Shape Curvature Glossary
- Discrete Gaussian Curvature : The diference of 2 PI and the sum of the angles between adjacent faces adjoining a vertex.
2Pi - (a1 + a2 + a3 +...+an) , where n is the number of faces adjoining the vertex, and a1,...,an are the angles on the faces. At each vertex we consider the angles formed on the adjacent faces at that corner. If the sum of these angles is exactly 360 degreee then the collection of faces can be flattened to the plane withough a gap and without any overlap. Therefore, it has zero curvature. If the sume of the angle is smaller than 360 degree then the situation is like that at the tip of a cone, or at the corner of a convex polyhedron. Here the curvature should be positive since such a polyhedron is similar to a round ball. Negative curvature arises if the sum of the angles is bgger than 360 degreee which happens, for example, at a saddle point.
Orientation Specification:
Look at surface from outside such a way that the pivotal vertex is
located upward. Then I call a triangle in the left is left triangle
and the other triangle in the right is right triangle.
Up pivotal vertex
*
* * *
* * *
* * *
* * * * * * * *
Left Right
Down
Computation Algorithim Summary:
Xo - pivot vertex
X1,..,Xj,..,Xn - the first ring neighbor vertices of the vertex o.
n - number of the first ring neighbor vertices of the vertex o.
Xo
* * * * o
* * * * * * *
* * * * * * *
* * * * O * * * * Xj+1 Xj-1 * Aoj * Boj* Xj+1
* * * * * * *
* * * * * * *
* * * * *
Xj-1 Xj Xj
Aoj is angle at the vertex Xj-1
Boj is angle at the vertex Xj+1
Mean Curvature Normal Operator is calculated by following 2 steps:
First, calculate summation of (cot Aoj + cot Boj)(Xo - Xj) over j
from 1 to n, where Xo is the pivot vertex and Xj's are elements of
first ring neighbors vertices of the pivot vertex.
Second, divide above value by 2*AreaMixed.
Area Mixed can be calculated by either Barycentric method or
Voronoi method.
1. Barycentric method calculate AreaMixed by adding one third of each
triangle area.
2. Voronoi method is follwing
if there is obtuse angle at pivot vertex, add half of the triangle
area.
else if there is obtuse angle beside pivot vertex, add one fourth of
triangle area.
else if there is no obtuse angle, add Voronoi area of triangle.
Finally the Mean Curvature value is computed by taking half of the
magnitude of the Mean Curvature Normal Operator.
- Discrete Principal Curvature : Let k1 and k2 to be two principal curvature. Since Mean curvature is (k1+k2)/2 and Gaussian curvature is k1*k2,
we can calculate k1 an k2 by solving two unknown and two equation system.
- Principal Curvature : For a two-dimensional surface embedded in R3, consider the intersection of the surface with a plane containing the normal vector and one of the tangent vectors at a particular point. This intersection is a plane curve and has a curvature. This is the normal curvature, and it varies with the choice of the tangent vector. The maximum and minimum values of the normal curvature at a point are called the principal curvatures, k1 and k2, and the directions of the corresponding tangent vectors are called principal directio
- Mean Curvature : Mean Curvature is equal to the sum of the principal curvatures, k1+k2, over 2. It has the dimension of 1/length.
- Gaussian Curvature : Gaussian curvature is equal to the product of the principal curvatures, k1k2.
