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Name: Page 1 Perspective Homework Problems Exploring Mathematics, Spring 2011 Unless the problem specifically says otherwise, please complete your work on these pages and turn it in. P1. In each of parts (a), (b), and (c), we are considering a rectangular box with its faces Day 1: 3D Coordi- nates parallel to the coordinate planes. Some of the coordinates of the eight corners (A, B, C , D, E, F , G, and H ) of the box are given; your job is to fill in the rest. (a) A = (1, 1, 1), B = (1, 1, 5), C = (4, 1, 1), D = (4, 1, 5), E = (4, 7, 1), F = (4, 7, 5), G = (1, 7, 1), H =( , , ) (b) A = (1, 2, 3), B = (2, 3, 4), C =( , , ), D =( , , ), E =( , , ), F =( , , ), G =( , , ), H =( , , ) (c) A = (1, 1, 1), B = (3, 4, 5), C =( , , ), D =( , , ), E =( , , ), F =( , , ), G =( , , ), H =( , , ) (d) Which of the boxes in parts (a), (b), and (c) is a cube? How long is each side of the cube? P2. A pyramid has a square base whose corners are the four points (1, 1, 4), (5, 1, 4), (1, 1, 9), and (5, 1, 9). The height of the pyramid is 3. What are the coordinates of the apex (highest point) of the pyramid?

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Page 1: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

Name: Page 1

Perspective Homework ProblemsExploring Mathematics, Spring 2011

Unless the problem specifically says otherwise, please complete your work on these pagesand turn it in.

P1. In each of parts (a), (b), and (c), we are considering a rectangular box with its faces Day1: 3DCoordi-nates

parallel to the coordinate planes. Some of the coordinates of the eight corners (A, B,C, D, E, F , G, and H) of the box are given; your job is to fill in the rest.

(a) A = (1, 1, 1),B = (1, 1, 5),C = (4, 1, 1),D = (4, 1, 5),E = (4, 7, 1),F = (4, 7, 5),G = (1, 7, 1),

H = ( , , )

(b) A = (1, 2, 3),B = (2, 3, 4),

C = ( , , ),

D = ( , , ),

E = ( , , ),

F = ( , , ),

G = ( , , ),

H = ( , , )

(c) A = (1, 1, 1),B = (3, 4, 5),

C = ( , , ),

D = ( , , ),

E = ( , , ),

F = ( , , ),

G = ( , , ),

H = ( , , )

(d) Which of the boxes in parts (a), (b),and (c) is a cube? How long is eachside of the cube?

P2. A pyramid has a square base whose corners are the four points (1, 1, 4), (5, 1, 4), (1, 1, 9),and (5, 1, 9). The height of the pyramid is 3. What are the coordinates of the apex(highest point) of the pyramid?

Page 2: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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P3. The rectangle ABCD is parallel to the yz-plane, so that all of its x-coordinates are Day 2the same positive number. Also, the y-coordinates of A and B are the same.

(a) Draw the Top View of the picture.

(b) On the Side View below, draw the lines of sight and draw the images A′, B′,C ′, and D′.

E

BA

C D

y

z

(c) Are the x′-coordinates of A′, B′, C ′, and D′ the same or different? If they aredifferent, list the points in order from smallest x′-coordinate to largest.

(d) Are the y′-coordinates of A′, B′, C ′, and D′ the same or different? If they aredifferent, list them in order from smallest y′-coordinate to largest.

(e) Are the z′-coordinates of A′, B′, C ′, and D′ the same or different? If they aredifferent, list them in order from smallest z′-coordinate to largest.

Page 3: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

Name: Page 3

P4. Think of the image below, from the Pixar movie Up, as being painted on the pictureplane, with the real characters and landscape behind it. Let C denote the top of Carl’sreal head, and let R denote the top of the real rock formation. The points C ′ and R′

are the images of C and R in the picture plane. For each of the following, circle whichis the larger number:

(a) the x-coordinate of C the x-coordinate of R they’re the same

(b) the y-coordinate of C the y-coordinate of R they’re the same

(c) the z-coordinate of C the z-coordinate of R they’re the same

(d) the x′-coordinate of C ′ the x′-coordinate of R′ they’re the same

(e) the y′-coordinate of C ′ the y′-coordinate of R′ they’re the same

(f) the z′-coordinate of C ′ the z′-coordinate of R′ they’re the same

C

R

Page 4: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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P5. Assume our usual setup for perspective drawing, with a viewing distance of 5.

(a) Suppose a fly is located at F1 = (2, 3, 5). What are the values of the x′- andy′-coordinates of the image F ′ of the fly on the picture plane?

(b) Now suppose the fly flies away from the viewer to the point F2 = (2, 3, 95). Whatare x′ and y′ now?

(c) Suppose the fly now flies to F3 = (2, 3, 995). What are x′ and y′?

(d) Draw one Top View and one Side View of this situation, including the pointsF1, F2, and F3 from parts (a)–(c), along with their images F ′

1, F′2, and F ′

3 in thepicture plane. Also draw light rays to the viewer’s eye. Your drawings need notbe to scale.

