DTU Discrete Differential Geometry Miniworkshop 16 July 2009
Faceting of curved surfaces using the curvature coordinate system
Henrik Almegaard (DTU Department of Civil Engineering)
In many situations, a curved surface has to be approximated by a facetted surface, i.e., as a network with planar meshes. Most often this is done by triangulation of the surface. Points are chosen on the surface and the points are connected by straight lines so that these lines make a network with triangular facets.
But faceting a curved surface can also be done using the plane as the basic geometrical element instead of the point. One way of doing this is by tangent faceting. Tangent points are chosen on the surface and the tangent planes at these points are connected along lines of intersection so that these lines make a network with planar meshes and so that no normal to the curved surface intersect more than one facet. The result is a faceted surface with facets and - unless special effort are made - three-way vertices.
On facetted surfaces the Gaussian curvature is concentrated at the vertices. If the curvature coordinate system [1] is used to determine triangulated as well as tangent-faceted surfaces, both the topology of the system and the sign of the Gaussian curvature of the facetted surface can be decided by the designer.
This paper describes the faceting processes and the geometric rules that determine the topology and the curvature of the facetted surface when given form in the curvature coordinate system.
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