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'Awith 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.