# misschuer

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## Transforming Planes

If we have a plane vector n = [a, b, c, d] which describes a plane then for any point p = [x, y, z, 1] in that plane the follow equation holds:

nt p = ax + by + cz + d = 0

If for a point p on the plane, we apply an invertible transformation R to get the transformed point p1, then the plane vector n1 of the transformed plane is given by applying a corresponding transformation Q to the original plane vector n where Q is unknown.

p1 = R p
n1
= Q n

We can solve for Q by using the resulting plane equation:

n1t p1 = 0

Begin by substituting for n1 and p1

(Q n)t (R p) = 0
nt Qt R p = 0

If Qt R = I then nt Qt R p = nt I p = nt p = 0 which is given.

Qt R =
Q
t = R-1
Q = (R-1)t

Substituting Q back into our plane vector transformation equation we get:

n1 = Q n = (R-1)t n

posted on 2017-12-06 11:39 此最相思 阅读(194) 评论(0)  编辑 收藏 引用 所属分类: mathematics

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