Generalized offsetting of planar structures using skeletons

Abstract

We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams.

Ramp Approach Parameter Correction Method for 3-axis Web Machining

“…All input segments and input points are called sites. For the sake of descriptional simplicity, it is assumed that no point in R 2 has the same weighted distance to more than three sites of S. Every input site s ∈ S is associated with a so-called offset circle c(s, t) which includes all points in R 2 that are at weighted distance t to s. We find it convenient to regard c(s, t) as a function of either time or distance since at time t every point on c(s, t) is at Euclidean distance t • w(s) from s, i.e., at weighted distance t. As discussed in [12], the offset circle c(ab, t) of a straight-line segment ab is formed by two circular arcs, which are induced by its end-points a and b, and two straight-line segments; see Figure 5c. (Of course, the offset circle of a variably-weighted straight-line segment is no genuine circle but we prefer to use the same term for the offsets of both points and straight-line segments.)…”

“…Thus, every input site is associated with an algebraic bivariate distance function. Held et al [12] argue that the graph of such a distance function is a sub-surface of a right conoid. Lemma 4.1 summarizes the fact that the general-purpose strategy of Agarwal et al [3] can be utilized to compute GWVDs.…”

“…Even though lifting a problem to higher dimensions can be enlightening from a theoretical point of view, experience tells us that it often complicates matters when it comes to developing a practical implementation. And, indeed, the proof-of-concept implementation by Held et al [12] that is based on this approach cannot go beyond tiny input sets. (And this limitation would hardly go away if their code were tuned for speed.)…”

“…Hence, the larger the weight of a point the quicker its fire wavefront spreads. Held et al [12] generalize this concept and introduce the generalized weighted Voronoi diagram (GWVD) of a set S of weighted points and variably-weighted straight-line segments: They assign weights to the endpoints a and b of an input segment ab and then obtain the weight of a point q on ab by a linear interpolation of the weights of a and b. Then d w (p, ab) := min q∈ab d w (p, q), and the corresponding GWVD VD w (S) is defined accordingly.…”

Computation of Implicit Representation of Volumetric Shells with Predefined Thickness

“…Venkatesh applied the marching cube algorithm to generate volume models from triply periodic minimal surfaces [2]. Held et al [3] used generalized Voronoi diagrams to compute offset functions of variable thickness. Pham presented an overview on offsetting methods [4], which was later extended by Maekawa [5].…”