Lecture 8: Mesh Representations and Geometry Processing (18)
somaniarushi
I'm still struggling a little bit with wrapping my head around a surface being manifold. Is there any shorthand way to be able to tell when a surface is not manifold? A bit hard to visualize cutting a surface with a sphere
SmurphySean
@rushi As far as cutting a surface goes, you can imagine taking the manifold in question and placing it through a sphere at some are along the manifold (the sphere should be sufficiently small so that the cross section is a disk.)
For instance, you can imagine taking a closed bottle adn placing asphere through the side walls. If the sphere is small enough, the cross section with the bottle will be a disk. This makes intuitive sense that the bottle is a 2D manifold since it's surface is two dimensional.
Things that aren't manifolds don't have this global property. If you imagine a cube connected to a plane then this set is not a manifold since on the one hand the cube is diffeomorphic to R^3 but the ajoined plane is diffeomorphic to R^2.
I'm still struggling a little bit with wrapping my head around a surface being manifold. Is there any shorthand way to be able to tell when a surface is not manifold? A bit hard to visualize cutting a surface with a sphere
@rushi As far as cutting a surface goes, you can imagine taking the manifold in question and placing it through a sphere at some are along the manifold (the sphere should be sufficiently small so that the cross section is a disk.)
For instance, you can imagine taking a closed bottle adn placing asphere through the side walls. If the sphere is small enough, the cross section with the bottle will be a disk. This makes intuitive sense that the bottle is a 2D manifold since it's surface is two dimensional.
Things that aren't manifolds don't have this global property. If you imagine a cube connected to a plane then this set is not a manifold since on the one hand the cube is diffeomorphic to R^3 but the ajoined plane is diffeomorphic to R^2.