Multi-degree reduction of Bézier curves using reparameterization

L²-norms are often used in the multi-degree reduction problem of Bzier curves or surfaces. Conventional methods on curve cases are to minimize ∫₀¹∥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 ∫₀¹∥A(φ(t))- C(t)∥²dt, where A(φ(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 φ(t) for a constrained multi-degree reduction of Bzier curves. Numerical examples show that this new reparameterization-based method has a much better approximation effect under Hausdorff distance than those of previous methods. © 2010 Elsevier Ltd. All rights reserved.