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Add the generalized conic with a triangular of directrices

Vectornaut 2025-05-23 23:19:55 +00:00
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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