\(C ^{1}\) spline implicitization of planar curves. (English) Zbl 1202.65023

Shalaby, Mohamed; Jüttler, Bert; Schicho, Josef

Winkler, Franz (ed.), Automated deduction in geometry. 4th international workshop, ADG 2002, Hagenberg Castle, Austria, September 4–6, 2002. Revised papers. Berlin: Springer (ISBN 3-540-20927-1/pbk). Lecture Notes in Computer Science 2930. Lecture Notes in Artificial Intelligence, 161-177 (2004).

Summary: We present a new method for constructing a low degree \(C ^{1}\) implicit spline representation of a given parametric planar curve. To ensure the low degree condition, quadratic B-splines are used to approximate the given curve via orthogonal projection in Sobolev spaces. Adaptive knot removal, which is based on spline wavelets, is used to reduce the number of segments. The spline segments are implicitized. After multiplying the implicit spline segments by suitable polynomial factors the resulting bivariate functions are joined along suitable transversal lines. This yields a globally \(C ^{1}\) bivariate function.
For the entire collection see [Zbl 1088.68010].

MSC:

65D17	Computer-aided design (modeling of curves and surfaces)
68U07	Computer science aspects of computer-aided design

Keywords:

implicitization; B-spline; approximation; knot removal