A-net
Chinese explaination refers to the Section 3.6 in the PhD thesis.
Definition
A-net is a curve network formed by two families of asymptotic curves. Its discretization [1] is defined by a quad mesh with all planar vertex starts, i.e. each vertex \(v\) and its 4 neighbouring connected vertices \(v_i(i=1,\cdots,4)\) are planar.
Constraint
To represent the planar vertex stars, additional vertex normals \(v_n\) are used as auxiliary variables. \(v_n\) should be unit normals that are orthogonal to vectors \(v_i-v (i=1,\cdots,4)\):
Suppose the number of vertex star of valence 4 is \(|V_4|\), then the number of all variables is \(|X| = 3|V| + 3|V_4|\) and the number of hard constraints is \(N = |F| + 5|V_4|\).
| Variable | Symbol | Number |
|---|---|---|
vertices |
\(v \in R^3\) | \(3\vert V \vert\) |
normals |
\(v_n\in R^3\) | \(3\vert V_4 \vert\) |
The function for the quad mesh being A-net is DOS/archgeolab/constraints/constraints_net.py/con_anet(),
while the function ./con_anet_diagnet() is used for the mesh diagonal net being A-net.
Minimal net
Orthogonal A-net is a parametrization of minimal surface [1]. The corresponding discretization can be defined by planar vertex stars and equal-diagonal-length quad faces.
Let the sparse matrix and the array of A-net are \(H_A\) and \(r_A\), respectively. For minimal net optimization, one just stacks the sparse matrix \(H\) and \(H_A\), and concatenate the array \(r\) and \(r_A\) to the final solver.
[1] Alexander Bobenko, Suris Yuri. 2008. Discrete differential geometry: Integrable structure. Vol. 98. American Mathematical Soc.