Re: [Algorithms] transforming a plane

[Charles B. <cb...@cb...>]

Silly me, that comes out of the math trivially.  Assuming an
orthonormal matrix, so Transpose(Inverse(M)) = M, then a point
on the plane initially is 

P = d N

Transformed, you get

P' = MP = M d N = d N' + T

where T is the translation in M.  Then d' is just

d' = N' * P' = d N' * N' + T * N' = d + T * N'

In other words, d += T * N' is all you need to transform a plane
(and rotate the normal) by an orthonormal matrix.

At 11:18 PM 9/6/2000 -0700, you wrote:
>> Is there a fast way to transform a plane?
>> 
>> Right now I'm doing it by rotating the normal and tranforming
>> a point on the plane, then re-generating the 4d-vector form of
>> the plane.  It seems there should be a way to do it with a
>> single 4x4 matrix multiply in some funny coordinate space.
>
>Yes, planes can be transformed with a single 4x4 matrix multiply.  Planes
>transform contravariantly like normal vectors do, meaning that you need to
>transform it using the inverse transpose of the matrix that you would
>transform normal points with.
>
>Suppose you have a 4D column vector V representing a plane and a 4x4
>transformation matrix M.  Let V = <Nx, Ny, Nz, -d> where N is the plane's
>normal vector and d = (P dot N) for any point P on the plane.  Then the
>transformed plane V' is given by
>
>   V' = Transpose(Inverse(M)) * V
>
>-- Eric Lengyel
>
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--------------------------------------
Charles Bloom           www.cbloom.com