2PP Viewpoint for Set Rectangle

Step 1

Draw an arbitrary rectangle in 2-point perspective and the viewing semicircle through the corresponding vanishing points (VPs).

Step 2

Next, extend the rectangle's diagonals to the horizon line and mark off the corresponding diagonal vanishing points (DVPs).

Step 3

Now we will define what kind of rectangle we want to represent with our perspective image (using the view on the right) and plot the angle 2α with the line bisecting the line segment DVP₁,DVP₂ perpendicularly.

Step 4

Next, we plot a circle with the center located at the point where the angle 2α is formed through DVP₁ and DVP₂. For our purposes we will only need the semicircular arc below the horizon line.

Step 5

Now we will make use of the Inscribed Angle Theorem to find the semicircular arc for which the angle ∠BDC is always equal to half the angle ∠BAC. (Move the point D around in the applet below to see how it works).

Now back to our image.

Step 6

Now all we need to do is intersect the blue viewing semicircle (representing all viewpoints for which lines extended to VP₁ and VP₂ are at right angle to each other in the perspective image) with the pink semicircular arc (representing all viewpoints for which lines extended to DVP₁ and DVP₂ are at angle α to each other in the perspective image) to find the viewpoint (EYE) satisfying the two conditions.

Step 7

Now let's check that the viewpoint we have found is the correct one. We can see that the angle DVP₁,EYE,DVP₂ is precisely equal to the angle α.

And the same goes for the angle VP₁,EYE,VP₂, which is precisely equal to 90°.

Step 8

Perhaps you are wondering why the rectangle does not always resemble the one pictured in the plan view (on the right). To answer the question, let's first plot the 60° cone of vision...

...and move the rectangle roughly inside the cone of vision. You see now that the rectangle in the perspective image does indeed resemble the one in the plan view much more closely.