Mapping L in two-point perspective
Recall the “T” from the Drawing ART exercise. This is a nice way of introducing a class of
mappings from a plane to a plane (possibly the same plane) that preserve certain properties, like
collinear points mapping to collinear points and the intersection point of two lines mapping to
the intersection point of the mapped lines. We’ll investigate this more in depth in this worksheet.
Instead of “T”, let’s consider the simpler 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. Next we choose a random point D on the front face of the L, and challenge the reader to
locate its image D0 on the back face, using a particular method. In Figure 2 we keep only the
vertices A, B, C and A0 , B 0 , C 0 of the L, and locate D as shown.
In terms of perspective drawing, O is the vanishing point of lines AA0 , BB 0 , and CC 0 . In
perspective drawing, the vertical line o is called the vanishing line of plane ABC and also of
plane A0 B 0 C 0 . The vanishing line of a plane is the image of its line at infinity in space. We
assume these planes to be parallel in the real world, so o is the image of their common line
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at infinity.
The point U is the vanishing point of lines AC and A0 C 0 (which represent parallel lines in
space in space), and similarly W is the vanishing point of lines BC and B 0 C 0 , and the vertical
lines AB and A0 B 0 also meet on o at an ideal point (not shown) Because U , V , and W belong
to both planes ABC and A0 B 0 C 0 , we write U = U 0 , V = V 0 , and W = W 0 . Through the
point D we extend the line CD to its vanishing point V ; we also draw V C 0 . Lines CD and
V C 0 represent lines that are parallel in the real world; CD lies in the plane ABC (the front
plane of the L) and V C 0 lies in the plane A0 B 0 C 0 (the back plane of the L). Now here is the
challenge: Locate D0 by drawing just one line in Figure 2.
U(= U’)
o
V(= V’)
A
A’
W(= W’)
D
O
B’
B
C’
C
Figure 2: Sparser image of the L in two point perspective, with a random point D.
3. In the previous problem, did we really need to be given O and o, or can we construct what
we need starting with just A, B, C, A0 , B 0 , C 0 , and D?
4. In Figure 2, what is the relationship of triangle ABC, triangle A0 B 0 C 0 , the point O, and the
line o in the language of Desargues’ theorem?
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Observe that we can think of the the mappings A 7→ A0 , B 7→ B 0 , C 7→ C 0 , D 7→ D0 , as taking place
in the plane of the paper. In fact, by adding some natural requirements, the three point mappings
A 7→ A0 , B 7→ B 0 , C 7→ C 0 determine such a mapping that is also consistent with the mapping
D 7→ D0 .
Definition: A collineation is a mapping from one plane to another, or from a plane to itself, that
bijectively maps points to points and lines to lines, and preserves incidence. (For example, if a
point such as A in Figure 2 belongs to a line such as U C, then the image A0 of A belongs to the
image U 0 C 0 of U C.)
Definition: We say that an ordered pair of distinct points is homologous (with respect to a
particular collineation) if the collineation maps the first point to the second point.
Definition: A perspective collineation is a collineation that has the following properties.
• There exists a point O called the center , such that any point A and its image A0 are collinear
with O. Each line through O is therefore fixed.
• There exists a line o called the axis, such that any line a and its image a0 are concurrent with
o. Each point on o is therefore fixed.
Theorem: A perspective collineation is determined by three pairs of homologous points, as long as
the three points are non-collinear.
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 in Figure 3.
Theorem: A perspective collineation is determined by a center, an axis, and one pair of homologous
points.
Figure 3: Figure for Problem 6.
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