3()`
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- `CGAL::surface_neighbors_3()`
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*/
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