When we are talking about manifold, we were saying that we want to zoom in very closely to see whether we can have a plane. In this case, does that mean if we are given any objects that have any sharp corner?it is not a manifold?
motoole2
@chenj This is a good question. The local geometry associated with a point on a sharp corner cannot be approximated with a plane. And yet, it may still be a manifold, because the geometry may still satisfies the definitions on this slide. To be a bit more precise with our definition, a manifold requires that local geometry can be flattened out into a plane; a corner still satisfies this definition.
TheNumbat
Why can't the self-intersecting shape be flattened into a plane?
Are we assuming the surface is "attached" to itself along the intersection? If not, then it could just be represented by a halfedge mesh where some faces happen to intersect.
motoole2
@TheNumbat Referencing the top-right figure, the assumption here is that the surface is attached to itself, such that the edge at this intersection is shared for four faces; in this case, it is not a manifold But you are also correct in your assumption that if the object were not attached to itself, then it is a manifold.
Similarly, referencing the top-left mesh with the two cones, we can represent this geometry with two separate meshes, whose vertices touch at a particular point in space.
When we are talking about manifold, we were saying that we want to zoom in very closely to see whether we can have a plane. In this case, does that mean if we are given any objects that have any sharp corner?it is not a manifold?
@chenj This is a good question. The local geometry associated with a point on a sharp corner cannot be approximated with a plane. And yet, it may still be a manifold, because the geometry may still satisfies the definitions on this slide. To be a bit more precise with our definition, a manifold requires that local geometry can be flattened out into a plane; a corner still satisfies this definition.
Why can't the self-intersecting shape be flattened into a plane?
Are we assuming the surface is "attached" to itself along the intersection? If not, then it could just be represented by a halfedge mesh where some faces happen to intersect.
@TheNumbat Referencing the top-right figure, the assumption here is that the surface is attached to itself, such that the edge at this intersection is shared for four faces; in this case, it is not a manifold But you are also correct in your assumption that if the object were not attached to itself, then it is a manifold.
Similarly, referencing the top-left mesh with the two cones, we can represent this geometry with two separate meshes, whose vertices touch at a particular point in space.