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Update Test problems

Glen Whitney 2025-08-14 23:45:19 +00:00
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Test-problems.md

@ -309,6 +309,8 @@ In a plane, draw 3 points on each of two lines and connect them in a hexagon tha
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.
Note that in Geometry Expressions, if you set up the hypotheses of Pappas' Theorem, and then attempt to add the constraint representing the conclusion (i.e., that one of the three new intersection points of the sides of the hexagon is incident to the line through the other three), the option for adding that constraint is greyed out. Apparently it realizes that the relative position of the third intersection point and the line between the other two is determined by all of the givens. However, it does not seem to realize that this determined situation is one of incidence. On the other hand, since it is able to compute algebraic expressions for the positions and (e.g.) cosines of angles at determined points in terms of variables representing the coordinates of given points, it should actually have enough information to generate and analytic-geometry-style proof of Pappas' Theorem. As long as we are sticking with numerical methods in Dyna3, we won't be able to generate actual _proofs_ but it would of course be nice nevertheless to notice that the relationship between that point and line is determined, and even better, that it always seems to be incidence so that if we have a sufficiently generic occurrence of incidence, we can be sure that it will always happen. (Jurgen R-G did mention in one Cinderella presentation that they could show in some generality that if an incidence held in a sufficient neighborhood of a configuration, then in fact it was a necessary occurrence in all configurations with the construction sequence of that configuration.)
## 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.