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Add bubble flowers

Vectornaut 2025-05-27 03:10:47 +00:00
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Test-problems.md

@ -112,6 +112,16 @@ Equivalently, fix three “directrix” lines in 3-space that intersect pairwise
To get an ellipse or an ellipsoid, use two pins in 2- or 3-space and fix the area of the triangle whose vertices are the pins and the pencil.
### Bubble flowers in 2d
#### Statement
In a plane, draw two concentric $n$-gons, and choose a congruence between them. Connect each vertex of the inner $n$-gon to the corresponding vertex of the outer $n$-gon using a line segment “spoke.” Rotate the $n$-gons so that the “spokes” don't intersect them or each other. We now have a planar graph whose faces are an inner $n$-gon surrounded by $n$ quadrilaterals.
Relax the spokes and the edges of the $n$-gons from line segments to constant-curvature arcs. Constrain the arcs to meet at 120° angles at the vertices of the graph. This turns the graph into a 2d bubble cluster obeying Plateau’s laws.
The configuration space should be an algebraic variety with two components. On one component, the assembly should be $n + 5$ degrees of freedom: 4 from the Euclidean motions and $n + 1$ from motions that vary the pressures in the bubbles. The other component intersects this one at the configuration where the inner and outer $n$-gons are regular, and their edges have the same curvature. Physically, this means that the inner $n$-gon is at the same pressure as the exterior. From this configuration, the assembly can move like a necklace of $n$ beads while the pressures in all the beads are held constant. On the “necklace component,” the assembly should have $2n + 1$ degrees of freedom: 4 from the Euclidean motions, $n$ from motions that vary the pressures in the necklace bubbles, and $n - 3$ from the constant-pressure necklace motions. Note that on most of the necklace component, the pressure in the inner $n$-gon can’t vary.
## Hierarchical constraints
These problems impose various kinds of *soft constraints* on top of the *hard constraints* that an assembly must satisfy to qualify as a solution. Here are some possible kinds of soft constraints.