proportions of a perspective-deformed rectangle

Question

Given a 2d picture of a rectangle distorted by perspective:

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Answer 1

Draw a right isosceles triangle with those two vanishing points and a third point below the horizon (that is, on the same side of the horizon as the rectangle is). That third point will be our origin and the two lines to the vanishing points will be our axes. Call the distance from the origin to a vanishing point pi/2. Now extend the sides of the rectangle from the vanishing points to the axes, and mark where they intersect the axes. Pick an axis, measure the distances from the two marks to the origin, transform those distances: x->tan(x), and the difference will be the "true" length of that side. Do the same for the other axis. Take the ratio of those two lengths and you're done.