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Locally Exact Smooth Reconstruction of Lines, Circles, Planes, Spheres, Cylinders and Cones by Blending Successive Circular Interpolants

Author(s): Richard Liska | Mikhail Shashkov | Blair Swartz

Journal: Selçuk Journal of Applied Mathematics
ISSN 1302-7980

Volume: 3;
Issue: 2;
Start page: 81;
Date: 2002;
Original page

Keywords: Smooth reconstruction | line | circle | plane | sphere | cylinder | cone | interpolation.

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.
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