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ion1/Vertex distance from planes.md

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Vertex distance from planes
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Vertex distance from planes.md

Plane normal dot products

P = [ 1  c ]
    [ c  1 ]

Plane normals

N N^T = P

[ N1x   0 ] [ N1x N2x ] = P
[ N2x N2y ] [   0 N2y ]

N1x^2 = 1
N1x N2x = c
N2x^2 + N2y^2 = 1

N1x = 1
N2x = c
N2y = sqrt(1 - c^2)

Vertex distance from planes

d = [ d1 d2 ]^T

Vertex

N v = d

v = N^-1 d

[ N1x   0 | 1  0 ]
[ N2x N2y | 0  1 ]

[ 1  0 |          1/N1x      0 ]
[ 0  1 | -N2x/(N1x N2y)  1/N2y ]

v = [          1/N1x      0 ] [ d1 ]
    [ -N2x/(N1x N2y)  1/N2y ] [ d2 ]

v = [               1                0  ] [ d1 ]
    [ -c/sqrt(1 - c^2)  1/sqrt(1 - c^2) ] [ d2 ]

v = [                        d1  ]
    [ (-c d1 + d2)/sqrt(1 - c^2) ]
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