Welcome to another Mathologer video. The shoelace formula is a super simple way to calculate the exact area inside any convoluted curve made up of straight line segments, like my cat head curve over there. Even the great mathematician Carl Friedrich Gauss was impressed by this formula and mentioned it in his writings. The formula was certainly not invented by him, however it’s often also referred to as Gauss’s area formula, probably because a lot of people first learned about it from Gauss (and not because someone calculated Gauss’s area with it 🙂 In today’s video I’ll show you how and why this formula works.
The visual proof I’ll show you is just as pretty as the formula itself and along the way I can promise you a couple of very satisfying AHA moments to make your day. I’ve got a special treat for you at the end of the video: a simple way to morph the shoelace formula into a very famous and very powerful integral formula for calculating the area enclosed by really complicated curvy curves, like for example this deltoid rolling curve here. Now obviously we call this crazy formula the shoelace formula because it reminds us of the usual crisscross way of lacing shoes. Now let’s make sense of the shoelace formula and use it to calculate the orange area. I start by filling in the coordinates of the blue points. Take one of these points and move its coordinates to the right.
Now we traverse the curve in the counterclockwise direction and do the same for the other blue points we come across. Here, there, there, there. Now we’re back at the point we started from and include its coordinates one more time at the end of our list. Now draw in the crosses. Okay this green segment stands for the product of the two numbers at its ends. So 4 times 1 equals 4.This red segment stands for minus the product of the number at its two ends. So 4 times 0 equals 0. Minus that is – 0. Oh, well obviously the “minus” is not important here but it will be later. Green again. So 0 times 5 equals 0. Red again, we need to calculate minus the product, so 1 times – 2 equals -2. Minus that, and so on. So we get two products for every cross, one taken positive and one negative. Now adding up all the numbers gives 110. Okay, almost there. The formula tells us to divide by two. So half of 110 is 55, and that’s the area of my cat head.
Really pretty and super simple to use. And this works for any closed curve in the xy-plane no matter how complicated. The only thing you have to make sure of is that the curve does not intersect itself like this fish curve here. And it will become clear later on why you have to be careful in this respect. Okay now for the really interesting bit, the explanation why the shoelace formula works. It turns out that the individual crosses in the formula correspond to these triangles which cover the whole shape. Note that all these triangles have the point (0,0) in common. Okay, so the area of the first triangle here is just 1/2 times the first cross. So, again, the first cross is equal to 4 times 1 minus 4 times 0 equals 4, and half that is 2. And it’s actually easy to check that this is true using the good old 1/2 base times height area formula for triangles. Now the area of the second triangle is 1/2 times the second cross, and so on. But why is the area of one of these triangles equal to 1/2 times the corresponding cross? Here’s a nice, really really nice visual argument due to the famous mathematician Solomon Golomb. What we want to convince ourselves of is this.
So let’s calculate the area of this triangle from scratch. Actually what we’ll do is to calculate the area of this parallelogram here whose area is double that of the triangle. Okay let’s start with the special rectangle here. Then the coordinates translate into the side lengths of these two triangles. First (a,b) turns into these two side lengths, and then (c,d) into these. Color in the remainder of the rectangle and shift the green triangles like this, and like that, Now do you see the second small rectangle materializing? Right there. The two triangles overlap in the dark green area and so we can pull the colored bits apart so that they fill exactly the parallelogram and the little rectangle. Since we started out with the colored bits filling a large rectangle this means that “large rectangle area” equals “parallelogram area” plus “small rectangle area”. But now the areas of the rectangles are ad and bc. That’s almost it. Now, without any words..
. Pure magic, right? And, of course, all of you who are familiar with vectors and matrices will realize that another way of expressing what we just proved is the mega famous result from elementary linear algebra that the area of the parallelogram spanned by the two vectors (a,b) and (c,d) is equal to the determinant of the 2 x 2 matrix a,b,c,d. Anyway, back to the shoeless formula. At this point we just need to divide by 2 to get the area of the triangle and that’s it, right? That completes the proof that the shoeless formula will always work, right? Well, not quite. We are still missing one very important very magical step. Let’s have another look at my cat hat, but let’s shift it so that the point (0,0) is no longer inside and again move around the curve and highlight the triangles whose area the shoeless formula adds. This time let’s start here.
