Techniques for modeling a high-quality B-spline curves by S-polygons in a float format

TL;DR

This paper introduces a stable method for modeling high-quality B-spline curves using S-polygons in float format, improving accuracy and control in curve approximation and editing.

Contribution

It presents a novel technique for geometrically stable B-spline modeling with S-polygons, enhancing curve quality and control compared to traditional methods.

Findings

Superlative efficacy of Mineur-Farin configuration for curvature curves

High-quality curve approximation with shape proximity

Effective control of curve end points and tangents

Abstract

This article proposes a technique for the geometrically stable modeling of high-degree B-spline curves based on S-polygon in a float format, which will allow the accurate positioning of the end points of curves and the direction of the tangent vectors. The method of shape approximation is described with the purpose of providing geometrical proximity between the original and approximating curve. The content of the notion of a harmonious, regular form of B-spline curve's S-polygon in a float format is revealed as a factor in achieving a high-quality of fit for the generated curve. The expediency of the shape modeling method based on S-polygon in a float format at the end portions of the curve for quality control of curve modeling and editing is substantiated. The results of a comparative test are presented, demonstrating the superlative efficacy of using the Mineur-Farin configuration for constructing constant and monotone curvature curves based on an S-polygon in a float format. The findings presented in this article confirm that it is preferable to employ the principle of "constructing a control polygon of a harmonious form (or the Mineur-Farin configuration) of a parametric polynomial" to a B-spline curve's S-polygon in a float format, and not to a B-polygon of the Bezier curve. Recommendations are given for prospective studies in the field of applying the technique of constructing a high-quality B-spline curves to the approximation of log-aesthetic curves, Ziatdinov's superspirals, etc. The authors of the article developed a technique for constructing smooth connections of B-spline curves with ensuring a high order of smoothness of the composite curve. The proposed techniques are implemented in the FairCurveModeler program as a plug-in to engineering CAD systems.