diff --git a/Test-problems.md b/Test-problems.md index f77e6e9..702f7e6 100644 --- a/Test-problems.md +++ b/Test-problems.md @@ -57,7 +57,23 @@ Fix two or more “pin” points in 2-space or 3-space. Constrain a movable “p #### Notes -To avoid topological issues, it would be simplest to model the length of the string as a whole-number linear combination of the distances from the points to the pencil. To get an ellipse, use two pins and fix the sum of the distances (with unit coefficients). +To avoid topological issues, it would be simplest to model the length of the string as a whole-number linear combination of the distances from the points to the pencil. To get an ellipse, use two pins and fix the unweighted sum of the distances. + +### Generalized conic: triangle of directrices + +#### Source + +No source found yet, but it seems to be in the spirit of [generalized conics](https://en.wikipedia.org/wiki/Generalized_conic) as described on Wikipedia. + +#### Statement + +Fix three “pin” points in 3-space. Constrain a movable “pencil” point by fixing the surface area of the tetrahedron whose vertices are the pins and the pencil. + +Equivalently, fix three “directrix” lines in 3-space that intersect pairwise, forming a triangle. Constrain a movable “pencil” point by fixing the sum of the distances from the directrices to the pencil. + +#### Notes + +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. ## Hierarchical constraints