# Distance Functions and Isometries

In the last module you encounter many different notions of distance. In this module we will codify what makes different notions of distance "work" in a predictable way. We will also see what it means for two apparently different notions of distance to be the "same" in the sense that even though they measure different things, the measurements they make are exactly the same.

## Distance Functions

What are some of the most important properties of Euclidean distance? To start out, we know that the distance between any two points is always greater than or equal to zero, and that it is zero precisely when the two points are the same. Also, if we have two points, say A and B, then the distance from A to B is the same as the distance from B to A. Finally, if we have three points, say A, B and C, then the distance you travel going from A then to B then to C is at least as long as going from A to C directly. (Can you show that all of these properties actually hold for the Euclidean distance function on the plane?)

All of these properties seem pretty reasonable for any measure of distance, so let's codify them:

A function d(A, B) that takes as inputs two points, and gives as output a real number is called a distance function if the following properties are satisfied:
1. d(A, B) = d(B, A) for all possible choices of A and B
2. d(A, B) = 0 if and only if A = B
3. d(A, B) > 0 for A and B not equal
4. d(A, B) + d(B, C) is at least d(A, C) for all possible choices of A, B, and C.

Problem 1: Consider the function d(A, B) = 2max{|x1 - x2|, |y1 - y2|}, where A = (x1, y1) and B = (x2, y2) are points in the Cartesian plane. Show that this function is a distance function.

Problem 2: Sketch a picture of the points that are distance 1 from the origin with respect to the distance function given in the previous problem. Which situation from the previous module does this function model?

Problem 3: Consider the function d(A, B) = min{|x1 - x2|, |y1 - y2|}, where A = (x1, y1) and B = (x2, y2) are points in the Cartesian plane. Is this function a distance function? If so, verify all the properties. If not, give an example where one of the properties fails.

Problem 4: Sketch the points B for which d((0,0), B) = 1, where d is as in the previous problem. Which situation from the previous module does this function model?

Problem 5: Consider the function d(A, B) =|x1 - x2| + |y1 - y2|, where A = (x1, y1) and B = (x2, y2) are points in the Cartesian plane. Verify that this is a distance function.

Problem 6: Sketch the points which are distance 1 from the origin with respect to the distance function given in the previous problem. Which situation from the previous module does this function model? We call this function the cab driver distance function.

As you saw above, functions that model distance traveled are not always distance functions. Where this failed above, the function was good at modeling distance traveled between two points, but not between a succession of three or more points. Do you see why this was the case? Can you relate it to the situation from the previous module that the function modeled?

## Isometries

Say that you have two distance functions on the plane, d1 and d2. Another function, f, that takes points in the plane to points in the plane is called an isometry if it satisfies the following two properties:
1. d1(A, B) = d2(f(A), f(B)), where A and B are points in the plane
2. For any point B in the plane, there is a point A in the plane such that f(A) = B

How does the existence of an isometry say that the two notions of distance that you have are essentially the same?
If d1 and d2 are the same function, what does the fact that a given function f is an isometry say about f as a rearrangement of the plane?

Problem 7: Let d1 and d2 both be the Euclidean distance function on the plane. Is reflection across the x-axis an isometry? What about reflection across the y-axis?

Problem 8: Let d1 and d2 both be the Euclidean distance function on the plane. Is rotation by a set angle an isometry? Why or why not? Think geometrically!

Problem 9: Let d1 and d2 both be the Euclidean distance function on the plane. Is translation by a fixed amount in a fixed direction an isometry?

Problem 10: Suppose that d1 is the distance function given in problem five, and d2 is the Euclidean distance function on the plane. Is there an isometry of the plane with respect to these two distance functions?
Hint 1: Using problem nine, you can assume that if an isometry existed it would send the origin to itself.
Hint 2: Using problem eight, you can assume that if an isometry existed it would send the point (1, 0) to itself.
Hint 3: How many points on the circle of points around the origin that are Euclidean distance 1 away are cab driver distance 2 away from (1, 0)? How many are Euclidean distance 2 from (1, 0)?