Some paramterization definitions

Let S(u,v) -> R^3 be a surface, parameterized by u and v.

• If S is a triangular mesh we can use barycentric coordinates to parameterize the surface. First, assign a location (ui,vi) in the plane for each vertex. Each face is then parameterized by Sum Bj (uj,vj) where the Bj are the barycentric coordinates for the point in the face, and the (uj,vj) are the 2D points for each vertex.
  • If S is a general mesh then we can create something similar to barycentric coordinates for each non-3-sided face. There are a variety of ways to do this: Mark Meyer, Haeyoung Lee, Al Barr, and Mathieu Desbrun cover previous work and introduce a simple formula based on the angles at each vertex. Levy, in Siggraph 2001, formalizes the notion of placing a point in the center of the polygon to triangulate it, then defining the barycentric coordinates for each created triangle.
  • If S is a mesh then the first derivatives are constant across a face and can be found by (S(u+h,v) - S(u,v)) / (h). The first derivatives are, in general, discontinuous at the edges and vertices.
• If S is a smooth, analytical surface than we can calculate the derivatives analytically. Usually we will be interested in the first derivatives in the u and v direction (dS(u,v)/du and dS(u,v)/dv) which are the tangent vectors. Their cross product is the surface normal. We may also be interested in the second derivatives.
• Riemann Theorem states, first, that any simply connected region of a surface (i.e., cut a piece out of the surface) can be mapped to the unit disk. Second, given two disks, we can find a conformal map between the two of them.

  We can't do the same with triangulations if we want to keep the straight edges - we have to let the edges of the triangles "bend".

• First fundamental form. This is the inner, or dot, product of the two tangent vectors. It measures, in some sense, how much an image would be distorted if it were mapped to the surface using the parameterization. If the two tangent vectors are orthogonal and unit length than the dot product is zero. As the two tangent vectors point more towards each other (or away), or get longer, the absolute value of the dot product gets bigger. (See conformal mapping below.)
• Atlas. Several papers partition the mesh, creating an atlas. We will use atlas to refer to meshes that have been partitioned into non-overlapping, connected sets of faces, with no continuity guaranteed across the boundaries. Each partition is called a chart. Note that an atlas is a manifold with all of the overlapping subsets being empty.
• Developable surface is a surface that can be flattened into the plane without introducing any distortion. For example, a cylinder can be split vertically and "unrolled" without changing the interior angles of the faces.
• Curvature is the measure of how much a surface is "bending". Curvature is calculated using the first, second, and mixed derivatives of the surface. It is closely related to the first and second fundamental form. Curvature can be completely defined by two principal curvature directions (two orthogonal vectors in the tangent plane) and their corresponding curvature values. These two vectors define the directions of maximum and minimum curvature. The two principal curvatures, k1 and k2, can be combined to produce some useful measures:
  • k1 * k2, Gaussian curvature. If the Gaussian curvature is negative then you have a saddle shape, zero and it's a flat plane or a cylinder side, positive and it's a "bump" or a "valley".
  • (k1 + k2)/2 Mean curvature.

  There are a variety of ways to calculate curvature on a mesh.

• The Gauss map associates a point on the sphere with a point on the surface, via the surface normal. The surface normal picks a unique point on the sphere.
• Harmonic map. A map u (in our case, the map from the mesh to the plane) that minimizes the integral of the norm of the differential of the map, squared.
• Iso-parametric curve: Same parameter curve. If you fix s (or t) and walk along t (or s) you trace out a curve for that s = s0 (or t = t0) parameter.
• I'm still trying to figure out what this is.
• Conformal map. Essentially, angle preserving. The tangent vectors are perpendicular and of the same length. There are a variety of ways to optimize for conformal mappings. These will give different results (for non-developable surfaces) because of how you measure the distortion.
  • Tangent in the u direction is orthogonal to the tangent in the v direction, AND the two tangents have the same length.
  • Normal cross tangent in the u direction = tangent in the v direction. This says the above, but with one equation.
  • Conformal mapping by constraining edge length; Floater has shown that this is similar to his shape-preserving metric. Here, we indirectly optimize for the conformal mapping by making edges be the same (relative) length. I.e., we can scale all of the edges of the triangle up or down, which preserves angles and the same length constraint.
  • Express the constraint in the complex plane.
  • Co-tangent constraint...
  • Sheffer...