# Different Notions of Distance

Sometimes knowing the straight line distance between two points does not do you much good. For example, as you saw in the last module, the straight line between two points on a sphere doesn't reveal how far you must travel along the surface of the sphere to get from one point to the other. As another example, it is about the same distance from Chicago, IL to New York, NY, as it is from Chicago, IL to Ogallala, NE, but it takes far less tim to get from Chicago to New York than it does to get from Chicago to Ogallala because there is frequent plane service between New York and Chicago. In this module we will discuss some different notions of distance, and how to measure them.

**Problem 1:** Suppose that you are a cab driver in a grid-plan city. You can only drive on city streets, and all the blocks in the city are perfectly square. Let's annotate position within the city in a two coordinate system, where the coordinate (*x*, *y*) denotes being x blocks east and *y* blocks north of the city center. If *x* is negative, you are to the west of the city center; if *y* is negative you are south of the city center. If you pick someone up at the city center and they ask you to drive them to the city zoo, which is located at (8, 10), how far must you drive? What if they want you to drive them to city hall, which is located at ( -1, 2)? How far must you drive if you pick someone up at city hall and they want to go to the zoo?

**Probelm 2:** Can you generalize your results from problem 1? That is, how far must you drive if you pick someone up at (*x*_{1}, *y*_{1}) and they want to go to (*x*_{2}, *y*_{2})? How many different ways can you drive between the two points and travel only the minimum distance?

**Problem 3:** Now suppose that you are a photographer working for the newspaper in the same city. You have a super-zoom lens for your camera, so that if a news event happens anywhere in the city, you only have to be on the same cross-street to get an image for the newspaper. Say that you are in your office at the newspaper building, which is located at (3, -4), and the mayor is making an appearance outside of city hall. How far must you drive to get a picture? Where will you be when you take your picture? Suppose that immediately after the mayor makes his appearance, you hear that a rhino has broken out of its enclosure at the zoo. How far must you travel to get a picture?

**Problem 4:** Can you generalize your results from problem 3? That is, if you are at location (*x*_{1}, *y*_{1}) and you want to take a picture of a news event happening at (*x*_{2}, *y*_{2}), how far must you drive? How many ways can you travel to get your picture and only travel the minimum distance?

**Problem 5:** Now suppose that instead of living in a large grid-plan city you live in a very small town, so small, in fact, that there is only one street. Pretend that you live at the very edge of town. Everyday you run at most two errands -- at most one on the left side of the street and at most one on the right side of the street (a little strange, I know). You run your errands in the most efficient manner possible, and only make one trip. Let's annotate the situation in a two coordinate system, where the coordinate (*x*, *y*) denotes that the errand on the left side of the street was *x* blocks from your house and the errand on the right side of the street was *y* blocks from your house. How would you annotate the situation in which you ran no errands? What if you only ran one errand and it was on the right side of the street? What if it was on the left? If you went to the bank, which is four blocks from your house on the left, and the hardware store, which is 3 blocks from your house on the right, how far did you travel?

**Problem 6:** Can you generalize your results from problem 5? That is, if the errands you ran today were annotated (*x*, *y*), how far did you travel to complete them?

**Problem 7:** Now suppose that you a physical geographer and all that you care about is elevation change. You denote points on the earth by (*x*, *y*, *z*) where *x* denotes latitude, *y* denotes longitude, and *z* denotes elevation. Given two points (*x*_{1}, *y*_{1}, *z*_{1}) and (*x*_{2}, *y*_{2}, *z*_{2}), what is the change in elevation?

**Problem 8:** How can you show which points on a map have the same elevation? Come up with both discrete and continuous ways of doing this. What are the advantages and disadvantages of each way? Who may want to use a continuous depiction? Who may want to use a discrete depiction?