Calculate area with cross ratio

Question

Suppose that I am given a drawing of a table, on which a book lies in one of the corners. The measures of the book are known, how can I find the measures of the table using cross ratios?

Answer 1

In the real world, the table edges are parallel. So if you extrend the edges in your drawings, and connect them, you obtain the images of the points at infinity. Now you have four points along the two edges common to book and table: the corner common to book and table, the corner where the book ends, the corner where the table ends, and the point at infinity. Now observe that

$$\operatorname{CR}(\infty,0;1,x)=x$$

which is an abbreviated notation for the more complete formulation

$$\operatorname{CR}\left( \begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}; \begin{pmatrix}1\\1\end{pmatrix}, \begin{pmatrix}x\\1\end{pmatrix} \right)=x$$

So if you take $A$ to be the point at infinity, $B$ the common corner, $C$ the corner where the book ends and $D$ the corner where the table ends, then $\operatorname{CR}(A,B;C,D)$ will be the length of the table, measured in multiples of the length of the book. Or, if you prefer that formulation, the cross ratio will be the length of the table divided by the length of the book.