# Inverse plane equation

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If I put the plane equation, Ax+By+Cz+D=0, in a 4x4 matrix, and took the inverse of that matrix, what would that give me? Doing a search for inverse plane equation on Google gives me nothing useful.

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Exactly then inverse of the matrix. Try multiplying them to see if the product is the identity matrix just to be sure. :*)

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What do you mean by "put the plane equation in a 4x4 matrix"?

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Yeah, I know it will give me the inverse of the matrix. What I'm trying to get at, is I would like to understand
the meaning behind it.

Is there such a thing as the inverse plane equation?

I'm trying to transform/rotate a plane to some neutral state/position.

What I mean is, given, Ax + By + Cz + D = 0, if I put this in matrix form, M:

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

M[1] = plane_normal.x;
M[5] = plane_normal.y;
M[9] = plane_normal.z;
M[13] = -DotProduct( plane_normal, plane_vertex );

Is the inverse of this matrix useful for anything? Or is my logic flawed?

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I think you may be on the wrong track with the 4x4 matrix. However:
Quote:
 I'm trying to transform/rotate a plane to some neutral state/position.
There is probably a solution to this problem, if you can describe more clearly what exactly you're trying to do.

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I want to rotate a plane (e.g. the vertices that make up the plane) so it has a neutral normal, like (0,1,0).

Can this be done by plugging the plane equation in a matrix and taking its
inverse?

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Quote:
 Original post by raydogI want to rotate a plane (e.g. the vertices that make up the plane) so it has a neutral normal, like (0,1,0).Can this be done by plugging the plane equation in a matrix and taking its inverse?

I don't think so. Although if you want you can rotate your plane on the z axis with 0 angle to make it orthogonal to your y axis.

ie:

x = (1,0,0)
y = (0,1,0)
z = (0,0,1)

So the 3 axes are orthogonal to each other.

If you want your plane to have a normal of let's say (0,1,0) then all you have to do is put your plane on the x,z axis so that it's orthogonal to the y plane.

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It is easy to transform a plane equation by a transformation matrix. You just need to write the equations symbolically and make the substitutions.
He is how for direct tranfomation.
Say M is a rotation matrix composed of a rotation R (3x3) matrix plus translation vector T

Matrix M will transform a point Q to P by:

1) P = Q * M = Q * R + T

Now say you have a plane equation

2) N % Transp(P) + d = 0;

N = a vector perpendicular to the plane surface,
d = the distance from the plane to the origin
Transp (P) is the transpose of vector P
% is the operation Dot product

Replacing equation 1 in 2 you get

N % [Transp (Q * R + T)] + d = 0

Now with some algebraic manipulation

N % [Transp (Q * R) + Transp (T)] + d = 0
N % Transp (Q * R) + N % Transp (T)] + d = 0
N % Transp (R) * Transp (Q) + N % Transp (T) + d = 0
[N % Transp (R)] * Transp (Q) + N % Transp (T) + d = 0

but Nr = N % Transp (R) is the inverse rotation of N by rotation matrix R.
dt = N % Transp (T) + d is the evaluation of the translation part of the matrix on the

Nr * Transp (Q) + dt = 0

I will let you do your own Inverse Transformation as an exercise

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Quote:
 Original post by Anonymous PosterNr * Transp (Q) + dt = 0

If Q is a point, and Transp() means Transpose, then how exactly does one take the Transpose
of a point? Matrix transposition is swapping rows for cols, but what is a point transposition?

Maybe a real-number example might clear things up.

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Quote:
 If Q is a point, and Transp() means Transpose, then how exactly does one take the Transpose of a point? Matrix transposition is swapping rows for cols, but what is a point transposition?
It's the same thing, but you have only one row or column, which becomes one column or row. Example:

[x y z]

Transposed, becomes:

[x]
[y]
[z]

Choosing one or the other has to do with rules of matrix multiplication. It looks like the AP is using row vectors in his example, so:

A*transpose(B)

Is the same as A.B (dot product). Also:

transpose(A)*B

Is a tensor product, but I don't think that came up in the example.

I'm not positive about this, but it looks like the AP is using dot product notation (% in his example) already, so the transposes may be redundant, in which case you can probably just ignore them. (I would have to look through the example more carefully to be sure, though...)

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