When I saw Entagmas great tutorial about minimal surfaces I remembered an old file I did to help a friend some years ago. He wanted to generate a smooth surface between some given boundaries and it should behave similar to elastic textile. To simulate it he used the same method which had been invented about 40 years ago by Frei Otto. He built a physical model out of thin wire and dripped it into soap film. After taking it out of the fluid the result was a perfectly smooth surface in the form of a soap film spanning between the boundaries.
After some experimentation and many models later, the question was finally how to digitize the surface. 3D-scanning was obviously not possible. He tried to model it based on photos but that didn’t work out very well. Repeatedly subdividing a coarse mesh wasn’t successful either. Thereby the solution is quite simple. Don’t try to digitize the surface, just digitize the whole process and skip the physical model.
Mathematically such a surface is a minimal surface or more precisely, it is a surface of minimal area for given boundary conditions. Another important definition is that it has zero mean curvature at all points. What he had to do is to minimize the overall area of the surface until the mesh reaches equilibrium. The resulting surface is a minimal surface, or at least a very good approximation of it. So, what’s the best method to compute such a surface?
Well, one way is of course to use a spring model, set the rest length to 0 and run the simulation. That’s basically what Entagma did in their tutorial. The problem with this method is that it´s rather slow and works best only on a fairly regular and clean mesh. On irregular meshes it’s highly likely that it won’t converge to a minimal surface.
A probably better method is laplacian smoothing. Again this works best on a regular mesh but it’s readily adaptable by using proper weights. While both methods might be slow due to the fact that they use an iterative approach, it’s generally better to rely on laplacian smoothing. This will result in a better approximation of a minimal surface.
Of course, there is a third method which again relies on the laplacian but uses a slightly different approach to solve the same problem. It´s based on the paper “Computing Discrete Minimal Surfaces and Their Conjugates” by Pinkall and Polthier and works by minimizing the Dirichlet energy. This leads to an efficient numerical method to compute the desired minimal surface. The basic idea is simple – just move the vertices of the mesh in a specific direction for a specific distance and the result is a minimal surface.
To do this, we first need to find the direction in which the area changes the most. In other words, we need the gradient of the area. Apparently this is the mean curvature normal which we get fairly easy by using the Laplace-Beltrami operator. The length of the mean curvature normal is the mean curvature and as we already know, it should be 0 for minimal surfaces. With this information it´s easy to set up a linear system in which we set the curvature to 0 and simply solve the system of equations for the surface with harmonic coordinate functions. The solution we get are the new point positions for the minimal surface we were looking for, or at least a very good approximation to it.
When using a fast sparse matrix solver, this method is not just quite accurate but also fast. For the examples below I had no problems to interactively modify an average dense mesh in realtime on my rather old laptop.
The same setup could be used for membrane and tension surfaces since they share many similarities with minimal surfaces, which is quite nice for playing around with the shape.
U. Pinkall, K. Polthier: Computing Discrete Minimal Surfaces and Their Conjugates