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!

Update:

Of course it is easy to transform a plane if you have it expressed in terms of a normal and position:

$P := (n,a)$

where $$a$$ is a point on the plane. The trick is then to find $$a$$ given just the normal and offset description.

$n \cdot x = w$

After looking at this for a minute, it appears that if I use the point:

$a := wn$

then

$n \cdot wn = w(n \cdot n) = w$

so it is easy to find a point on the plane given the normal and offset. Then we can just transform $$n$$ and $$a$$.