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Journal for Geometry and Graphics 11 (2007), No. 2, 199--208
Copyright Heldermann Verlag 2007

An Angle Criterion for Conical Mesh Vertices

Wenping Wang
Dept. of Computer Science, University of Hong Kong

Johannes Wallner
Institut für Geometrie, Technische Universität, Kopernikusgasse 24, 8010 Graz, Austria

Yang Liu
Dept. of Computer Science, University of Hong Kong

A vertex in a mesh surface with planar faces may have the property that offsetting all the face planes incident with the vertex by a constant distance leads to planes which intersect again in a common point. This is equivalent to the property that the planes, consistently oriented via the connectivity of the mesh, are tangent to an oriented cone of revolution. We show that for vertices of valence 4, this conical property is characterized in terms of the interior angles of the faces adjacent to the vertex: The two sums of opposite angles are equal. For a convex vertex this angle criterion follows directly from known results in spherical geometry concerning convex spherical quadrilaterals. For other types of vertices, however, the occurrence of non-convex spherical quadrilaterals makes it necessary to exhaustively enumerate and study a number of cases. The present short note resolves this combinatorial difficulty and proves that all conical vertices are characterized by this same angle criterion. This result is especially relevant in the context of modeling with conical meshes.

Keywords: Spherical geometry, conical meshes.

MSC: 51M04; 51N20, 52B70

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