(e) If P = (x, y, z) is a point, and the x and y do not change but the z gets biggerand bigger, what happens to the picture plane image P ′ of P?

(f) Our everyday experience tells us that objects appear smaller as they get fartheraway. Explain how this is related to your answers to parts (a)–(e).

Page 5: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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P6. In this problem, we will use a computer to help draw a dollhouse in perspective.First, look at the dollhouse below to understand the general shape. The dollhouse issymmetric.

Next, consider the following Top View and Side View:

TopView z

x

E

G

18

SideView z

y

E

C(−12, 3, 99)

F

D(−12,−18, 117)A(−12,−18, 81)

B(−12,−6, 93)

(a) What are the coordinates of point F?

(b) What are the coordinates of point G?

(c) How high is the dollhouse?

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(d) The coordinates of all vertices of the house are located in the table below. Theviewer’s eye is at (0, 0,−15), so the viewing distance d = 15. Use Excel or an-other spreadsheet program to calculate the x′- and y′-coordinates for each vertex.Use the scatterplot function to display the images; print out the scatterplot andconnect the dots with a pencil to draw the house.

x y z−12 −18 81−30 −18 81−30 −6 81−21 3 81−12 −6 81−12 −18 117−30 −18 117−30 −6 117−21 3 117−12 −6 117−12 −6 93−30 −6 93−30 3 99−21 3 99−12 3 99−12 −6 105−30 −6 105

Your final picture should look something like this:

P7. Improve the house from the previous exercise by adding correct x, y, and z coordinates (XC)

to your Excel spreadsheet 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. Thebottom vertices of the chimney should lie on the roof, not above or below it.

Then use Excel to compute the new x′ and y′ values, make a scatterplot, print it, andfinally connect the dots to draw the picture of the house with the new details as in theprevious exercise.

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P8. Below, a start has been made on drawing a cube in one-point perspective. The front Day 3face is a square and V is the vanishing point. The viewing distance should be 7 inches.Finish the drawing of the cube.

V

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P9. Consider the box below. Its front face is a square below, but the top face is twice aslong as it is wide from left to right. Thus the viewing distance is not the distancebetween V ′ and V . What is the viewing distance? Explain why.

V ′ V

width

length

Page 9: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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P10. Now let’s draw a box that is not a cube. Its front face is the square below, but thetop face is three times as long as it is wide from left to right. The viewing distance is9 inches, and the vanishing point is V . Draw the box on the diagram below.

V

Page 10: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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P11. Photographs of the Moon are usually shot with telephoto lenses to make the Moonseem dramatically large. However, when we see the Moon in ordinary photographs, itappears quite small. To see why, suppose you want to make a drawing of the Moonrising over the plains, with a viewing distance of two feet. What should the diameterof your moon image be? (You will have to consult astronomical sources to find therelevant numbers. Be sure to show your work.)

P12. On another sheet of paper, make a sketch of a street, alleyway, hallway, walkway, orcityscape, using one or more vanishing points to help make your drawing look realistic.(It will help to pick a view that includes lots of rectangular objects.)

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P13. Draw an 8× 8 chessboard in one-point perspective. Turn in your answer on a separate Day 4piece of paper.

P14. The box below is a package that needs to be wrapped up with string, like in thispicture:

Draw the string around the package, so that each face is divided by the string into fourequal rectangles. Observe that the package has two vanishing points, not just one.

Page 12: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

Name: Page 12

P15. Here is a picture of a road stretching off into the distance. I have started to draw asidewalk next to the road. Finish drawing the rest of the sidewalk tiles, until they gettoo small for you to continue.

Page 13: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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P16. Solve the following drawing problems about a fence panel.

(a) Draw 7 more fenceposts inside the fence panel to divide the panel into 8 equalsections.

Page 14: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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(b) Draw 2 more fenceposts inside the fence panel to divide the panel into 3 equalsections. (This is harder, but would be necessary to draw the Italian or Mexicanflag in perspective.)

Page 15: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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(c) Draw 4 more fenceposts inside the fence panel to divide the panel into 5 equalsections, without any measuring. (XC)

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P17. On the blank billboard next to the highway in the picture below, draw a word of atleast four block letters in true one-point perspective, as in the sample billboard shown. Day 5All your letters should appear to be the same height and width.

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P18. Here is a picture of a cube. Make it into a Rubik’s Cube by dividing each side into 9equal squares.

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P19. Here is a picture of a square lying on the ground in 2-point perspective. The viewingtarget is T , and the viewing distance is the distance between T and U . Draw a cubethat has this square as its bottom. Be sure to use the techniques we discussed to makeit the right height!

V1 V2T

U

Page 19: Perspective Homework Problems - Concordia Collegefaculty.cord.edu/ahendric/2011Spring105/PerspectiveHomework.pdf · Perspective Homework Problems Exploring Mathematics, Spring 2011

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P20. Below is the start of a drawing of a child’s backyard playhouse. Add another window Day 6to the right-hand wall. The new window should represent one in the real world that is

(a) the same size and shape as the existing window, AND

(b) located at the same height as the existing window, AND

(c) separated from the back wall by the same distance that the existing window isseparated from the front wall.