Comparing offset curve approximation methods

Abstract

Offset curves have diverse engineering applications, which have consequently motivated extensive research concerning various offset techniques. Offset research in the early 1980s focused on approximation techniques to solve immediate application problems in practice. This trend continued until 1988, when Hoschek [1, 2] applied non-linear optimization techniques to the offset approximation problem. Since then, it has become quite difficult to improve the state-of-the-art of offset approximation. Offset research in the 1990s has been more theoretical. The foundational work of Farouki and Neff [3] clarified the fundamental difficulty of exact offset computation. Farouki and Sakkalis [4] suggested the Pythagorean Hodograph curves which allow simple rational representation of their exact offset curves.

Computational Methods for Geometric Processing. Applications to Industry

“…In general, we deal with offset curves and surfaces, which are also curves and surfaces at a constant distance d from a given initial curve or surface. Several methods for the computation of the offsets for curves are compared in [15]. As pointed out in [59], offsetting general surfaces is more complicated, and an offset surface is often approximated [21], although this approximation becomes inaccurate near its selfintersecting area [2,59].…”

New Approximation Methods for Planar Offset and Convolution Curves

“…Elber et al [10] compared conventional offset approximation methods from quantitative, as well as qualitative, point of view. The difference function ψ(t) of (5) was used as a criterion to measure the performance of each method.…”

“…Elber et al [10] reviewed the current state-of-the-art of offset approximation methods and classified conventional approaches according to the following two categories: (i) control polygon based approach [5,7,8,31], and (ii) interpolation based approach [19,20,23,29]. Control polygon based methods are usually much simpler.…”

“…For example, the offset of a cubic Bézier curve has an algebraic degree of 10 [12]. Consequently, these fundamental limitations have led offset research to develop various offset approximation techniques [10,22,25]. Based on quadratic curve approximation and tangent field matching, we present new methods that approximate offset and convolution curves with polynomial or rational curves.…”

On rational Minkowski Pythagorean hodograph curves

“…Details about offset curves trimming can be found e.g. in Elber et al (1998); Maekawa (1999); Pekerman et al (2008); Seong et al (2006). Hence, an alternative approach to the problem based on the medial axis transform of a planar domain plays a crucial role.…”