§ Repulsive curves
§ Gradient depends on norm
- consider a function and an energy .
- We want to optimize .
- however, what even is ?
- Recall that is the differential which at a point on the space , in a tangent direction , computes .
- Now the gradient is given by . So the gradient is the unique vector such that the inner product with the gradient produces the value of the contangent evaluated in that direction.
- Said differently, . This is a Reisez like representation theorem.
- Note that asking for an inner product means we need a hilbert space.
- One choice of inner product is given by , where .
- More generaly, we can use a Sobolev space, where we define the inner product given by , which can also be written as .
- Similarly, for the sobolev space , we would use . which is equal to .
- In general, we can write our inner product as something like .
§ Solving heat equation with finite differences
- Solving .
If we try to solve this equation using, say, explicit finite differences with grid spacing h, we will need a time step of size O(h 2) to remain stable—significantly slowing down computation as the grid is refined
§ Different norm is good for different situations
§ Tangent point energy
- Key intuition: want energy that is small for points that are "close by" in terms of on the knot, want energy that repels points on the knot that are far away in terms of by close by in terms of .