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, Břehová 7, 115 19 Prague 1, Czech Republic;
   email: liska@siduri.fjfi.cvut.cz

² Group T-7, Los Alamos National Laboratory, Los Alamos, NM 87545, USA;
   email: shashkov@lanl.gov
   email: bks@lanl.gov

Received: July 16, 2002
 

Summary
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 points.  Then one blends the two segments continuously between the middle pair of nodes.  The blend is relatively linear in terms of arc-length along each segment.  The union of such successive curve-sections forms a G¹ curve. Wire-frames of such curves define cell edges.  Similar 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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