L"2-norms are often used in the multi-degree reduction problem of Bezier curves or surfaces. Conventional methods on curve cases are to minimize @!"0^1@?A(t)-C(t)@?^2dt, where C(t) and A(t) are the given curve and the approximation curve, respectively. A much better solution is to minimize @!"0^1@?A(@f(t))-C(t)@?^2dt, where A(@f(t)) is the closest point to point C(t), that produces a similar effect as that of the Hausdorff distance. This paper uses a piecewise linear function L(t) instead of t to approximate the function @f(t) for a constrained multi-degree reduction of Bezier curves. Numerical examples show that this new reparameterization-based method has a much better approximation effect under Hausdorff distance than those of previous methods.

Multi-degree reduction of Bézier curves using reparameterization

Hybrid chameleon swarm algorithm with multi-strategy: A case study of degree reduction for disk Wang-Ball curves

Let G be the group M(n) generated by all orthogonal transformations and translations of the n-dimensional Euclidean space or G be the subgroup of M(n) generated by rotations and translations of \(R^{n}\). This paper presents the conditions of G-equivalence for two Bezier curves in \(R^{n}\) of degree m, where \(m\ge 1\). Moreover, two minimal complete systems of control G-invariants of a Bezier curve are obtained. The definition of a non-directional Bezier curve is introduced and a complete system of control G-invariants of this curve is obtained. Correlations between elements of the second minimal complete system of control M(n)-invariants are investigated.

Equivalence Conditions of Two Bézier Curves in the Euclidean Geometry

We present a novel approach for dealing with optimal approximate merging of two adjacent Bezier curves with G2-continuity. Instead of moving the control points, we minimize the distance between the original curves and the merged curve by taking advantage of matrix representation of Bezier curve’s discrete structure, where the approximation error is measured by L2-norm. We use geometric information about the curves to generate the merged curve, and the approximation error is smaller. We can obtain control points of the merged curve regardless of the degrees of the two original curves. We also discuss the merged curve with point constraints. Numerical examples are provided to demonstrate the effectiveness of our algorithms.

Optimal approximate merging of a pair of Bézier curves with G2-continuity

Disk B-spline curve provides an alternative shape modeling tool to standard control point based modeling techniques by extending control points to control disks, resulting in an easy and straightforward way of modeling regions or shapes with adjustable thickness. Due to this feature, disk B-spline curves are frequently applied in artistic shape modeling applications, such as brushstroke representation, calligraphy, and animation, but they are also suitable for engineering purposes. However, the boundary curves of the described shapes tend to have unintentional self-intersections or cusps, negatively affecting the usability and artistic value of the resulting shape and texture. In this paper, we introduce an iterative algorithm to give a solution to this problematic issue with the help of an approximating circle skinning method, while preserving the advantageous properties of the disk B-spline curves. Our method also results in smoother texturing of the shape around the more sharply curved sections. 

A skinning technique for modeling artistic disk B-spline shapes

Degree reduction of disk Bézier curves

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Degree reduction of disk Bézier curves