Cartesian Plane – Distance Formula
First, we will review the distance formula and then introduce some new concepts. First, the distance formula: If we have two points in the Cartesian plane, with coordinates (×1, y1) and (×2, y2), then the distance between them is given by the Pythagorean Theorem. Distance in the Plane – The distance formula – Nearest neighbors – Clustering The Pythagorean Theorem is a statement of the mathematical relationship between the lengths of the sides of a right triangle and its hypotenuse; specifically, that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship can be written as z^2 = x^2 + y^2, which is equivalent to saying that z is equal to V(×^2 + y”2). This formula can be used to calculate distance by subtracting y from both sides, which will yield z= x^2 – y^2 [Triangle(z, x, y)] Pythagorean Theorem z^2=x^2+y^2 z=sqrt(x^2 +y^2) Suppose we have a right triangle (a triangle where one of the angles is 90 degrees). Let’s assume that we’ve drawn it so that it’s very close to being a right angle. We will now consider two points on this plane: Point A, which is equal to (a, b), and Point C, which is equal to (c, d). We will draw a line segment between these two points so that it passes through the point D, which is equal to (a+c)+(b+d). Now we want to ask: How far apart are Points A and C? What the distance formula says is that the distance between A and C–that is, Dist(A,C)-is given by the square root of the mathematical expression “difference in x values” plus “difference in y values.” Why would that be true? Let us draw a right triangle. Let us take this point here and draw a dotted line, and then draw another dotted line there. If you recall the Points in the Plane, we can prove that this point has the same y-coordinate as A and has the same x-coordinate as C; this point is indeed (c,b). Therefore, the length of this is c minus a, and the length of this is d minus b. So we have a right triangle with sides lengths c minus a and d minus b. Therefore, the length of this hypotenuse, once this hypotenuse is, of course, is the distance between A and C, is given by this formula down here. [Graph] Let’s work through some examples. For this problem, we have some points on the plane and we’ll compute the distance between them. So let’s start with point A is (1, 1) and let’s take point B way up here, not to scale. It is (5, 4). We can compute the distance between A and B by using the distance formula: square root of the difference in x values squared times x values squared. Now we have to do a little bit of arithmetic: x value squared minus 1^2 plus 4^2 equals 16+9 equals 25 which magically works out to be 5. So that means that the length of this line between A and B, let’s draw it in, the length of that line is five.It is five units apart in the × direction, which is interesting because it does not follow that you need to go five units in the x direction to get from A to B. You also do not need to go five units in the y direction. They are fairly far away from each other. Let’s also draw the origin. This is the point big O, this is (0,0). And let’s compute the distance between A and the origin. It’s equal to the square root, so the distance between the x-values, so (1-0)^2 + (1-0)^2. Stop for a second by the way and point out that (1-0)^2 is the same thing as (0-1)^2. That is, it doesn’t matter whether you do the x-value of A minus the x-value of 0 or the x-value of O minus the x-value of A, which makes sense because the distance from A to zero should be the same as distance from O to A, should be symmetric.So if we work this out, this is just the square root of two. In other words, for the fans of the Pythagorean Theorem, that length there is square root of two, there’s a right triangle. [Graph] OK. let’s make one more point. Let’s look at D equals (1,3/2), so the distance of that line. Now, here you don’t really need : fancy formula; you notice the only difference between them is an x-value. It’s pretty clear the distance between A and D is just 3/2-1, just a half. So let’s use the distance formula for this problem: the set S, which is equal to the origin, B, and D. Notice I just computed the distances from A to these three points O, B and D. The distance from A to O is 1.4; the distance from A to D is approximately 0.5; and the distance from A to B is 5. The distance between A and D iS one half. The square root of two equals approximately 1.4, so we have these three distances here. Here is the key concept: consider the set S, which is equal to the origin, B and D. Notice that we computed the distances from A to these three points O, B and D: The distance from A to O is 1.4; the distance from A to D is one half or 0.5; and the distance from A to B is five. This shows that the nearest neighbor of A in S is D because it’s the nearest point; the second nearest neighbor of A in S is O, since it’s not so far away; and finally, B is farthest away from A in S. That’s something we often use in data science when we have these three points O, D and B–we want to say if A had to be most like one of them, which one would it be? In this case we see that if you choose point D, you’d be correct because that’s what was used in this example. [Graph] One last little use of distance formulas that we use in data science is the idea of clustering. Let’s suppose we have a set of points in the plane. So here are many, many points that look like this, and let’s say here is another bunch of points that look like that, and say another clump over here. Visually, we might say there are three clusters of points, or clumps. We did not define what a cluster or clump is; however, it looks like we have three groups. Over here there is cluster one, cluster two and cluster three. If these were people measured by some blood measurement or something like that, we could say that there are three groups, group one, group two and group three. A distance measurement can be used to express the degree of separation between two points. If the points A, B and C are all in cluster one, but point D is in cluster three, we might say that the distance between A and B is much less than the distance between A and C, which is also much less than the distance between A and D. So having this distance formula, this distance metric. often allow you to break points up into stereotypical clusters or clumps, and somehow, whateve these are measuring, A and B are much, much more similar than A is to C and A is to D.
