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Alberti's Perspective Construction

**Feature Column Archive**

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3. The underlying geometry

Why does the pavement construction work? Since the lines separating thecolumns are all perpendicular to the picture-plane, their imagesmust pass through the vanishing point `C` and soare determined by their intersection with the lower edge of theframe. The crucial piece of information is the location ofthe image of the far left-hand corner of the checkerboard, becauseonce this is found the diagonal can be drawn; and since the relationbetween rows, columns and diagonals is preserved by the perspectiveprojection, the lines separating the rows can then all be correctlyconstructed. The following figure can be JAVA-animated by clickingon 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 inthis illustration), on a level with the vanishing-point C, and such that the horizontal distance `O'C'`tothe frame is equal tothe distance `OC` from the eye to C. Let `M` be the point where the far edge of the checkerboardintersects 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, itis enough to know the height of `H`. Since the checkerboardis square, the figure `O'C'H'AP'` is congruent to thefigure `OCHMP`: the height of `H` is the same as theheight 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 collapsesinto the 2-dimensional (red) one, and the perspective problemadmits an elementary geometric solution.

3. The underlying geometry

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