Description: |
RiemannMapper : A Mesh Parameterization Toolkit. Background: Mesh parameterization refers to the process to map a 3D triangle mesh onto a planar domain, most mesh parameterization algorithms are based on the theories in differential geometry. A conformal mapping maps a 3D surface onto a 2D planar domain, such that the mapping preserves angles, or equivalently, the mapping maps infinitesimal circles on the surface to the infinitesimal circles on the plane. As shown in the following figure, the 3D human face surface is parameterized onto the unit planar disk by a conformal mapping. We put the checker board texture on the disk, and pull it back onto the 3D face, then all the right corner angles of the checkers are well preserved. Similarly, we put a circle packing texture on the disk, then pull it back onto the 3D face, then all the small circles are well preserved. According to uniformization theorem, all surfaces in real life can be conformally mapped to one of 3 canonical shapes, the sphere, the plane and the hyperbolic space. Details can be found in the following books written by the developer of this toolkit.The toolkit demonstrates the theories and computational algorithms in the books for education and research purposes. |