Villanova University, Department of Mathematical Sciences

Cartographiometry (MAT 1210/ GEO 1700)

(©  copyright 1996-1999, Timothy G. Feeman and Elaine F. Bosowski.)

Laboratory Exercise: Tools of the trade/Coordinate systems

(toolslab.htm)

This lab consists of four sections. Each of the first three sections focuses on a different type of coordinate system for the plane. The fourth section is easier with the help of a friend. For each section, you must use the appropriate tools to construct the coordinate system described and answer the stated questions about that system.

Section I : Cartesian coordinates.  Using your straight edge and protractor, and any other tools you feel might be helpful, construct two perpendicular axes to form a  Cartesian coordinate system. Mark each axis with a scale, using the same unit size for both axes.

1. In the Cartesian coordinate system you have just constructed, locate and label the following points: (0, 0), (2, 1), and (-3, 2). Locate and label four additional points of your own choosing, one in each of the four quadrants.
   
2. In the Cartesian coordinate system you have just constructed,
   • (a) sketch all points having y-coordinate equal to -1, and
   • (b) sketch all points having x-coordinate equal to 3.
   • (c) What are the equations that describe the sketches you made in parts (a) and (b)?
3. In a Cartesian coordinate system, a circle can be described as the curve consisting of all those points whose x- and y- coordinates satisfy a relation of the form x²+y²=r². The number r in the equation represents the radius of the circle. For instance, a circle of radius 3 has the equation x²+y²=3² (=9).  Draw this circle. The point (3, 0) is on this circle because 3²+0²=9 is a true equation. Similarly, the point (0, -3)  is on this circle because the equation 0²+(-3)²=9 is true.
   • (a)  Show that the points (-3, 0) and (0, 3) are both on the circle x²+y²=9.
   • (b)  Is the point (2*sqrt{2}, 1) on this circle? What about the points (1, 2), (-2, 2), and (3/2, 3*sqrt{3}/2)? Explain your answers. (Notation: "sqrt{v}" means the positive number whose square is equal to v.)
   • (c)  Can any point on the circle x²+y²=9 have an x-coordinate greater than 3? Explain.

Section II: Polar coordinates.  Construct a polar coordinate system by marking a point to serve as the origin and a half-line emanating from the origin to serve as the reference direction. All angles should be measured clockwise from the reference direction. Mark a scale along the reference line.

1. In the polar coordinate system you have just constructed, locate and label the points (1, 0°), (2, 90°), and (3, 180°). Locate and label four additional points of your own choosing all at different angles from the reference direction.
2. In the polar coordinate system you have just constructed,
   • (a)  sketch all points having an angle coordinate equal to 80°, and
   • (b)  sketch all points having a distance coordinate equal to 3.
   • (c)  What is the full range of angle coordinates for the points in the sketch for part (b)?
3. Suppose that the center of your polar coordinate system represents the North Pole, that the reference direction represents the Prime (Greenwich) Meridian, and that each of your distance units represents 10° of latitude. Then find and label the location of Villanova University (40° north lat., 75° west long.) in the polar coordinate system you have just constructed.

Section III: Grid coordinates.  Construct a three-by-three square grid of horizontal and vertical segments (like a portion of a checkerboard). From left to right, label the four nodes along the lower horizontal edge with the numbers 12, 13, 14, and 15. From bottom to top, label the four nodes along the left vertical edge with the numbers 21, 22, 23, and 24.

1. Locate and label the points 1323, 135225, 142230. Locate and label three additional points of your own choosing.
2. Suppose each small square in the grid system you have just constructed represents 10 paces on each side and suppose you have received directions telling you that, to find the secret treasure, you should start at the point 1221, walk six (6) paces right, then fifteen (15) paces up, then twelve (12) paces right, then three (3) paces down. What are the coordinates of the point where the treasure is located?
   

Section IV: Field work!   With a partner, construct a polar coordinate system outside, using some landmark as the center together with another landmark to define your reference direction. Take the length of your stride as the distance unit (so distances will be measured in 'paces'). Locate and label (give coordinates for) at least three additional landmarks. (A protractor will be useful.)  For this part of the laboratory exercise, submit a sketch of your field work.