The Research Of Laplacian Coordinates For Mesh Deformation

In this thesis, we introduced the Laplacian Coordinates for mesh editing. We considered the different weights of Laplacian Coordinates and discussed their geometric meaning. The meaning of Laplacian Coordinates with unit weights is the vector from the vertex to the center of its adjacent vertices.In recent years, differential information has been greatly used for mesh editing. Given certain constraints, a deformed mesh is reconstructed by minimizing the changes in the differential information. Since the differential information is encoded in the global coordinate system, it must be transformed to fit the orientation of details in the deformed surface. Otherwise distortion will appear. We observe that visually desired deformed meshes should preserve both local parameterization and geometry details.To find suitable representations for the two information, the special property of Laplacian coordinates is useful.Specially, we can consider the coefficients of Laplacian operator as the parametrization information and the magnitudes of the Laplacian coordinates as the geometry information. Both sets of informations are non-directional and non-linearly dependent on the vertex positions. The meaning of Laplacian Coordinates with cotangent weights is the mean value curvature of the vertex. The cotangent weights may be negative and problematic to define for very large angles due to the properties of cotangent nearπ;convex weights that mimic the cotangent weights are called mean-value coordinates.We also introduced the method of Laplacian Editing in recent years.The main problem is that reconstructing the mesh by minimizing deviation from the original Laplacian Coordinates would lead to undesired distortion, specifically, shearing and stretching distortion. There are many people to study this problem. We chose one of their method in our experiment and got the comparison between unit weights and cotangent weights. From the experiment results, we can see there are same With theory.