Q:

determine cube orientation given one side in a perspective projection

Suppose that we are given an arbitrary quadrilateral T that does not have any parallel edges. I want to draw a cube in a three-point perspective projection such that T is one of its sides. The projection parameters are not given. Is the solution always unique? How to find it? 

A:

Two opposite sides of the quadrilateral, if prolonged, meet at a vanishing point. So two of your three perspective points are easy to find and the third one should be arbitrary. But it is not true that every quadrilateral can be the face of a cube, see for instance the image below. If $ABCD$ is your quadrlateral, $E$ and $F$ are two vanishing points, while $H$ and $I$ are the corresponding measuring points ($FH=FG$ and $EI=EG$). But for the edges $BC$ and $CD$ to be equal, you should have $KC=JC$, which is not true.

