Principal curvature net
Chinese explaination refers to the Section 3.5 in the PhD thesis.
Definition
Principal curvature net is a curve network formed by principal curvature lines, whose directions at each point follow the maximum and minimum curvature of the surface. Any net that is conjugate and orthogonal is a principal curvature net [1]. The corresponding discretization is orthogonal planar quad mesh, i.e. orthogonal PQ mesh.
Other discretizations include circular mesh [1] and conical mesh [2].
Constraint of PQ mesh
The representation of planar quad faces is similar to the planar vertex stars, but adding additional quad face normals \(f_n\) as auxiliary variables. \(f_n\) are unit normals orthogonal to vectors \(v_{i+1}-v_i (i=1,2,3)\):
The number of all variables is \(|X| = 3|V| + 3|F|\) and the number of hard constraints is \(N = |F| + 4|F|\).
| Variable | Symbol | Number |
|---|---|---|
vertices |
\(v \in R^3\) | \(3\vert V \vert\) |
normals |
\(f_n\in R^3\) | \(3\vert F \vert\) |
The function for PQ mesh is DOS/archgeolab/constraints/constraints_basic.py/con_planarity_constraints().
[1] Alexander Bobenko, Suris Yuri. 2008. Discrete differential geometry: Integrable structure. Vol. 98. American Mathematical Soc.
[2] Yang Liu, Helmut Pottmann, Johannes Wallner, Yongliang Yang, Wenping Wang. 2006. Geometric modeling with conical meshes and developable surfaces. ACM Trans. Graphics 25, 3, 681--689.