Multi-degree reduction of Bézier curves using reparameterization

L2L2-norms are often used in the multi-degree reduction problem of Bézier 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)L(t) instead of tt to approximate the function φ(t)φ(t) for a constrained multi-degree reduction of Bézier curves. Numerical examples show that this new reparameterization-based method has a much better approximation effect under Hausdorff distance than those of previous methods.

Research highlights
► The Hausdorff distance is a good measure for the approximation effect between two curves.
► To minimize ∫01‖A(φ(t))−C(t)‖2dt, where A(φ(t)) is the closest point to point C(t), will produce a similar effect as that of the Hausdorff distance.
► To use a piecewise linear function L(t)L(t) instead of tt to approximate the function φ(t)φ(t), leads to a much better approximation effect under the Hausdorff distance.
► When L(t)L(t) is determined, the control points of the approximation Bézier curve can be explicitly computed under the L2L2-norm.