Math 113 Linear Perspective Homework
1. As mentioned in the notes, all points in the xy -coordinate plane have z -value equal to 0. In fact, z = 0
is the equation of the xy -plane.
a) What is the equation of the xz -coordinate plane?
b) What is the equation of the yz -coordinate plane?
2. In each of parts a), b), and c), we consider a rectangular box with sides parallel to the coordinate
planes. Some of the coordinates of the eight corners of the box are given; find the missing coordinates.
a) (1, 1, 1), (1, 1, 5), (4, 1, 5), (4, 1, 1), (4, 7, 1), (4, 7, 5), (1, 7, 5), (?, ?, ?)
b) (1, 2, 3), (2, 3, 4), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?)
c) (1, 1, 1), (3, 4, 5), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?)
d) Which of the above boxes is a cube? How big is it?
3. As a prelude to the Perspective Theorem, suppose the art director was at a viewing distance, d = 3,
from the window (xy -plane); and suppose that the point P (x, y, z) in the real world had coordinates
x = 0, y = 4, z = 5.
a) Which coordinate plane would P (x, y, z) lie in?
b) What would x0 of its projection P 0 (x0 , y 0 , z 0 ) onto the window be? (Easy!)
c) What would y 0 be?
1
4. The rectangle ABCD in the figure below is parallel to the yz-plane, so that all of its x-coordinates are
the same. The y -coordinates of A and B are the same; and the y -coordinates of C and D are also equal.
With the viewer located as shown, what can we say about the x0 , y 0 , z 0 -coordinates of the image points
A0 , B 0 , C 0 , and D0 ? [This problem is harder than it looks; you need to consider each coordinate x0 , y 0 , z 0
individually. In particular, for the x0 -coordinate there are three different cases.]
y
A
B
z
B
C
D
↑
edge of picture plane
5. Think of the Jurassic Park image below as being the view out of a window. Let P (x, y, z) be the
actual top of the womans head (person in middle), and let Q(x, y, z) be the actual top of the head of
the dinosaur drinking water. The points P 0 and Q0 are the perspective images (on the window) of these
points.
Which is bigger?
a) the x-coordinate of P , or the x-coordinate of Q?
b) the y -coordinate of P , or the y -coordinate of Q?
c) the z -coordinate of P , or the z -coordinate of Q?
d) the x-coordinate of P 0 , or the x-coordinate of Q0 ?
e) the y -coordinate of P 0 , or the y -coordinate of Q0 ?
f) the z -coordinate of P 0 , or the z -coordinate of Q0 ?
2
6. This exercise deals with a point P whose x- and y coordinates do not change (they are equal to 2
and 3, respectively), but whose z -coordinate gets bigger and bigger. That is, the point moves farther and
farther away from the picture plane and the viewer.
(a) Referring to Perspective Theorem, suppose the viewing distance d is 5 units, and suppose P =
(2, 3, 5). What are the values of x0 and y 0 ?
(b) Now suppose P = (2, 3, 95). What are x0 and y 0 ?
(c) Suppose P = (2, 3, 995). What are x0 and y 0 ?
(d) Draw one TOP VIEW and one SIDE VIEW, as we did in class. Include in each VIEW all the points P
and P from Parts (a)(c). Then draw the light rays from the viewers eye to each point (the drawings need
not be to scale). Describe whats happening to the points P 0 .
(e) Consider a point P (x, y, z). If x and y do not change, but z gets larger and larger, where does the
picture plane image P 0 of P go?
(f) Our everyday experience tells us that objects appear to get smaller as they move away from the viewer.
How do Parts (a)(e) explain this?
7. Our first perspective drawing will be of a box with its faces parallel to the coordinate planes, and a
viewing distance of d = 15 units.
(a) If two corners of the box have coordinates (−4, −2, 6) and (−6, −4, 3), then:
i. How wide is the box in the x-direction?
ii. How high is the box in the y -direction?
iii. How deep is the box in the z -direction?
(b) List the coordinates of the other 6 corners.
(c) Make a table listing all 8 corners and their x0 - and y 0 -coordinates.
(d) Plot the (x0 , y 0 ) in (c) in the xy -plane and connect the dots with straight lines to obtain the perspective image of the box. (Use dashed lines to indicate hidden edges. Note that all of the image points
have negative coordinates.)
3
8. In this more involved problem of drawing a dollhouse, we’ll use a spreadsheet program to make the
work easier. Below are a SIDE VIEW and a TOP VIEW of the situation; with this information (and a little
thought), you should be able to find all the coordinates of every defining corner of the dollhouse. Assume
that the viewer’s eye is at (0, 0, −15).
y
SIDE VIEW
C(−12, 3, 99)
z
B
F
B(−12, −6, 93)
A(−12, −18, 81)
↑
D(−12, −18, 117)
edge of picture plane
↓
TOP VIEW
G
18
F
Q
z
x
(a) What are the coordinates of the point F ? What are the coordinates of the point G? If the measurements are in inches, how high is the dollhouse?
(b) Find all seventeen corners of the house. [Hint: for x = −12, there are seven corners.]
(c) Using a spreadsheet program as we showed in class, calculate the (x0 , y 0 ) for each of the seventeen
points in (b). Then print out both the table of values and a scatterplot chart of these results with the dots
carefully connected.
4
9. The house you’ve made is a bit featureless, so your job in this problem is to add to the x, y , and z
columns of your Excel spreadsheet the correct 3-space coordinates for the vertices of the following items:
(a) two or more windows on the near wall;
(b) a door on the right-hand wall, centered under the dormer;
(c) a small, rectangular yard for the house;
(d) a chimney somewhere on the roof, at least partially visible to the viewer. The bottom vertices of
the chimney should lie on the roof, not above or below it. To complete the exercise, compute the new x0
and y 0 values, and print out the picture of the house as you did in the previous problem.
5