Last edited 12Jul by ortony@cm.math.uiuc.edu
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Proseminar Recap of "Rescalable Real-Time Interactive Computer
Animation" by John Sullivan
The representation of self-interesting shapes and surfaces is very
challenging. One way to achieve it is using optimal geometry,which uses
topological techniques. Dr. Sullivan introduced the concept of
manifolds as part of this concept. These are spaces which locally look like
R^n. They are divided
into 0-nfld (a point), 1-nfld (line segment or circle), and 2-nfld
(Euclidean plane or a torus kline). The interesting characteristics of the
manifolds are used to
ellaborate optimal surfaces. On these, the bending enery is minimized. One
important aspect of these surfaces is that the optimal metric (intrinsic
shape) should be symmetric, meaning that points and directions are
equivalent, locally. Also, there is an intrinsic idea of curvature in
manifolds. On a plane it is equal to 0, while for a sphere is greater than
0, and for an hyperbolic surface it is less than 0.
There are interesting examples of manifolds in different dimensions.
0-nfld and 1-nfld are not particullary interesting. Tori, Klein Bottles, and
Mobius Surfs are good examples of 2-nflds. They behave peculiarly. For
example, the torus doesn't have a completely symmetrical behavior, because
of its spheric and hyperbolic components. Other important fact about
manifolds is that there is always the possibility of finding a metric for
any kind of manifold: Euclidean space, Spherical space, Hyperbolic Space,
Sphere X Reals, Hyperbolic-plane X Reals, or three others related to Lei
Groups.