Suppose you have a plane equation in local space and you’d like to express that plane equation in world space. The plane in local space is written as:

[ P := (n, w) ]

where (n) is the plane normal and (w) is the plane offset.

A point (x) is on the plane if

[n cdot x = w ]

Now define a transform (A) as

[ A := (R, p) ]

where (R) is an orthonormal rotation matrix and (p) is a translation vector.

Suppose we have a transform (A) that transforms points in local space into world space. With our transform (A) we can convert any point (x_1) in local space (space 1) into world space (space 2):

[x_2 = R x_1 + p]

Also any vector (n_1) in local space can be converted to world space:

[n_2 = R n_1]

Also suppose we have a plane defined in local space (space 1). Then for any point (x_1) in local space:

[n_1 cdot x_1 = w_1 ]

The main problem now is to find (w_2), the plane offset in world space. We can achieve this by substitution. First invert the transform relations above:

[x_1 = R^T (x_2 – p)]

[n_1 = R^T n_2]

where (R^T) is the transpose of (R). Recall that the inverse of an orthonormal matrix is the equal to the transpose.

Now substitute these expressions into the local space plane equation:

[R^T n_2 cdot (R^T (x_2 – p)) = w_1]

Expand:

[R^T n_2 cdot R^T x_2 – R^T n_2 cdot R^T p = w_1]

The rotations cancel out since they are orthonormal. Also the dot product is equivalent to matrix multiplication by the transpose. For example:

[R^T n_2 cdot R^T x_2 = n_2^T R R^T x_2 = n_2^T I x_2 = n_2 cdot x_2]

Simplify:

[n_2 cdot x_2 = w_1 + n_2 cdot p]

From this we can identify the world space plane offset (w_2):

[w_2 = w_1 + n_2 cdot p]

Done!