From aca62731c27e940fbe66093dbd08303d3e0ad6e4 Mon Sep 17 00:00:00 2001 From: Vectornaut Date: Fri, 23 May 2025 23:07:41 +0000 Subject: [PATCH] Add the multifocal ellipsoid --- Test-problems.md | 19 ++++++++++++++++++- 1 file changed, 18 insertions(+), 1 deletion(-) diff --git a/Test-problems.md b/Test-problems.md index 7654251..f77e6e9 100644 --- a/Test-problems.md +++ b/Test-problems.md @@ -23,7 +23,7 @@ for some positive $r$, $s$. Show that $A$, $B$, $C$, $D$, $E$, $F$ must be the v ## Exploring configuration spaces -### Equiangular equilateral hexagons +### The configuration space of equiangular, equilateral hexagons #### Source @@ -42,6 +42,23 @@ O’Hara has described the configuration space of hexagons in 3-space which are According to Dan Piker, the configuration space of right-angled equilateral heptagons also has at least one one-dimensional component. +### Generalized conic: multifocal ellipsoid + +#### Source + +Two-dimensional version + +- **Author:** James Clerk Maxwell +- **Published:** ["Paper on the Description of Oval Curves"](https://www.google.com/books/edition/The_Scientific_Letters_and_Papers_of_Jam/zfM8AAAAIAAJ?hl=en&gbpv=1&pg=PA35) (February 1846), in *The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862* + +#### Statement + +Fix two or more “pin” points in 2-space or 3-space. Constrain a movable “pencil” point by running a fixed-length string through all the points in some sequence. Find the locus of possible positions of the pencil. + +#### 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). + ## 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.