CGAL users discussion list

[碰碰 <>]

--

At 2011-02-10，mturner <> wrote:

>
>Hello,
>
>We are using the CGAL libraries to help analyze 
>surface motions of geological features on the Earth.  
>The geometry of the features define the boundary 
>and interior of specific surface regions.
>
>Because our geographic regions are small, 
>and to keep things simple for now,
>we consider the positions and motions only in 2D, 
>using coordinates of Latitude and Longitude.
>
>Each geologic feature is defined by a basic geometry (point, line, or
>polygon),
>and each geometry contributes to either the boundary of the region, or the
>interior.
>
>Following the example code in Chapter 35 "2D Triangulations", 
>we use the vertices from the feature geometry, to build a series of
>triangulations:
>
>- Delaunay Triangulation    ( CGAL::Delaunay_triangulation_2<...> ),
>- Constrained Triangulation (
>CGAL::Constrained_Delaunay_triangulation_2<...> ),
>- Mesh Triangulation        ( CGAL::Delaunay_mesher_2<...> ).
>
>We have posted images showing an example region,
>with the various feature geometry components,
>and the resulting triangulations here:
>
>http://web.gps.caltech.edu/~mturner/cgal_post/
>
>Figure 1 shows the basic feature data:
>the boundary of the region is defined by intersections of five lines
>(two red lines at the top, purple line on the right, red line at the bottom,
>green line on left).
>The Interior elements are defined by three points, three lines, and two
>polygons inside that region.
>
>Figure 2 shows the Delaunay Triangulation from that data in black,
>including the convex hull triangles formed where the boundary is concave.
>
>Figure 3 shows the Constrained Triangulation in gray.
>
>Figure 4 shows the Mesh Triangulation in light green, 
>with areas outside the total outer boundary un-meshed;
>and areas inside the interior polygons un-meshed.  
>
>The interior holes were generated by adding seed points and calling 
>the constrained mesher's set_seeds(...) function.
>
>Then the Mesh was generated with a call to:
>constrained_mesher->refine_mesh();
>
>This aspect of using CGAL to create triangulations and meshes 
>has been very successful for both test examples and real world data.
>
>
>Now we would like to use the final meshes to perform interpolation.
>
>We have measurements of surface velocity for the basic feature data,
>and we would like to use a refined CGAL mesh to interpolate 
>that velocity data for any given test point inside the region.
>
>Following the example code in Chapter 68 "2D and Surface Function
>Interpolation",
>we have successfully used the interpolation functions on the Delaunay
>Triangulation, 
>using the functions such as:
>CGAL::natural_neighbor_coordinates_2, 
>CGAL::sibson_gradient_fitting, 
>CGAL::Data_access, etc.
>
>Figure 5 shows the measured velocity at the feature vertices with white
>arrows, 
>and the interpolated velocity at the sample test points with black arrows.
>
>Unfortunately, interpolating with the Delaunay Triangulation 
>allows for interpolation outside the region (as shown by the left black
>arrow),
>and inside the holes of the mesh (as shown inside).
>
>We would like to use the Mesh Triangulation for interpolation, 
>not only because it is more refined,
>but also because the mesh is able to exclude, or 'mask out', 
>the interior polygon areas and external concave areas.
>
>However, I cannot find an example using a CGAL Mesh to interpolate.
>
>Is there a way to connect the CGAL interpolation code with the mesh code?
>
>It seems like there must be a way to utilize the refined mesh 
>to perform interpolation, but I could not find a way to make it work...
>
>
>Thanks for any help or guidance you can provide,
>
>Mark Turner
>California Institute of Technology
>Division of Geological and Planetary Sciences, Mail Code 100-23
>Pasadena, CA 91125
>Phone: 626-395-4543
>
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>
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