Anna drives a taxi in Grid City. All the streets are laid out in equal sized blocks, like a grid. In this picture, Anna is in her taxi at the intersection of 8th St. and Avenue H.

Anna’s city gives us an interesting geometry that’s a little different from the geometry you’re used to.

1. When a customer tells her a destination, Anna thinks about the nearest intersection to it and figures how many blocks she’ll have to drive to get there. Anna thinks of each intersection as a point--and there are no points other than intersections.

   Find the following points on the city above, and describe a route for Anna to take to each point.

   1. 3rd St. and Avenue L
   2. 10th St. and Avenue K
   3. 9th St. and Avenue B
2. Find three different routes for Anna to take to 4th St. and Avenue D. Make one a different length than the other two.
3. What is the shortest distance Anna would have to travel to get to 4th and D?
4. A route that Anna can travel to a destination without repeating a block is a line between where she is and that destination point.
   1. Can there be more than one line between two points?
   2. What kind of line should Anna use to figure the distance she has to travel to a destination?
5. A circle is the set of points that are a particular distance (called the radius) from another point (called the center).
   Anna would want to use a shortest path between two points, so by distance we mean the length of a shortest path.
   
   
   1. Mark all the points that are 1 block from 8th and H.
   2. Mark all the points that are 4 blocks from 8th and H.
   3. What does a circle look like in “taxicab geometry”? Describe it as fully as you can.
6. A parabola is the set of points whose distance to a fixed line (called the directrix) is the same as the distance to a fixed point (called the focus). The distance from a point to a line is the shortest distance from the point to any point on the line. For example, the distance from 8th and H to Avenue C is 5, because 8th and C is 5 blocks away. To get to any other point on Avenue C would require traveling more than 5 blocks.

   On the following grid, mark the points that form the parabola with 6th St. as the directrix and 8th and H as the focus.
   

   
7. An ellipse is the set of points whose distances to two fixed points (the foci) have a constant sum. (For example, a point A is on the ellipse with foci B and C and constant d if AB + AC = d.)
   Foci is the plural of focus.
   Mark the points whose distances to the marked intersections add to 9.
   
   
8. A hyperbola is the set of points whose distances from two fixed points (foci) have a constant difference.

   Mark the points whose distances to the marked intersections have a difference of 4.
   

   

1. Routes can vary greatly, as long as it begins at 8th and D and ends in the proper place. Students may draw the routes or describe them. (For example, go down 5 streets and then right 4. OR, go South 5 streets, turn left, and go another 4 streets.) An example is given for each.
   
   
2. Short routes will go down 4 streets and left 4 streets. They might do this directly (4 down and 4 left, or 4 left and 4 down) or in steps (for example, 3 down, then 1 left, then 1 down, then 3 left). Longer routes will include at least one right or down.
3. 8 blocks
4. 1. Yes. (The routes in problem 2 are examples.)
   2. One with at most two directions, one each of up/down and left/right.
5. Parts a and b are shown in the following grid. Part a points are white, part b points are grey.
   
   
   1. In this geometry, a circle forms a Euclidean square, with vertices on the four axis. The points making up the sides have slopes of either 1 or -1. If the radius is r, there will be r + 1 points on each side, including both vertices. (Thus the circle will have 4r points in it.)
6. The parabola looks like this:
   
   
7. The ellipse looks like this:
   
   
8. The points in the hyperbola form two parallel Euclidean lines: