Taxicab Activities in Sketchpad

Tools

Arrowhead      
Point              
Compass          
Straight Edge      
Alphabet      
Script View      
Creating a Taxicab Script to Measure Taxicab Distance
  • Open up Sketchpad and use the point tool to create two points that are not on the same vertical or horizontal line.
  • Use the arrowhead tool to select the point on the left and under Transform, release on Translate. Change the angle to 0 degrees. You will see that the translation is in the horizontal direction 1 cm and that this translated point is closer to the second point.
  • Construct a line passing through the point and its translate by first selecting both points and releasing on Line under Construct
    Construct the perpendicular to this line through the other point by selecting the line and the other point and releasing on Perpendicular Line under Construct
  • Select both lines and release on Intersection under Construct.
  • Using Display/Hide, hide the lines and the translated point. You now have 3 points remaining (which form a right triangle).
  • Using the straight edge tool, construct the horizontal part of the triangle.
  • Under Display, scroll down to Color and release on Aqua.
  • Using the straight edge tool, construct the vertical part of the triangle.
  • Under Display, scroll down to Color and release on Orange.
  • Using Measure/Length, select and then measure the horizontal and vertical distances.
  • Under Measure or Number, release on Calculate, and compute the sum of these distances (ie the taxicab distance) by clicking on measurement 1, then +, then measurement 2. Hit Ok. You'll see a measurement in cm.
  • Under Display, Hide the first two measurements and the horizontal and vertical distances. You now have 3 points, the aqua horizontal line, the orange vertical line, and the taxicab distance, like in the following picture:
  • Use the arrowhead tool in order to create a big rectangle that highlights everything.
  • Use the Script tool to Create New Tool and check the Show Script View box.
    You have just created a script that will measure the taxicab distance between two points.
    Save your Sketchpad file as yourname.gsp and leave the script open. Mail the file to yourself or use the personal storage space on ASULearn so that you can use this later.

    Square Circles Under File release on New Sketch. Under Graph release on Show Grid. Using Graph, plot points, precisely plot the point (-6,4). Create a taxicab circle of radius 2 about the point (-6,4) by using the graph/plot point feature to create the four vertices of your taxicab circle. Use the point tool (not the graph/plot point feature) to create an approximate point on the boundary. Use the arrowhead tool to select the point (-6,4) and then the boundary point you created, and then click on All Steps from your taxicab script tool. Notice that the script will now run and it will calculate the taxicab distance for you. Move the point approximately on the boundary to see that the taxicab distance to the center remains the same as the horizontal and vertical distances change.

    Finding All the Points Equidistant from A=(0,0) and B=(3,3) in the Taxicab Metric Under File release on New Sketch and nder Graph release on Show Grid. Carefully create the point (3,3) using the Graph/plot points feature. Use the point tool to create a point that is NOT on the axes and is not directly above, directly below, or directly to the left/right of A or B. Next use your Taxicab Script tool to measure the distances from this point to both A and B (click on 2 points and then All Steps in the script and then redo this). Move the point around and note the distance to A and B in order to find all of the points that are equidistant from A and B. Sketch a diagram of these equidistant points on a sheet of paper and show Dr. Sarah when you think that you have found them all.

    Precinct Problem In a town having perfect square blocks and equally spaced streets running north and south, east and west, two police stations are to be located at A=(0,0) and B=(3,3). The town officials want to divide the town into two precincts - Precinct 1 served by Station A and Precinct 2 served by Station B. What are the real-life issues that would go into deciding how the boundary should be drawn? Explain your ideas to Dr. Sarah when you are ready.

    3 Noncollinear Points In Euclidean Geometry, 3 noncollinear points determine a unique circle. Is this true in taxicab geometry?