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

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.

💡 Research Summary

The paper introduces a novel technique for the geometrically stable modeling of high‑degree B‑spline curves by employing an S‑polygon represented in a floating‑point (float) format. The authors argue that traditional approaches, which rely on Bézier control polygons (B‑polygons) and the Mineur‑Farin configuration, suffer from numerical instability and poor endpoint control when applied to high‑order curves. By shifting the control structure to an S‑polygon stored in float precision, the method achieves precise positioning of curve endpoints and accurate alignment of tangent vectors, while preserving a “harmonious, regular” shape of the control polygon.

The core of the method consists of three intertwined ideas. First, the Cox‑de Boor algorithm is used to define the B‑spline, and the vertices of the S‑polygon are placed on a regular polygon (e.g., a dodecagon for a ninth‑degree spline) inscribed in a circle. Because the float format retains high precision, the division of each segment into a fixed ratio remains numerically stable, and the differential characteristics of the curve (first and second divided differences) can be computed uniformly across the whole spline.

Second, the Mineur‑Farin configuration—originally devised for Bézier curves—is transplanted to the S‑polygon. In this configuration the lengths of the polygon legs vary monotonically (or stay constant for constant‑curvature curves) while the angles between successive legs remain fixed. The authors term a polygon that satisfies these constraints a “harmonious, regular form.” Such a form guarantees that the discrete curvature (first‑order divided differences) and the discrete curvature of curvature (second‑order divided differences) vary monotonically, mirroring the continuous curvature and torsion of the target curve up to the second level. This property is crucial for preserving visual quality, especially for class‑A curves where monotonic curvature is a design requirement.

Third, a detailed seven‑step procedure is presented for correcting endpoint position and tangent direction errors. After constructing an initial S‑polygon, the algorithm computes a position‑difference vector and a direction‑difference angle between the actual and desired endpoint data. It then translates the terminal vertices by the position‑difference vector and rotates them by the direction‑difference angle, iterating the process until the S‑polygon attains the harmonious, regular shape. Because all operations are performed in float precision and involve only ratio‑based divisions, the method is robust against rounding errors that would otherwise corrupt a closed B‑polygon.

The authors validate the approach through several experiments. In a circular‑arc approximation test, a ninth‑degree B‑spline built from an S‑polygon achieves an error of 5.9 × 10⁻⁸, whereas the corresponding Bézier curve (also ninth degree) exhibits an error of 3.2 × 10⁻². Moreover, rounding a single vertex of the closed S‑polygon leads to a dramatic increase in error for the Bézier case but only a modest change for the S‑polygon case, demonstrating superior stability. Additional tests on curves with monotonic curvature change show that the Bézier curve’s curvature becomes non‑monotonic, while the S‑polygon‑based B‑spline retains monotonic curvature throughout.

Beyond basic curvature, the paper introduces the concept of n‑level curvature and torsion. The first level corresponds to the ordinary curvature and torsion functions; the second level is the curvature of those functions (i.e., curvature of curvature and curvature of torsion). By ensuring shape equivalence up to the second level, the method guarantees that the number and order of inflection points and extrema are preserved between the original analytic curve and its spline approximation. An example is provided where an eighth‑degree B‑spline approximates a conical spiral; the curvature and curvature‑of‑torsion graphs of the spline match those of the analytic spiral up to the second level.

Finally, the technique has been implemented as a plug‑in for the FairCurveModeler system, allowing engineers to generate, edit, and evaluate high‑quality B‑splines directly within mainstream CAD environments. The plug‑in offers real‑time visualization of the S‑polygon, automatic endpoint correction, and diagnostics of curvature monotonicity, making the method practical for industrial design, automotive surface modeling, and other applications requiring class‑A curve quality.

In summary, the paper demonstrates that using a float‑based S‑polygon with a Mineur‑Farin harmonious configuration provides a numerically stable, endpoint‑accurate, and curvature‑preserving framework for high‑degree B‑spline modeling. The extensive comparative tests, theoretical justification, and CAD‑integrated implementation collectively establish the approach as a significant advancement over traditional Bézier‑based techniques, with promising extensions to log‑aesthetic curves, superspirals, and other specialized curve families.