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Engine prototype #13

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glen merged 133 commits from engine-proto into main 2024-10-21 03:18:48 +00:00
Conversation 9 Commits 133 Files Changed 20 +5 -5
3 changed files with 5 additions and 5 deletions
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2
engine-proto/gram-test/Engine.jl
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@ -41,7 +41,7 @@ rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), she
point(pos) = [pos; 0.5; 0.5 * dot(pos, pos)] point(pos) = [pos; 0.5; 0.5 * dot(pos, pos)]
plane(normal, offset) = [normal; 0; offset] plane(normal, offset) = [-normal; 0; -offset]
function sphere(center, radius) function sphere(center, radius)
dist_sq = dot(center, center) dist_sq = dot(center, center)

6
engine-proto/gram-test/circles-in-triangle.jl
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@ -58,9 +58,9 @@ gram = sparse(J, K, values)
guess = hcat( guess = hcat(
Engine.plane(BigFloat[0, 0, 1], BigFloat(0)), Engine.plane(BigFloat[0, 0, 1], BigFloat(0)),
Engine.sphere(BigFloat[0, 0, 0], BigFloat(1//2)), Engine.sphere(BigFloat[0, 0, 0], BigFloat(1//2)),
Engine.plane(BigFloat[1, 0, 0], BigFloat(1)), Engine.plane(-BigFloat[1, 0, 0], BigFloat(-1)),
Engine.plane(BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(1)), Engine.plane(-BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(-1)),
Engine.plane(BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(1)), Engine.plane(-BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(-1)),
Engine.sphere(BigFloat[-1, 0, 0], BigFloat(1//5)), Engine.sphere(BigFloat[-1, 0, 0], BigFloat(1//5)),
Engine.sphere(BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//5)), Engine.sphere(BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//5)),
Engine.sphere(BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//5)), Engine.sphere(BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//5)),

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notes/inversive.md
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@ -7,7 +7,7 @@ These coordinates are of form $I=(c, r, x, y, z)$ where we think of $c$ as the c
| Entity or Relationship | Representation | Comments/questions | | Entity or Relationship | Representation | Comments/questions |
| ------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | ------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| Sphere s with radius r>0 centered on P = (x,y,z) | $I_s = (1/c, 1/r, x/r, y/r, z/r)$ satisfying $Q(I_s,I_s) = -1$, i.e., $c = r/(\|P\|^2 - r^2)$. | Can also write $I_s = (\|P\|^2/r - r, 1/r, x/r. y/r, z/r)$ -- so there is no trouble if $\|E_{I_s}\| = r$, just get first coordinate to be 0. | | Sphere s with radius r>0 centered on P = (x,y,z) | $I_s = (1/c, 1/r, x/r, y/r, z/r)$ satisfying $Q(I_s,I_s) = -1$, i.e., $c = r/(\|P\|^2 - r^2)$. | Can also write $I_s = (\|P\|^2/r - r, 1/r, x/r. y/r, z/r)$ -- so there is no trouble if $\|E_{I_s}\| = r$, just get first coordinate to be 0. |
| Plane p with unit normal (x,y,z), a distance s from origin | $I_p = (2s, 0, x, y, z)$ | Note $Q(I_p, I_p)$ is still -1. Also, there are two representations for each plane through the origin, namely $(0,0,x,y,z)$ and $(0,0,-x,-y,-z)$ | | Plane p with unit normal (x,y,z) through the point s(x,y,z) | $I_p = (-2s, 0, -x, -y, -z)$ | Note $Q(I_p, I_p)$ is still −1. |
| Point P with Euclidean coordinates (x,y,z) | $I_P = (\|P\|^2, 1, x, y, z)$ | Note $Q(I_P,I_P) = 0$.  Because of this we might choose  some other scaling of the inversive coordinates, say $(\||P\||,1/\||P\||,x/\||P\||,y/\||P\||,z/\||P\||)$ instead, but that fails at the origin, and likely won't have some of the other nice properties listed below.  Note that scaling just the co-radius by $s$ and the radius by $1/s$ (which still preserves $Q=0$) dilates by a factor of $s$ about the origin, so that $(\|P\|, \|P\|, x, y, z)$, which might look symmetric, would actually have to represent the Euclidean point $(x/\||P\||, y/\||P\||, z/\||P\||)$ . | | Point P with Euclidean coordinates (x,y,z) | $I_P = (\|P\|^2, 1, x, y, z)$ | Note $Q(I_P,I_P) = 0$.  Because of this we might choose  some other scaling of the inversive coordinates, say $(\||P\||,1/\||P\||,x/\||P\||,y/\||P\||,z/\||P\||)$ instead, but that fails at the origin, and likely won't have some of the other nice properties listed below.  Note that scaling just the co-radius by $s$ and the radius by $1/s$ (which still preserves $Q=0$) dilates by a factor of $s$ about the origin, so that $(\|P\|, \|P\|, x, y, z)$, which might look symmetric, would actually have to represent the Euclidean point $(x/\||P\||, y/\||P\||, z/\||P\||)$ . |
| ∞, the "point at infinity" | $I_\infty = (1,0,0,0,0)$ | The only solution to $Q(I,I) = 0$ not covered by the above case. | | ∞, the "point at infinity" | $I_\infty = (1,0,0,0,0)$ | The only solution to $Q(I,I) = 0$ not covered by the above case. |
| P lies on sphere or plane given by I | $Q(I_P, I) = 0$ | | | P lies on sphere or plane given by I | $Q(I_P, I) = 0$ | |