What are some cases in which we might want to render something that is not a manifold?
ellenluo
@keirp More complex 3D meshes may not be manifold. For example, meshes with holes in them might be necessary if you want to render something that is infinitesimally thin.
cornrow-kenny
Here's a stack overflow post with some pictures/examples of modeling non-manifold geometry https://blender.stackexchange.com/questions/7910/what-is-non-manifold-geometry
CptTeddy
In a two-dimensional setting, the Euler's polyhedron formula can be reduced to a similar statement with constraints regarding whether the shape is convex.
CptTeddy
The three-dimensional Euler's polyhedron formula is amazing. Here's a thorough, if not a rigorous proof of the statement: http://www.ams.org/publicoutreach/feature-column/fcarc-eulers-formula. Intuitively, imagine that all faces of a topological manifold are defined by the number of vertices surrounding it; while trimming out redundant edges, the number of vertices becomes one more than the number of edges where it simply becomes a 'tree' structure; now the face is only divided up by a tree-shaped edge, which gives one 'pseudo' face, and thus leads to 1 + [(v+1) - v] = 2 is the number of faces plus the number of vertices minus the number of edges.
Can we have a formal definition of a face?
"In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object."
Source: https://en.wikipedia.org/wiki/Face_(geometry)
What are some cases in which we might want to render something that is not a manifold?
@keirp More complex 3D meshes may not be manifold. For example, meshes with holes in them might be necessary if you want to render something that is infinitesimally thin.
Here's a stack overflow post with some pictures/examples of modeling non-manifold geometry https://blender.stackexchange.com/questions/7910/what-is-non-manifold-geometry
In a two-dimensional setting, the Euler's polyhedron formula can be reduced to a similar statement with constraints regarding whether the shape is convex.
The three-dimensional Euler's polyhedron formula is amazing. Here's a thorough, if not a rigorous proof of the statement: http://www.ams.org/publicoutreach/feature-column/fcarc-eulers-formula. Intuitively, imagine that all faces of a topological manifold are defined by the number of vertices surrounding it; while trimming out redundant edges, the number of vertices becomes one more than the number of edges where it simply becomes a 'tree' structure; now the face is only divided up by a tree-shaped edge, which gives one 'pseudo' face, and thus leads to 1 + [(v+1) - v] = 2 is the number of faces plus the number of vertices minus the number of edges.