## Harmonic Maps and Minimal Surfaces

Discrete harmonic maps and their conjugates are used to minimize surface area.

**Find:** Minimal surface bounded by a given curve G

**Algorithm:** Use initial surface M_{0} and construct sequence
of surfaces M_{i+1} by finding (Laplace-Beltrami) harmonic maps F_{i}
with

F_{i} : M_{i} --> M_{i+1} with
boundary(M_{i+1}) = G.

Limit surface is a minimal surfaces under certain conditions.

During minimization boundary vertices are retained. Pick and drag vertices with the left mouse button by holding key "p" pressed to modify the initial surface. Set the number of iteration loops for the minimization algorithm by typing into the text field "Num Loops".

By the checkboxes "Tangential" and "Normal" you define, in which directions the minimizer is allowed to move vertices. The Checkboxes "Update Normals" and "Update Domain" appoint, if surface normals and domain are recomputed in every minimization step.

The Button "Step" invokes one minimization step, the button "Minimize" starts as many minimizing iterations as are specified in the "Num Loops" text field. By the "Resume" button you can stop and continue the iteration.