How do the key characteristics of a manifold in 2D geometry differ from non-manifold structures?
rcorona
My high-level understanding (pardon if I butcher this somewhat) about manifolds is that a manifold of n dimensions is a subspace whose every point has a local neighborhood that is bijective with R^n. In other words, for any point on the manifold, it should be possible to define a radius s.t. all the points on an (n-1)-sphere of that radius centered around the point are also in the manifold.
Relating that to the slide. For all the left examples it should be possible to define a bijection between any points' neighborhood and a small enough 2D circle.
For the examples on the right side of the slide -- the top left is not a 2D manifold because points along the shared edge require a third dimension for their neighborhood.
Top right and bottom examples cannot define a local neighborhood circle at the point where all the triangles meet.
Not 100% sure about this though, so please correct me if I'm wrong!
Mehvix
To add on, one characteristic of manifold geometry is that the twin of an edges is well defined (but ill defined for non-manifold, see top-left example)
Mehvix
Additionally, the bottom non-manifold examples shows that the consistent orientation property doesn't always hold for non-manifold geometry (e.g. it is ambiguous how we'd traverse about the point)
Michael-Equi
Are there any efficient algorithms for doing this kind of check? Is it done often in graphics or is it more a formalism for mathmatical proofs?
colinsteidtmann
I found the definition from the textbook helpful: a manifold is "a surface in which a small neighborhood around any point could be smoothed out
into a bit of flat surface."
Combining it with Ren, I suppose a 2D manifold must require that ANY point can have the neighborhood around it smoothed out to a continuous/unbroken piece.
JunoLee128
Making sure - a disc is some sort of topological object (like sphere <-> cube), and not an actual disc (circle) equation right?
JunoLee128
Also how/when would we extend this to surfaces beyond these dimensions?
How do the key characteristics of a manifold in 2D geometry differ from non-manifold structures?
My high-level understanding (pardon if I butcher this somewhat) about manifolds is that a manifold of n dimensions is a subspace whose every point has a local neighborhood that is bijective with R^n. In other words, for any point on the manifold, it should be possible to define a radius s.t. all the points on an (n-1)-sphere of that radius centered around the point are also in the manifold.
Relating that to the slide. For all the left examples it should be possible to define a bijection between any points' neighborhood and a small enough 2D circle.
For the examples on the right side of the slide -- the top left is not a 2D manifold because points along the shared edge require a third dimension for their neighborhood. Top right and bottom examples cannot define a local neighborhood circle at the point where all the triangles meet.
Not 100% sure about this though, so please correct me if I'm wrong!
To add on, one characteristic of manifold geometry is that the twin of an edges is well defined (but ill defined for non-manifold, see top-left example)
Additionally, the bottom non-manifold examples shows that the consistent orientation property doesn't always hold for non-manifold geometry (e.g. it is ambiguous how we'd traverse about the point)
Are there any efficient algorithms for doing this kind of check? Is it done often in graphics or is it more a formalism for mathmatical proofs?
I found the definition from the textbook helpful: a manifold is "a surface in which a small neighborhood around any point could be smoothed out into a bit of flat surface."
Combining it with Ren, I suppose a 2D manifold must require that ANY point can have the neighborhood around it smoothed out to a continuous/unbroken piece.
Making sure - a disc is some sort of topological object (like sphere <-> cube), and not an actual disc (circle) equation right?
Also how/when would we extend this to surfaces beyond these dimensions?