Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications.@en

Bézier and B-spline curves are foundational tools for curve representation in computer graphics and computer-aided geometric design, with their intersection computation presenting a fundamental challenge in geometric modeling. This study introduces an innovative algorithm that quickly and effectively resolves intersections between Bézier and B-spline curves. The number of intersections between the two input curves within a specified region is initially determined by applying the resultant of a polynomial system and Sturm’s theorem. Subsequently, the potential region of the intersection is established through the utilization of the pseudo-curvature-based subdivision scheme and the bounding box detection technique. The projected Gauss-Newton method is ultimately employed to efficiently converge to the intersection. The robustness and efficiency of the proposed algorithm are demonstrated through numerical experiments, demonstrating a speedup of 3 to 150 times over traditional methods@en

NURBS method is the standard mathematical method for describing the shapes of curves/surfaces/volumes, and it is extensively used in computer-aided design, computer-aided manufacturing, and computer graphics. The injectivity of NURBS volumes means that they do not have self-intersections. Since the injectivity of parameterizations depend on the signs of their Jacobian functions, and the Jacobian of a NURBS volume is determined by the determinant of its tangent vectors in three directions, we first propose a method to compute the bounding vectors of the tangent cones of NURBS volume in this paper. Then the sufficient condition for the injectivity of NURBS volume is proposed. A checking algorithm is also presented. Some examples are given to verify the effectiveness of the algorithm.

@en

Toric surface patches are a class of multi-sided surface patches that can represent multi-sided domains without mesh degeneration. In this paper, we propose an improved subdivision algorithm for toric surface patches, which subdivides an N-sided toric surface patch into N rational tensor product Bézier surface patches. By the proposed subdivision algorithm, a C ^k-continuous spline surface composed of piecewise toric surface patches is subdivided into a set of rational tensor product Bézier surface patches with G ^k-continuity. Additionally, after subdivision, toric surface patches are compatible with CAD systems. Combining the subdivision algorithm with the classical knot insertion algorithm of non-uniform rational B-splines, we develop a novel h-refinement scheme for isogeometric analysis with planar toric parameterizations. Several numerical examples are given to demonstrate the effectiveness and numerical stability of the presented method.

@en