# Laplacian Meshes

## Mesh Reconstruction

I use CSparse to solve the linear problem in mesh reconstruction. After anchoring the position of a vertex (v(0) = (0, 0, 0)), the original mesh is reconstructed, yet translated to a new position. Results are shown in Figure 1 and Figure 2.

## Mesh Deformation

By adding new entries into laplacian matrix, we put constraints on the positions of anchored vertices. We select the value of the maximum entry that appears in the diagonal of laplacian matrix as the weight of newly-added entries. Results are shown in Figure 3, 4, 5.

## Parameterization

After picking 4 vertices on the meshes (forming two neighboring triangles), I replace the corresponding 4 rows in the laplacian matrix, so that I put constraints on the positions of the four vertices instead of their sigma coordinates. If new entries are added into the matrix insteading of replacing, the result will be problematic, since the sigma coordinates on the corners tend to pull vertices out of the square. Results are shown in Figure 6, 7, 8.

## Membrane Surface

Fixed some vertices, and keep laplacian coordinates to be 0 on other vertices. Results are shown in Figure 9, 10.

## Function Interpolation

Take the vertices with given value as constraints, and set all sigma coordinates as 0 (make the value on a vertex tend to be the average of the values on its neighbors). In the following examples, I pick 3 vertices on each mesh, and set the values on the three vertices to be 100, 200, 300. Results are shown in Figure 11, 12, 13.

## Mean Curvature

Use cotangent formula to mimic mean curvature. Note that this method is problematic when triangle is obtuse, therefore I try my best to find good meshes for this application, where most triangles are normal. Results are shown in Figure 14, 15, 16.

## Texture Mapping

Uae the result of parameterization, to map checker boarder onto the mesh. Unfortunately, since parameterization given above is not an ideal one, it causes great distortion around the selected 'corner vertices'. The size of boxes vary greatly. Results are shown in Figure 16, 17, 18, 19, 20.