Download Image of a line

Image of a line
A reason to expand our horizons
In this class (as elsewhere in Projective Geometry), we use the convention of naming points with
italicized capital letters, lines with italicized lower-case letters, and planes with lower-case Greek
letters. Unless we state otherwise, when we project objects onto a plane we will let O stand for the
center of the projection and π 0 to denote the picture plane. (We often think of O as the viewer’s
eye; as Figure 1 shows, there are other physical interpretations of O.) For any object X, we’ll let
X 0 be the projection via O onto the picture plane π 0 .
π0
π0
O
P0
O
P0
P
P
Figure 1: Two versions of a projection: on the left, as with an artist looking though a window; on
the right, as with a camera projecting onto film. In each version, O is the center of the projection,
π 0 is the picture plane, and the point P projects to its image P 0 .
We should be cautious here. We have not yet defined exactly what we mean by “the projection
via O onto the picture plane π 0 ”, and (as you will see below) there are several different things we
might mean by this phrase. For now we’ll leave the meaning ambiguous.
1. In Figure 1, we see that it is possible for the image of a point to be another point. That is,
if P is a point, P 0 could be a point.
(a) Is it possible that P 0 = ∅? Draw a plan view explaining your reasoning.
(b) Is it possible that P 0 is a line? Again, draw a plan view.
2. Now consider a line `. Draw plan views to show whether it is possible for the image `0 to take
the following forms:
(a) ∅;
(b) a point;
(c) a line segment;
(d) a ray;
(e) a line;
(f) a line with one point missing;
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(g) a line with two points missing; or
(h) an ellipse.
Homework
Questions
1. Consider Figure 2, which shows a vertical picture plane π and two intersecting vertical planes
α and β, which meet π in parallel lines `1 and `2 , respectively.
(a) The blue line lies in plane α. Locate the perspective images X 0 , Y 0 , Z 0 of points X, Y ,
and Z, respectively. (Hint: To locate Z 0 you may have to extend a line of the drawing.)
(b) What is the perspective image of the entire blue line?
(c) The green lies in plane β. What is the image of the entire green line?
(d) Are the blue line and the green line parallel? Do they intersect? How can you tell?
(e) The two red lines meet at point P . What is the relationship of line OP to the picture
plane π?
(f) If a line lies in plane α and does not pass through O, its perspective image is
line lies in plane β and does not pass through O, its perspective image is
.
Figure 2: Three planes, with lines and points, for homework question 1.
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. If a
Art Assignment
A2. Find a rectangular solid in your environment (for example, a tissue box or a bookshelf). You
should take three photographs of this object, each showing three faces of the object.
• One photograph should be in 1-point perspective, meaning one face of the object should
be parallel to the picture plane and one set of edges of your object will appear to converge
to a vanishing point1 .
• A second photograph should be in 2-point perspective, meaning one edge of the object
should be parallel to the picture plane and two sets of edges will appear to converge to
vanishing points.
• The third photograph should be in 3-point perspective, meaning no edge of the object
should be parallel to the picture plane.
Print out these pictures and draw on top of them. Extend the edges of your object and
determine the location of the vanishing point(s). For the 2- and 3-point perspective pictures,
you may need to shrink the picture or tape on extra paper, because the vanishing points could
possibly be off the edges of the picture.
Proof/Counterexample
1. none.
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Hint: The vanishing point should be straight ahead, so if your photograph is un-cropped, the vanishing point
will be exactly in the middle of the picture.
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Instructors’ notes for “Image of a Line”
Materials needed:
• pencil
• straightedge (optional)
• [instructor only] drafting tape, if you’re going to display students’ hallway sketches.
Lessons of the worksheet:
• notation for points, lines, planes, and their projected images
• images of lines in a picture plane
• importance of considering “special cases” and careful definitions
• the difference between artistic applications and mathematical definitions
The main point of this worksheet is to begin to introduce mathematical notation and rigor into the
definitions of “image”.
A sneakier goal is to get the students to the point of begging you to allow them to use the notion
of infinity to make their mathematical lives easier. (Usually, projective geometry texts just declare
the existence of “points at infinity”, leaving most of the students twitchy. We have discovered that
our approach lays the groundwork for a greater acceptance, even a sense of welcome relief, that
such points get to join the party.)
Questions 1 and 2 on the worksheet are designed to get the students thinking about various scenarios. I’ve found that I occasionally have to play a little bit of “what if?” myself, but that students
get quite animated and come up with many possibilities on their own.
They will probably realize that very bad things happen if P = O or if O ∈ π 0 ; likewise they will
probably discover on their own that the empty set is a possibility if either P or ` lies in the plane
parallel to π 0 containing O.
Many of the possibilities depend on what we mean by “projection”—for example, if we think of
the picture plane as a finite artist’s canvas (as makes sense in real-world applications), then it is
possible for `0 to be a line segment. As another example, for a mathematical O ⊂ R3 , it is possible
that `0 could be a line with one point missing, but in the physical world, it is hard to imagine
a scenario where O could act both like a viewer and a camera. It is worth exploring the various
possibilities in order to convince students that definitions and axioms matter.
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