How to draw a tiled floor in two-point perspective
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The diagrams below explain how to draw a grid of rectangles in two-point perspective. The grid consists of rectangles as in the diagram below (this diagram is the Euclidean picture with no perspective).
When we look at this from a viewpoint above the plane of the rectangle, we see the rectangular grid in two-point perspective, giving us a picture like the diagram below. The two perpendicular sides \(AB\) and \(AC\) extend out to meet the horizon at the two points of perspective \(X\) and \(Y\). The horizon line is coloured in red.
On this page we will describe how to construct this picture using a ruler.
Begin with a horizon line \(XY\) as in the diagram below, let \(A\) be another point in the plane, and choose points \(B\) and \(C\) on \(AX\) and \(AY\) respectively. Draw the lines \(BY\) and \(CX\). This creates the two-point perspective picture of the rectangle \(ABCD\) from the previous picture. We would like to know how to construct the two-point perspective picture of the other rectangles using just a ruler.
From the first picture we can see that the diagonals are all parallel. Therefore, when we draw them in perspective they should all meet on the horizon line \(XY\). Now draw the diagonal \(AD\), let \(P\) be the point where this meets the horizon, and draw the lines \(BP\) and \(CP\) as in the picture below. The lines \(BP\) and \(CP\) will be the diagonals of the new rectangles.
Now let \(B'\) be the intersection of \(BP\) with \(CX\). This will be the opposite corner of the new rectangle. Similarly, let \(C'\) be the intersection of \(CP\) with \(BY\). Now draw the lines \(C'X\) and \(B'Y\). As shown in the picture below, we have now drawn a \(2 \times 2\) grid of rectangles.
The construction in the picture seems to work perfectly, but there is one piece of information missing. How do we know that the lines \(AP\), \(B'Y\) and \(C'X\) all intersect at the point \(D'\)? The answer is that they do always intersect at \(D'\), but the proof is not obvious, and uses Desargues' theorem. The proof is explained on the next page.
Historical note. Desargues was an architect, and it was this exact question of drawing a grid of rectangles in two-point persective that led him to prove his famous theorem.
