Mapping L in two-point perspective
Recall the “T” from the Drawing ART exercise. As alluded to earlier, this is a nice way of introducing the idea of particular mappings from a plane to a plane (possibly itself) that preserves certain
properties, like collinear points map to collinear points and intersection point of two lines map to
the intersection point of the mapped lines. We’ll investigate this more in depth in this worksheet.
Instead of “T”, let’s consider a more simple letter, “L”, but this time in two point perspective. This
means that not only do the images of the parallel lines indicating depth converge to a vanishing
point on the horizon, but also the images of the parallel lines parallel to the ground along the front
and back face of the L converge to another vanishing point along the horizon. In Figure 1 below,
we see the front face of the block L, with three corners labeled A, B and C, as well as the mapping
of those three corners to the corners of the back of the block L, A0 , B 0 and C 0 .
Figure 1: Can we construct the back face of the block L given just this information?
1. Is this enough information to draw the back of the block L? If so, draw a transparent block
L, so you can see the entire back face.
2. Let’s make sure we’re not using any extra information from this being a L in two point
perspective. Consider the sparser image in Figure 2. Given the random point D, which we
can assume is somewhere on the front face of the L, is there enough information regarding
the perspective drawing to draw D0 on the back face?
3. Can we get away with less information? Why or why not?
4. How is Desargues’ Theorem involved?
1
Figure 2: Sparser image of the L in two point perspective, with a random point D
ABC and A0 B 0 C 0 are perspective from a point. When view this as a mapping from ABC to A0 B 0 C 0
that obeys the rules of perspective, then these three pairs of homologous points define a mapping
from the entire plane to the plane. Given any other point D in the plane, we can find its image D0 .
So what do we mean when we say that it “obeys the rules of perspective”?
5. Write down the rules of perspective used to find the image D0 of D.
2
This motivates to the following definition and theorem.
Definition: A perspective collineation is a bijective mapping from points to points and lines to
lines in a plane (possibly itself) that has the following properties.
• The mapping preserves collinearity and concurrence, i.e., collinear points get mapped to
collinear points and concurrent lines get mapped to concurrent lines.
• Given a point A and its image A0 , the line AA0 goes through a point O, called the center.
• Given a line a and its image a0 , the intersection point of lines a and a0 lies on line o, called
the axis.
Theorem: A perspective collineation is determined by three pairs of homologous points, as long
as the three points are non-collinear.
Given this definition, we can prove other properties, like a perspective collineation always has a
linewise fixed point, the center, and a pointwise fixed line, the axis.
6. Since Desargues’ theorem is involved, perhaps we can determine the mapping with a different
set of information. Verify the following theorem by mapping D to its image D0 .
Theorem: A perspective collineation is determined by a center, axis and one pair of homologous points.
3