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Specify "real"

Vectornaut 2024-02-13 21:52:12 +00:00
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Engine-prototype.md

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### Challenging examples ### Challenging examples
With three mutually tangent spheres, `HomotopyContinuation` doesn't seem to find solutions, even though we know there are some. One solution, in $[r, s, x, y, z]$ coordinates, is: With three mutually tangent spheres, `HomotopyContinuation` doesn't seem to find real solutions, even though we know there are some. One solution, in $[r, s, x, y, z]$ coordinates, is:
* $[0, 0, 0, 0, 1]$ (plane perpendicular to $z$ axis) * $[0, 0, 0, 0, 1]$ (plane perpendicular to $z$ axis)
* $[1, 0, 0, 0, 1]$ (unit sphere above plane, touching at $[x, y] = [0 ,0]$) * $[1, 0, 0, 0, 1]$ (unit sphere above plane, touching at $[x, y] = [0 ,0]$)
* $[1, 4, 0, 2, 1]$ (unit sphere above plane, touching at $[x, y] = [0, 2]$) * $[1, 4, 0, 2, 1]$ (unit sphere above plane, touching at $[x, y] = [0, 2]$)