# Alberti's Perspective Construction

**Feature Column Archive**

## 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.