# Point Based Methods

CREATED: 200701021627 Author: Leonidas J. Guibas

** Point based methods

- physical simulation ** phyxels (sparse) ** surfels (dense)
- good for highly dynamic situations - mesh does not get in the way

** Dealing with point clouds

- given point cloud, connect nearby (dist < $\epsilon$ points to get topology
- Q: How to set $\epsilon$ to get the actual topology?
- for a range of epsilon, observe birth/death of topological features, persistent features are part of the shape
- persistent bar codes captures topology and geometry of point clouse
- use local properties such as neighbourhood, curvature to produce a series of filtration to get bar code
- topology is coarse but robust
- use tangent space to get geometry
- increase curvature of tangent to get closer approximations to the actual shape
- stability, efficiency can be obtained by using a derived set of points

** Simulations

- point based elasticity from continuum mechanics
- Q: How to derive global properties, displacement field form local properties of points
- Phyxels (dense), surfels (dense but not involve in physics, contribute to appearance)
- Add new simulation nodes, when crack passes through to improve accuracy
- No mesh restructuring required for simulation of cracks

** Proximity

- interactions between points are short range
- geometric spanner - stable combinatorial structure which captures proximity across time
- space graph such that shortest path ~ euclidean distance
- maintain this property as graph changes
- replace geometric queries (who is near) by graph search

** Summary

- particles interact with neighbours
- resampling
- topological features are created and destroyed
- maintain some combinatorial structure behind the scenes to make things fast