b2

b1

O

P

a2

Y

b1

⇡

`a

Z

↵

a1

Figure 3.0: Linear Perspective set up from [T]; three vertical planes ↵, , and ⇡; Shadow of a Tree by Federal Art Project photographer George Herlick [H]; a camera obscura as illustrated in the Universal Magazine [U]. [For use with the What’s the image of a line? module.]

X

a1

`b

Chapter 3 What’s the image of a line? Overview: When we taped windows, we had an artist, and then a picture plane, and then the object, in that order. But that configuration is not the only possible way to project the image of an object onto a plane. Figure 3.0 shows that there are many possible kinds of projections. In this module, we’ll explore many possible notions of “projection onto a plane”. Our goal will be to try to create a more general mathematical model for projection that might include any or all of these types of projections.

Let us think about possible ways of projecting the images of points and lines onto a plane. In every projection we will consider, there are • an object being projected, • an image of the projection, • and a point that we will call the center from which the object and image line up with each other. 1. The top left drawing of an artist drawing a cube in Figure 3.0 is from Brook Taylor’s New Principles of Linear Perspective [T], published in 1715. (Some math students will have seen a calculus concept called “Taylor series”; this is the same Taylor). In this drawing, we see a projection like the kind we did ourselves in class. In this drawing, what is the object? What is the image? What is the center of the projection? 2. The top right photograph taken by George Herlick in 1937 shows, among other things, the shadow of a tree [H]. If we think of a shadow as a projection, then what is the object of the projection? What is the image? What is the center of the projection? 3. The lower right drawing in Figure 3.0 shows a camera obsucura—translated literally as “dark room”. In this drawing, the projection is caused by light rays passing from outside of the room and being cast upon a wall inside the room. In this drawing, what is the object? What is the image? What is the center of the projection?

29

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CHAPTER 3. WHAT’S THE IMAGE OF A LINE?

In this class (as elsewhere in Projective Geometry), we will 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 denote the center of the projection and ⇡ denote the picture plane. For any object X, we will let X ′ be the projection via O onto the picture plane ⇡.

⇡

⇡

O P′

O P

P′

P

Figure 3.1: 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, ⇡ is the picture plane, and the point P projects to its image P ′ . We should be cautious here. We have not yet defined exactly what we mean by “the projection via O onto the picture plane ⇡”, and (as you will see below) there are several di↵erent things we might mean by this phrase. For now we’ll leave the meaning ambiguous. 4. In Figure 3.1, we see that the image of a point could be another point. That is, if P is a point, P ′ could be a point. (a) Could we have P ′ = �? (The symbol � is the empty set, the set that contains nothing.)

(b) Could P ′ be a line?

(c) Could P ′ be a plane?

For each of your answers above, draw a top view and side view explaining your reasoning.

31 5. We know that the image of a line in R3 could be another line. The physical world does not always correspond to the abstract mathematical setting, however. Draw top and side views for a geometrical setting, or give physical examples from a real-world setting, to show whether the image `′ could take the following forms: (a) �;

(b) a point;

(c) a line segment;

(d) a ray;

(e) a line with one point missing;

(f) a line with two points missing; or

(g) an ellipse.

Eventually, for reasons that will make our mathematical lives easier, we will want to be able to assume that the image of a point is always a point and that the image of a line is always a line. That is, eventually that we will need a way to rule out any of the counter-examples you might have discovered above. But that will be the topic of a future module.

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CHAPTER 3. WHAT’S THE IMAGE OF A LINE?

In the meanwhile, here are some questions that might help you visualize projections in three-dimensional space. 6. Consider Figure 3.2, which shows two intersecting vertical planes ↵ and , which meet a vertical picture plane ⇡ in parallel lines `a and `b , respectively. From the center O, we project objects in R3 to their images on ⇡. (a) The line a2 (containing the points X, Y , and Z) lies in plane ↵. Locate the perspective images X ′ , Y ′ , Z ′ of points X, Y , and Z, respectively. (Hint: To locate X ′ you may have to extend a line of the drawing.) (b) What is the perspective image of the entire line a2 ? (c) The line b2 lies in plane . What is the image of the entire line b2 ? (d) Are a2 and b2 parallel? Do they intersect? How can you tell? (e) What is the image of a1 ? What is the image of b1 ? (f) The two lines a1 and b1 meet at point P . What is the relationship of line OP to the picture plane ⇡? (g) If a line lies in plane ↵ and does not pass through O, its perspective image is lies in plane and does not pass through O, its perspective image is .

`b

a1

b1

P

`a

b1

a1 a2

X

Z

Y

b2 ↵

O

⇡

Figure 3.2: Three planes, with lines and points, for question 6.

. If a line

33 P

S O

T

8

Perspective Geometry

Figure 3.3: An aerial view of an outpost.

HOMEWORK ○ 3.1. Figure 3.3 shows an aeral view of a fenced-in area. Around this area there are low stone walls, and Questions at each of the four corners is a flag on a tall post. Each flag has one of four letters (P , O, S, or T ).

1. Consider showsthe twostone sketches of hallways. For each Figure 3.4 shows how a person standing atFigure point5,Xwhich outside walls would read the flags, sketch, determine whether the artist was sitting or standing, from left to right, as “STOP”, whereas a person standing at point Y would read the flags and from left explain how to right as “PSOT”. What other “words” canyou weknow. read by standing in di↵erent locations? Can we

X

P Y S

O

T

Figure 3.4: From X, we read “STOP”; from Y , we read “PSOT”. find a location where we can see . . . (a) POST? Figure 5: Two sketches of hall(b) OPTS?

ways for homework question 1.

(c) POTS? 2. Suppose a red door and a blue door, two doors along the same wall, are exactly X units apart. Both Ryan and Barb are on the other side Draw a map that shows regions from which various “words” are visible to an onlookers1 . of a window, looking down the hallway at these doors, but Ryan is X units further from the window than Barb. That is, the order is

(d) TOPS?

Proof/Counter-example 1

Ryan - Barb - window - Red - Blue.

This problem is adapted from one that appeared in the problem section of The Emissary [MSRI].

Each of the two artists creates a picture of the doors on the window as s/he sees them. Is the image of the red door (according to Ryan) the same size as the image of the blue door (according to Barb)? Use a plan view to answer this question. [This question was suggested by

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CHAPTER 3. WHAT’S THE IMAGE OF A LINE?

� 3.1. An artist stands exactly on top of the intersection of a pair of perpendicular, horizontal lines. (Perhaps one is the east-west track for a trolley, and another is the the north-south track of a trolley). That is, the artist’s feet are on these lines, and her head is directly above them. The artist then sets up a large, vertical canvas in front of her so that it crosses both lines (perhaps it runs southwest to northeast). Describe the images of these two lines. Prove that your description is correct by drawing a top view and side view and referring to those diagrams in your explanation.

This work was supported by NSF TUES Grant DUE-1140135. “What’s the image of a line?”, Perspective Geometry Modules, Crannell, Frantz, and Futamura, available via www.fandm.edu/annalisa-crannell, June 13, 2017 version.