Straight skeletons for general polygonal figures in the plane

Abstract

{ oai oh, auren}%igi, tu-graz, ar at I n t r o d u c t i o nA planar straight line graph, G, on n points in the Euclidean plane is a set of noncrossing line segments spanned by these points. A skeleton of G is a partition of the plane into faces that reflect the shape of G in an appropriate manner. The well-known and widely used examples of skeletons are the medial axis of a simple polygon or, more generally, the (closest-point) Voronoi diagram of G. Skeletons have numerous applications, for example in biology, geography, pattern recognition, robotics, and computer graphics; see e.g. [Ki, L, Y] for a short history.The Voronoi diagram of G consists of all points in the plane which have more than one closest object in G. Typically, it contains curved arcs in the neighborhood of the vertices of G. This is considered a disadvantage in the computer representation and construction, and sometimes also in the application, of this type of skeleton.There have been several attempts to linearize and simplify Voronoi diagrams of planar straight line graphs, mainly for the sake of efficient point location and motion planning [CD, KM, MKS]. The compact Voronoi diagram for convex polygons in [MKS] is particularly suited to these applications as its complexity is linear in the number of polygons rather than in the number of edges. However, its faces do not reflect much of the shape of the polygons which might restrict its application when being used as a skeleton for polygonal figures.In the present paper, a novel type of skeleton, the straight skeleton of G, is introduced and discussed. Its arcs are pieces of angular bisectors of the edges of G. Its combinatorial complexity is in general is even less than the complexity of the Voronoi diagram of G. Still, G can be reconstructed easily from its straight skeleton. This fact is considered important in certain applications of skeletons [PR]. Beside its use as a skeleton, we describe two applications that come from a spatial interpretation of straight skeletons. One concerns the question of constructing a roof above a general polygonal outline of ground walls. The other application is the reconstruction of a geographical terrain from a given map that delineates coasts, lakes, and rivers.

Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

“…We generalize the proof in [3] to one more dimension. Take some line in the hyperplane W 0 and normal to the supporting plane H i ⊃ f i , and consider the restriction ψ| of ψ to .…”

“…A vertex v is pointed if there exists an open (geometric) disk whose intersection with Q is exactly v. A saddle vertex is incident to edges that positively span 3-space. 3 These two types are exclusive, but not exhaustive among the non-touching vertices. If not already present in Q's boundary, touching vertices and saddle vertices will be created generically in the polytope offsetting process, including such having coplanar facets, or some facet with a reflex angle.…”

“…Negative edge weights [13] may occur now, however, because f can have edges that shift toward its exterior. A simple and usually fast implementation is by triangulating all the facets of Q beforehand, and detecting facet changes by the collapse of triangles [3]. This works in O(h) overall time for all facets per increase of , when Q has h facets and each of them is of constant size.…”

“…This structure is not defined via distances but rather in a procedural way, by means of a mitered boundary offsetting process which shrinks the object in a self-parallel way till it vanishes. In two dimensions, the shrinking process is combinatorially trivial, and the geometry of the straight skeleton is well understood [3,4]. It is a unique graph whose leaf nodes are polygon vertices like with the medial axis, but whose arcs stem from angle bisectors and thus are straight-line segments.…”

Route Planning in 3d Environment with a Multivariant Model

“…Рельеф поверхности функции на графическом образе (рисунок 9в) формирует скелет общего контура описания сцены, широко применимый при анализе его формы и ко многим другим прикладным задачам [28,29].…”