@1:03:20 one simple example where it would fail to work if the 2 faces are in the same plane. So the cross product would be 0. I think we could handle this by checking other faces containing these 2 vertices of the edge, however, if the whole loop of faces around a vertex sets in the same plane, there would be no way to recover that vertex position.
I don't think even having other edges around the same vertex helps: one example, this will always fail to reconstruct the length of a prism. If you imagine a rectangular prism, then each face is a quad made up of two coplanar triangles, you can never determine the angle of the diagonals of those quads, so you can't figure out their aspect ratios, and it doesn't help that every vertex does have 3 planes adjacent. In fact, if you take any convex solid and shift the bounding planes along their normal, you end up with a different shape with all of the same normals. This same reasoning can be applied to some non-convex shapes by considering a subset of the surface that corresponds to a subset of some convex surface. Ah, and if I had been more patient: the examples shown later of discrete surfaces that aren't simplicial complexes give the same result! Those can all be triangulated into simplicial complexes where the normals aren't sufficient, in the exact same way I'm describing. Being a discrete surface with some non-triangular face is sorta equivalent to being a simplicial surface where some triangles are coplanar.
Professor Keenan, there is a slight error minutes 49:00. There is indexing error on the first cotan on the same formula, should be alpha_k^{ij}. Overall great lectures. I have been following your series and starting a Discrete Geometry group in Indiana University Bloomington with your material as the basis, teaching it to undergrad (I'm a PhD student in Math dept here). Hope to get in touch one day.
I'm a bit confused. The video is called discrete surfaces but - as far as I have seen - it only provides a definition for simplicial surfaces. maybe I need to watch the video again or maybe I should know beforehand that they are one and the same, i don't know.
I don't understand how you can recover the shape of any mesh from its normals. What if, for example, you have a single triangle with normal in the z direction; I don't believe theres a way to recover the triangle angles from that normal?
24:59 Does it matter that the allowed immersive mapping has more vertices in the image than the original? Doesn't that break the definition of injectivity?
@1:03:20
one simple example where it would fail to work if the 2 faces are in the same plane. So the cross product would be 0. I think we could handle this by checking other faces containing these 2 vertices of the edge, however, if the whole loop of faces around a vertex sets in the same plane, there would be no way to recover that vertex position.
I don't think even having other edges around the same vertex helps: one example, this will always fail to reconstruct the length of a prism. If you imagine a rectangular prism, then each face is a quad made up of two coplanar triangles, you can never determine the angle of the diagonals of those quads, so you can't figure out their aspect ratios, and it doesn't help that every vertex does have 3 planes adjacent.
In fact, if you take any convex solid and shift the bounding planes along their normal, you end up with a different shape with all of the same normals. This same reasoning can be applied to some non-convex shapes by considering a subset of the surface that corresponds to a subset of some convex surface.
Ah, and if I had been more patient: the examples shown later of discrete surfaces that aren't simplicial complexes give the same result! Those can all be triangulated into simplicial complexes where the normals aren't sufficient, in the exact same way I'm describing. Being a discrete surface with some non-triangular face is sorta equivalent to being a simplicial surface where some triangles are coplanar.
Great lecture professsor! Even I don't fully understand the math equations, I can learn English from you. Many thanks
Thank you for these videos
Professor Keenan, there is a slight error minutes 49:00. There is indexing error on the first cotan on the same formula, should be alpha_k^{ij}.
Overall great lectures. I have been following your series and starting a Discrete Geometry group in Indiana University Bloomington with your material as the basis, teaching it to undergrad (I'm a PhD student in Math dept here). Hope to get in touch one day.
I'm a bit confused. The video is called discrete surfaces but - as far as I have seen - it only provides a definition for simplicial surfaces. maybe I need to watch the video again or maybe I should know beforehand that they are one and the same, i don't know.
I don't understand how you can recover the shape of any mesh from its normals. What if, for example, you have a single triangle with normal in the z direction; I don't believe theres a way to recover the triangle angles from that normal?
24:59 Does it matter that the allowed immersive mapping has more vertices in the image than the original? Doesn't that break the definition of injectivity?