= Solution
The <vector> $\mathbf r-(\mathbf r\cdot\mathbf u)\mathbf u$ is the perpendicular component of $\mathbf r$ relative to the axis $\mathbb R\mathbf u$, so its length is the distance from that axis. Hence \b[the surface is an infinite <circular cylinder> of radius $\beta$ about the line through the origin parallel to $\mathbf u$]. Here $\beta$ is the radius and $\mathbf u$ fixes the axis direction; reversing $\mathbf u$ leaves the same <circular cylinder>.
If $\mathbf v_1,\mathbf v_2$ are an <orthonormal basis> of the <plane> perpendicular to $\mathbf u$, a parameterization is
$$
\boxed{\mathbf r=s\mathbf u+\beta\bigl(\cos\varphi\,\mathbf v_1+\sin\varphi\,\mathbf v_2\bigr),\qquad s\in\mathbb R.}
$$
This verifies that all heights and all angular positions, but no interior points, belong to the surface.
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