What about a small cylinder? Wouldn't a spherical cross section yield a torus and not a circle?
moridin22
This is only a rough definition, but the keyword here is "small" - in the case that you're describing, the sphere would have to be about the same size as the cylinder in order to produce a torus-shaped cut. For it not to be a manifold, you have to be able to use an infinitesimally small sphere to make the non-disk cut.
RichardChen9
Does "with border" mainly apply to 2D surfaces as opposed to the 3D ones on the top row?
StaffJakeHoles
It actually applies in 3D as well. Manifolds with boundary have the relaxed condition that the area around a point can be described by a half open disk. The ones in the top row are actually 2d manifolds submersed in 3d since their surface can be locally parameterized by a 2d disk. Interestingly enough, the boundary of an n-dimensional manifold is an n-1 dimensional manifold. The ones in the bottom row are examples of 2d manifolds with 1d manifold boundaries.
raghav-cs184
What is the motivation behind this definition for a manifold? What aspect of geometric algorithms depend on this specific property? I was thinking it would allow some sort of continuity in shapes, but I'm not sure why that continuity would be useful for triangle based algorithms, as they seem to be inherently discrete?
mnicoletti15
I am not a scientist but my best attempt as an undergrad at answering this question is the following: First, as noted above this is a very rough definition. Secondly, somehow, maybe for deep reasons, this the "natural" definition of a sort of generalized 2-D (or n-D) surface that does not have to be smooth. Another way to say this is that these objects are useful to study because they show up a lot, including in this context of computer graphics. So I would say that the motivation for the definition is not in its applications, but rather it is more that this definition is describing something we care about in our study of graphics so it is useful for us to understand it and related objects so that we can use any existing knowledge about these objects as tools.
avinashnandakumar
I'm still a bit confused on the examples and non examples of the manifold property. In the first example on the left, it seems like this disk that is being cut from the sphere is bent? So what would make this different than the first non example where there are three semi-circular disks stuck together? Am I just misinterpreting the first example image?
letrangg
You got it partly right. From my little exposure with manifold in a topology-like math course, manifold is where the surrounding of a point is all smooth (where we can find the derivative of a point and the points around it). Whereas at non-manifold points (like in the three intersection points that we saw), we can't really decide the derivative of that point as the function that yield those shapes are changing abruptly in many different directions. Actually I kinda am confused with the bottom right image on the "with border, manifold" category and the "with border, not manifold" image too. Is it because the change in the bottom non-manifold image is discrete and not continuous? Either way I need to brush up on topology and the basic definitions, it's been a while so I can't remember what is the formal definition of a manifold.
AronisGod
I like to simply manifold as something that can be described by 1 continuous surface, either open or closed.
jchen12197
I am also confused by the "With border" examples. The one on the left is a half disk and the one on the right isn't a full circle either. So what makes these qualified as manifold but not the "With border" example for not manifold?
rachelthomas7
One good way of telling if an edge is non-manifold is if it connects more than two faces.
What about a small cylinder? Wouldn't a spherical cross section yield a torus and not a circle?
This is only a rough definition, but the keyword here is "small" - in the case that you're describing, the sphere would have to be about the same size as the cylinder in order to produce a torus-shaped cut. For it not to be a manifold, you have to be able to use an infinitesimally small sphere to make the non-disk cut.
Does "with border" mainly apply to 2D surfaces as opposed to the 3D ones on the top row?
It actually applies in 3D as well. Manifolds with boundary have the relaxed condition that the area around a point can be described by a half open disk. The ones in the top row are actually 2d manifolds submersed in 3d since their surface can be locally parameterized by a 2d disk. Interestingly enough, the boundary of an n-dimensional manifold is an n-1 dimensional manifold. The ones in the bottom row are examples of 2d manifolds with 1d manifold boundaries.
What is the motivation behind this definition for a manifold? What aspect of geometric algorithms depend on this specific property? I was thinking it would allow some sort of continuity in shapes, but I'm not sure why that continuity would be useful for triangle based algorithms, as they seem to be inherently discrete?
I am not a scientist but my best attempt as an undergrad at answering this question is the following: First, as noted above this is a very rough definition. Secondly, somehow, maybe for deep reasons, this the "natural" definition of a sort of generalized 2-D (or n-D) surface that does not have to be smooth. Another way to say this is that these objects are useful to study because they show up a lot, including in this context of computer graphics. So I would say that the motivation for the definition is not in its applications, but rather it is more that this definition is describing something we care about in our study of graphics so it is useful for us to understand it and related objects so that we can use any existing knowledge about these objects as tools.
I'm still a bit confused on the examples and non examples of the manifold property. In the first example on the left, it seems like this disk that is being cut from the sphere is bent? So what would make this different than the first non example where there are three semi-circular disks stuck together? Am I just misinterpreting the first example image?
You got it partly right. From my little exposure with manifold in a topology-like math course, manifold is where the surrounding of a point is all smooth (where we can find the derivative of a point and the points around it). Whereas at non-manifold points (like in the three intersection points that we saw), we can't really decide the derivative of that point as the function that yield those shapes are changing abruptly in many different directions. Actually I kinda am confused with the bottom right image on the "with border, manifold" category and the "with border, not manifold" image too. Is it because the change in the bottom non-manifold image is discrete and not continuous? Either way I need to brush up on topology and the basic definitions, it's been a while so I can't remember what is the formal definition of a manifold.
I like to simply manifold as something that can be described by 1 continuous surface, either open or closed.
I am also confused by the "With border" examples. The one on the left is a half disk and the one on the right isn't a full circle either. So what makes these qualified as manifold but not the "With border" example for not manifold?
One good way of telling if an edge is non-manifold is if it connects more than two faces.