Computer Lab One Taxicab Geometry

Transcription

1 Computer Lab One Taxicab Geometry MA 408, Spring 2011 Will be discussed on January 27, and needs to be submitted on February 2. Introduction and objective: Loosely speaking, two dimensional geometry can be thought of as studying the relationship between the geometric objects occurring in a given model of geometry. The geometry will be determined by how the different geometric objects relate to each other. The geometric objects we will be most concerned with include points, lines, distance measure ( i.e., distance between points), and angle measure. When one of the geometric objects changes, the entire geometry can change as a result. In this lab, we will investigate the geometry of R 2 with a new distance measure, the taxicab metric. Specifically, we will consider the points of R 2, the usual lines of Euclidean geometry in R 2, and the usual angle measure of Euclidean geometry in R 2 all subject to a new distance measure (i.e., a metric) between points. The goal is to see how altering the distance measure between points will change the usual Euclidean geometry in R 2. 1

2 Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by the taxicab metric. The taxicab distance between two points is the sum of the (absolute) differences of their coordinates. The taxicab metric is sometimes referred to as the Manhattan metric or the L 1 -distance. For a given point P 1 (in the plane) with coordinates (x 1, y 1 ) and the point P 2 at (x 2, y 2 ), the taxicab distance is defined as: ρ(p 1, P 2 ) = x 2 x 1 + y 2 y 1. P 1 = (x 1, y 1 ) C = (x 1, y 2 ) P 2 = (x 2, y 2 ) Problem 1. a.) What is the distance between P 1 and P 2 in Euclidean geometry? b.) In general, which distance function is greater? That is, given two points, is the Euclidean or Taxicab distance between the points greater? Under what circumstances will the Euclidean distance between to points be equal to the Taxicab distance between two points? 2

3 1 Exploring Distance 1. Open the program Geometer s Sketchpad and maximize the screen. Open the file Taxicab Lab.gsp. Maximize this file. You should have a blank screen with four tabs at the bottom. Click on the tab labeled Task 1 (it should be the left most tab). We will need to see the coordinate grid, so from the Graph menu, click on Show Grid. There are two points visible on the grid a point at the origin and and a point at (1, 0). PLEASE DO NOT MOVE THESE POINTS. Moving these points would change the scaling. 2. To plot two points, select the Graph menu and then Plot Points. Now enter the coordinates ( 5, 3) and click Plot. The window will stay open. Plot the point (6, 7), and then click on Done. Two points should appear on the grid. On the left hand toolbar, click on the Text Tool. Now clicking on each point will label them. We want to measure the distance between these two points. Click on the arrow on the left hand toolbar. Highlight each point by clicking on them. The points will turn pink when you highlight them. Select the Measure menu and then select Distance. The distance U V should appear on the screen. This is the Euclidean distance. Problem 2. Record the Euclidean distance: 1. To find the taxicab distance, hold down the Custom Tool button on the left hand toolbar and click on the Taxicab tool. Click on your first point and then drag to your second. The taxicab distance should appear. Problem 3. Record the taxicab distance: (b) Using the taxicab tool, create two other points that have the same taxicab distance from point U. Problem What are the coordinates of point 1? 2. What are the coordinates of point 2? 2

4 2 Exploring Circles A circle is the set of points equidistant from a special point (the center). The Euclidean metric is d((x 1, y 1 ), (x 2, y 2 )) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. The Euclidean circle of radius two centered at (0, 0) is shown below. Points (x, y) on this circle satisfy d((0, 0), (x, y)) = (x 0) 2 + (y 0) 2 = 2. We can simplify the above equations (how?) to obtain a familiar equation: x 2 + y 2 = 4. Problem 5. Which of the points (0, 2), ( 2, 2), (1, 1) lie on the Euclidean circle of radius two centered at (0, 0)? Problem Write the equation of a circle with center at (0, 0) and radius 2 in the Taxicab geometry. 3

