LBO
Discrete Laplace Beltrami Operator (LBO) and related applications.
Applications:
Mean Curvature
A direct application of discrete LBO is calculate the mean curvature vector of a surface by simply multipling the Laplacian matrix on the point itself. Following is an example:
Harmonic Map
Disk Harmonic Map
Disk Harmonic Map maps a surface homeomorphic to the planar unit disk. It is a popular parameterization method. An example of disk harmonic map on a nefertiti model is shown below:
![](figures/harmonic_map00_crop.png)
![](figures/harmonic_map01_crop.png)
![](figures/harmonic_map02_crop.png)
Figure 2. An example of disk harmonic maps. Left to right: Original mesh, mapped mesh, stretched mesh on unit disk
Heat method
The heat method ([1]) is a discrete geodesic solution on various domains (grids, triangle/polygonal meshes, point clouds etc.). It's specially suitable for calculating distance fields with different sets of sources on the same mesh for several times. The implementation here is based on triangle mesh only. Here is an example output of heat method on a bunny model:
[1] Crane K, Weischedel C, Wardetzky M. Geodesics in heat: A new approach to computing distance based on heat flow[J]. ACM Transactions on Graphics (TOG), 2013, 32(5): 152.