Exact computation of the medial axis of a polyhedron

Culver, Tim; Keyser, John; Manocha, Dinesh

doi:10.1016/j.cagd.2003.07.008

Cited by 89 publications

(54 citation statements)

References 16 publications

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“…This instability of the medial axis bears two consequences. First, it makes the medial axis hard to compute exactly because of numerical instabilities; consequently, exact computation of medial axis has only been attempted for a few limited classes of shapes (see for example [7]). Second, the complete medial axis may be less interesting in practice than an approximation of it which carries the same topological type but is more stable under small perturbations of the shape.…”

Section: Introductionmentioning

confidence: 99%

Medial Axis Approximation and Unstable Flow Complex

Giesen

Ramos

Sadri

2008

Int. J. Comput. Geom. Appl.

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The medial axis of a shape is known to carry a lot of information about it. In particular a recent result of Lieutier establishes that every bounded open subset of R n has the same homotopy type as its medial axis. In this paper we provide an algorithm that, given a sufficiently dense but not necessarily uniform sample from the surface of a shape with smooth boundary, computes a core for its medial axis approximation, in form of a piecewise linear cell complex, that captures the topology of the medial axis of the shape. We also provide a natural method to freely augment this core in order to enhance it geometrically all the while maintaining its topological guarantees. The definition of the core and its extension method are based on the steepest ascent flow induced by the distance function to the sample. We also provide a geometric guarantee on the closeness of the core and the actual medial axis.

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Section: Introductionmentioning

confidence: 99%

Medial Axis Approximation and Unstable Flow Complex

Giesen

Ramos

Sadri

2008

Int. J. Comput. Geom. Appl.

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“…For a convex polyhedron, the medial axis and the straight skeleton are the same. For bounds on the size and the complexity of computing the medial axis, see [20,1,13,30]. For definitions and results on straight skeletons, see [4,3,19,21,8].…”

Section: The Straight Skeleton and The Medial Axismentioning

confidence: 99%

Continuously Flattening Polyhedra Using Straight Skeletons

Abel

Demaine

et al. 2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come into contact in the flat folding, which answers an open question in the book Geometric Folding Algorithms. The primary creases in our folding process can be found in quadratic time, though necessarily, creases must roll continuously, and we show that the full crease pattern can be exponential in size. We show that our method solves the fold-and-cut problem for convex polyhedra in any dimension. As an additional application, we show how a limiting form of our algorithm gives a general design technique for flat origami tessellations, for any spiderweb (planar graph with all-positive equilibrium stress).

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“…For a general domain with piecewise smooth boundary, the medial surface and the medial surface transform both consist of several components with different dimensions, see, e.g., [6]. Here we consider only two-dimensional components, which are called sheets.…”

Section: Medial Surface Transformmentioning

confidence: 99%

“…be the homogeneous coordinates of a point in IR 6 . The center of the projection is chosen at (1 : −1 : 0 : 0 : 0 : 0 : 0) ⊤ .…”

Section: Characterizing the Pilt Vectors Of Mos Surfacesmentioning

confidence: 99%

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MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries

Kosinka

Jüttler

Mathematics of Surfaces XII

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Abstract. We consider rational surface patches s (u, v) in the fourdimensional Minkowski space IR 3,1 , which describe parts of the medial surface (or medial axis) transform of spatial domains. The corresponding segments of the domain boundary are then obtained as the envelopes of the associated two-parameter family of spheres. If the Plücker coordinates of the line at infinity of the (two-dimensional) tangent plane of s satisfy a sum-of-squares condition, then the two envelope surfaces are shown to be rational surfaces. We characterize these Plücker coordinates and analyze the case, where the medial surface transform is contained in a hyperplane of the four-dimensional Minkowski space.

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Exact computation of the medial axis of a polyhedron

Cited by 89 publications

References 16 publications

Medial Axis Approximation and Unstable Flow Complex

Medial Axis Approximation and Unstable Flow Complex

Continuously Flattening Polyhedra Using Straight Skeletons

MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries

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