Euler operator
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In mathematics, Euler operators are a small set of functions to create polygon meshes. They are closed and sufficient on the set of meshes, and they are invertible.
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[edit] Purpose
A "polygon mesh" can be thought of as a graph, with vertices, and with edges that connect these vertices. In addition to a graph, a mesh has also faces: Let the graph be drawn ("embedded") in a two-dimensional plane, in such a way that the edges do not cross (which is possible only if the graph is a planar graph). Then the contiguous 2D regions on either side of each edge are the faces of the mesh.
The Euler operators are functions to manipulate meshes. They are very straightforward: Create a new vertex (in some face), connect vertices, split a face by inserting a diagonal, subdivide an edge by inserting a vertex. It is immediately clear that these operations are invertible.
Further Euler operators exist to create higher-genus shapes, for instance to connect the ends of a bent tube to create a torus.
[edit] Properties
Euler operators are topological operators: They modify only the incidence relationship, i.e., which face is bounded by which face, which vertex is connected to which other vertex, and so on. They are not concerned with the geometric properties: The length of an edge, the position of a vertex, and whether a face is curved or planar, are just geometric "attributes".
Note: In topology, objects can arbitrarily deform. So a valid mesh can, e.g., collapse to a single point if all of its vertices happen to be at the same position in space.
[edit] See also
[edit] References
- Sven Havemann, Generative Mesh Modeling, PhD thesis, Braunschweig University, Germany, 2005.
- Martti Mäntylä, An Introduction to Solid Modeling, Computer Science Press, Rockville MD, 1988. ISBN 0-88175-108-1.
