# Alberti's Perspective Construction

## 3. The underlying geometry

Why does the pavement construction work? Since the lines separating the columns are all perpendicular to the picture-plane, their images must pass through the vanishing point C and so are determined by their intersection with the lower edge of the frame. The crucial piece of information is the location of the image of the far left-hand corner of the checkerboard, because once this is found the diagonal can be drawn; and since the relation between rows, columns and diagonals is preserved by the perspective projection, the lines separating the rows can then all be correctly constructed. The following figure can be JAVA-animated by clicking on its surface.

The checkerboard is horizontal and abuts the edge of the (vertical) frame. A point O' is drawn in the picture-plane (to the right in this illustration), on a level with the vanishing-point C, and such that the horizontal distance O'C' to the frame is equal to the distance OC from the eye to C. Let M be the point where the far edge of the checkerboard intersects the vertical plane through O and C. The line of sight OM cuts the picture plane at H. To construct the image of the far edge of the checkerboard, it is enough to know the height of H. Since the checkerboard is square, the figure O'C'H'AP' is congruent to the figure OCHMP: the height of H is the same as the height of H' which is the intersection of O'A with the right-hand edge of the frame.

In this way the three-dimensional (blue) construction collapses into the 2-dimensional (red) one, and the perspective problem admits an elementary geometric solution.