5 2. Sketch this circle on the grid. 3. Which of the the points (0, 2), ( 2, 2), (1, 1) lie on the taxi-cab circle of radius two centered at (0, 0)? We will now look at the circle of radius 3 with the center at (2, 1). 1. Open the second tab named Task2. Again, you will need to see the coordinate grid. Go to the Graph menu and select Show Grid. 2. We will first plot the center of our circle. Using the same method as before, plot the point I at (2, 1). 3. Hold down the Custom Tool button and click on the Taxicab tool. 4. We want to find points on the taxicab circle of radius 3. Click on point I and drag the other point so that it is 3 units from the center. (Notice that the measurement shows up in the top left hand corner of the screen.) Continue plotting points until you see a geometric figure begin to form. Problem 7. Draw a picture of the taxicab circle of radius 3 centered at the point (2, 1) on the grid. 4

6 3 Exploring Points Equidistant From Two Given Points Problem 8. Describe the set of points that are equidistant from two given points in Euclidean geometry. (Hint: It may be helpful to draw a picture. The set is infinite.) Let us explore how the set of points that are equidistant from two given points look like the Taxi-Cab geometry. 1. Open the third tab named Task 3. Again, you will need to see the coordinate grid. 2. We will first need to plot the two points. Use the procedure from before to plot the points, O = ( 4, 3) and P = (2, 1). 3. Select the Custom Tool button and click on the Taxicab tool. Click 5

7 on O and drag the second point somewhere other than P. The taxicab measurement should pop up on your screen in the top left hand corner. 4. Click on P and drag your second point to the point you just created. A second measure should pop up. 5. Click on the arrow tool. You can now drag this point and the measures of the two distances should change. Drag this point and find a place at which the distances are equal. 6. Record the point you just found on the coordinate grid below. Now drag the point to find another coordinate for which the distances are equal. Record this point. Continue this procedure until you have enough points to make an educated guess about the entire infinite set of points equidistant from O and P. Problem 9. Sketch the set of all points that are equidistant from O = ( 4, 3) and P = (2, 1) on the following grid. 6

8 Problem 10. Now prove your answer algebraically: (Hint: You want find all points (x, y) such that x y 3 = x 2 + y 1. Explain why. To drop absolute values you need to consider several cases.) 7

9 4 Application Problems Click on the tab labeled Task 4. Bring up the grid. You may use the Sketchpad to answer the following question, or you may find the solution on paper. Problem 11. Two pizza parlors are located, one at X = (2, 1) and the other at Y = ( 1, 1), in an ideal city. You want a pizza delivered to A = ( 1, 4). From which parlor should it come? Problem 12. The telephone company wants to set up pay phone booths so that everyone living within six blocks of the center of town is within two blocks of a pay phone. How few booths can they get by with, and where should they be located? List the points where the phone booth should be placed and graph them below. (Hint: The taxicab circle below sets the boundary for living within six blocks of the center of town. You should partition this region using taxi-cab circles of radius two.) 8

10 5 Homework Problem 13. Find all points equidistant from A = (0, 0) and B = (3, 3). Sketch your solution on the grid and prove algebraicly that you indeed found all points. (Hint: You want find all points (x, y) such that x + y = x 3 + y 3. Explain why. To drop absolute values you need to consider several cases. You may attach extra paper if needed.) Note that the resulting set of points is very different from the one you obtained in Problem 9 of this lab. It includes 2-dimensional regions! 9

11 Definition 5.1. Two triangles are congruent if their corresponding sides and angles are congruent. To prove that two triangles are congruent, one needs to check that the six corresponding parts are equal. In Euclidean geometry the triangle congruence criteria: Side-Angle-Side, Side-Side-Side, Angle-Side-Angle, and Side- Angle-Angle, allow us to check just three corresponding parts instead of six. We want to determine if these triangle congruence criteria also hold in Taxicab geometry. Problem 14. Test the SAS postulate in Taxicab geometry using the triangles given below. Hint: Compute the distance of all three sides in both triangles. (Remember to use taxicab distance!) Problem 15. Does SAS hold in Taxicab geometry? 10

12 Problem 16. The other three triangle congruence criteria do not hold in Taxicab geometry. Provide a counterexample for each (on the given grids). 1. SSS 2. ASA 11

13 3. SAA Problem 17. Check that the Triangle Inequality holds in Taxicab geometry: ρ(p, Q) ρ(p, R) + ρ(r, Q) for any three points P, Q, R in the plane. (Use that a + b a + b for any real numbers a, b.) 12