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Update Test problems

Glen Whitney 2025-08-12 01:52:54 +00:00
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Test-problems.md

@ -299,6 +299,16 @@ Relax the spokes and the edges of the $n$-gons from line segments to constant-cu
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.
### Pappas' Theorem
(suggested by Aaron Abrams)
#### Statement
In a plane, draw 3 points on each of two lines and connect them in a hexagon that alternates lines. That produces three new intersection points of pairs of sides of the hexagon. Those three intersection points are collinear.
Therefore, ideally if we set this up (currently with a "drawing plane" and modeling lines as perpendicular planes and all points incident to the drawing plane) we can add all the hypotheses of the theorem. It should produce a configuration in which the intersection points are collinear of course. And now if we add the constraint that these three points are collinear, ideally Dyna3 should realize that this new constraint does not remove any degrees of freedom or even better that the configuration space is identical to what it was before.
## 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.