Perspective Homework Problems

Transcription

1 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 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? 1: 3D Coordinates

2 Name: Page 2 P3. The rectangle ABCD is parallel to the yz-plane, so that all of its x-coordinates are Day 2 the 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. y A B E z C D (c) Are the x -coordinates of A, B, C, and D the same or different? If they are different, 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 are different, 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 are different, list them in order from smallest z -coordinate to largest.

3 Name: Page 3 P4. Think of the image below, from the Pixar movie Up, as being painted on the picture plane, with the real characters and landscape behind it. Let C denote the top of Carl s real 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 which is 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

4 Name: Page 4 P5. Assume our usual setup for perspective drawing, with a viewing distance of 5. (a) Suppose a fly is located at F 1 = (2, 3, 5). What are the values of the x - and y -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 F 2 = (2, 3, 95). What are x and y now? (c) Suppose the fly now flies to F 3 = (2, 3, 995). What are x and y? (d) Draw one Top View and one Side View of this situation, including the points F 1, F 2, and F 3 from parts (a) (c), along with their images F 1, F 2, and F 3 in the picture plane. Also draw light rays to the viewer s eye. Your drawings need not be to scale. (e) If P = (x, y, z) is a point, and the x and y do not change but the z gets bigger and bigger, what happens to the picture plane image P of P? (f) Our everyday experience tells us that objects appear smaller as they get farther away. Explain how this is related to your answers to parts (a) (e).

5 Name: Page 5 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 is symmetric. Next, consider the following Top View and Side View: G Top View E x 18 z y Side View E A( 12, 18, 81) C( 12, 3, 99) B( 12, 6, 93) F D( 12, 18, 117) z (a) What are the coordinates of point F? (b) What are the coordinates of point G? (c) How high is the dollhouse?

6 Name: Page 6 (d) The coordinates of all vertices of the house are located in the table below. The viewer s eye is at (0, 0, 15), so the viewing distance d = 15. Use Excel or another spreadsheet program to calculate the x - and y -coordinates for each vertex. Use the scatterplot function to display the images; print out the scatterplot and connect the dots with a pencil to draw the house. x y z 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. The bottom 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, and finally connect the dots to draw the picture of the house with the new details as in the previous exercise.

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

8 Name: Page 8 P9. Consider the box below. Its front face is a square below, but the top face is twice as long as it is wide from left to right. Thus the viewing distance is not the distance between V and V. What is the viewing distance? Explain why. V V length width

9 Name: Page 9 P10. Now let s draw a box that is not a cube. Its front face is the square below, but the top face is three times as long as it is wide from left to right. The viewing distance is 9 inches, and the vanishing point is V. Draw the box on the diagram below. V

10 Name: Page 10 P11. Photographs of the Moon are usually shot with telephoto lenses to make the Moon seem dramatically large. However, when we see the Moon in ordinary photographs, it appears quite small. To see why, suppose you want to make a drawing of the Moon rising over the plains, with a viewing distance of two feet. What should the diameter of your moon image be? (You will have to consult astronomical sources to find the relevant numbers. Be sure to show your work.) P12. On another sheet of paper, make a sketch of a street, alleyway, hallway, walkway, or cityscape, 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.)

11 Name: Page 11 P13. Draw an 8 8 chessboard in one-point perspective. Turn in your answer on a separate Day 4 piece of paper. P14. The box below is a package that needs to be wrapped up with string, like in this picture: Draw the string around the package, so that each face is divided by the string into four equal rectangles. Observe that the package has two vanishing points, not just one.

12 Name: Page 12 P15. Here is a picture of a road stretching off into the distance. I have started to draw a sidewalk next to the road. Finish drawing the rest of the sidewalk tiles, until they get too small for you to continue.

13 Name: Page 13 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 equal sections.

14 Name: Page 14 (b) Draw 2 more fenceposts inside the fence panel to divide the panel into 3 equal sections. (This is harder, but would be necessary to draw the Italian or Mexican flag in perspective.)

15 Name: Page 15 (c) Draw 4 more fenceposts inside the fence panel to divide the panel into 5 equal sections, without any measuring. (XC)

16 Name: Page 16 P17. On the blank billboard next to the highway in the picture below, draw a word of at least four block letters in true one-point perspective, as in the sample billboard shown. Day 5 All your letters should appear to be the same height and width.

17 Name: Page 17 P18. Here is a picture of a cube. Make it into a Rubik s Cube by dividing each side into 9 equal squares.

18 Name: Page 18 P19. Here is a picture of a square lying on the ground in 2-point perspective. The viewing target is T, and the viewing distance is the distance between T and U. Draw a cube that has this square as its bottom. Be sure to use the techniques we discussed to make it the right height! V 1 T V 2 U

19 Name: Page 19 P20. Below is the start of a drawing of a child s backyard playhouse. Add another window Day 6 to 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 is separated from the front wall.