Selšuk Journal of Applied Mathematics

 Selšuk Journal of
  Applied Mathematics

  Summer-Autumn, 2002
  Volume  3
  Number 2

Research Center of 
  Applied Mathematics

 SJAM Summer-Autumn 2002, Volume 3 - Number 2

Locally exact smooth reconstruction of lines, circles, planes, spheres, cylinders and cones by blending successive circular interpolants

Richard Liska1, Mikhail Shashkov2 and  Blair Swartz2

1 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in
  Prague, Břehovß 7, 115 19 Prague 1, Czech Republic;

2 Group T-7, Los Alamos National Laboratory, Los Alamos, NM 87545, USA;

Received: July 16, 2002


G1-smooth curves and surfaces are developed to span a given logically cuboid distribution of nodes. Given appropriate data, they locally reconstruct the curves and surfaces of spherical  or cylindrical coordinates. Thus, if a set of nodes consists of a contiguous subset of a tensor product grid of points associated with a (possibly non-uniform) set of coordinate values of some rectangular, cylindrical, or spherical coordinate system; then the appropriate coordinate curves (linear or circular segments) and coordinate surfaces (segments of planes, cylinders, spheres and cones) that interpolate the subset are reconstructed exactly. The underlying construction uses four successive nodes to define a curve spanning the middle pair as follows: One interpolates each of the two successive triples of nodes with the segment of a circle or straight line going through these three pointsThen one blends the two segments continuously between the middle pair of nodesThe blend is relatively linear in terms of arc-length along each segmentThe union of such successive curve-sections forms a G1 curve. Wire-frames of such curves define cell edgesSimilar intermediate curvilinear interpolation of the wires defines cell faces, and their union defines G1 coordinate-like surfaces. The surface generated depends on the direction one interpolates the wires. If the nodes are a tensor product grid associated with a sufficiently smooth reference coordinate system, then the cell edges (and probably also the cell faces) are third-order accurate.


Key words
smooth reconstruction, line, circle, plane, sphere, cylinder, cone, interpolation

2000 Mathematics Subject Classification : 65D15

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