As we move around the curve in the counterclockwise direction the green radius which chases us also rotates around (0,0) in the counterclockwise direction. Something does not look right here. The yellow triangles are sticking out of the cat head and at this point the combined area of the triangle is larger than that of the cat head and should get even larger as we keep going. However, whereas up to now the radius has been rotating in the counterclockwise direction, at this point it starts rotating in the clockwise direction and this change in sweeping direction has the effect that the shoeless formula subtracts the areas of the blue triangles. And this means that the area calculated by the shoelace formula will be the total area of the yellow triangles minus that of the blue triangles which is exactly the area of our cat head again.
The same sort of nifty canceling of areas makes sure that no matter how convoluted a closed curve is as long as it doesn’t intersect itself the shoelace formula will always give the correct area. Here’s an animated complicated example in which I dynamically update what area the shoelace formula has arrived at at the different points of the radius changing sweeping direction. Real mathematical magic, isn’t it? It’s also easy to see why reversing the sweeping direction leads to negative area. Let’s see. Sweeping in the counterclockwise direction we first come across (a,b) and record it, followed by (c,d). When we sweep clockwise the order in which we come across (a,b) and (c,d) is reversed and this leads to these changes in the formulas. And the last swap obviously leads to the number turning into it’s negative.
And that’s really it. Now you know how the shoelace formula does what it does. In these videos we keep encountering really fancy curves like this cardioid in a coffee cup in the “Mandelbrot and times tables” video or this deltoid rolling curve whose area actually already played a quite important role in the video on the Kakeya needle problem. At first glance it looks like we won’t be able to use the shoelace formula to calculate the area of one of these curves because they are not made up from line segments. Well you can definitely approximate the area by calculating the area of a straight line approximation like this, with those blue points on the curve. And by increasing the number of points we can get as close to the true area as we wish.
In fact, by taking this process to the limit in the usual calculus way, we can turn the shoelace formula into a famous integral formula for calculating the exact area enclosed by complicated curves like the deltoid. Here’s how you do this. I’ve tried to make sure that even if you’ve never studied calculus you’ll be able to get something out of this. Well we’ll see, fingers crossed 🙂 A curve like this is often given in parametric form. For example this is a parametrizations of this deltoid. Here x(t) and y(t) are the coordinates of a moving point that traces the curve as the parameter t changes from, in this case 0 to 2 pi. Let’s have a look. So here’s the position of the point at t=0. And once it gets going the slider up there tells you what t we are up to. Right now we’ll translate all this into the language of calculus. Let’s stop the point somewhere along its journey.
A little bit further along we find a second point. A tiny, tiny little bit further on is usually expressed in terms of infinitesimal displacements in x and y. It’s a bit lazy to do it this way but mathematicians are a bit lazy and love doing this because it captures the intuition perfectly and in the end can be justified in a rigorous way. Anyway just add dx and dy to the coordinates of our first point to get the coordinates of our second point. Now, of course, these displacements are not independent of each other. The connection is most easily established in terms of the derivatives of the coordinate functions. So the derivative of the x coordinate with respect to the parameter t is dx/dt which I write at x'(t) and similarly for the y coordinate function. Solving for dx and dy gives this and this then links both dx and dy to an increment dt of the parameter t that’s changing, right? Now we substitute like this and now we’re ready to calculate the area of our infinitesimal triangle as before.
1/2 times a cross. And this evaluates to this expression here. And this we can write in a slightly more compact form like that. Okay now what we have to do is to add all these infinitely many infinitesimal areas and as usual in calculus this is done with one of those magical integrals. The little circle twirling in the counterclockwise direction says that we’re supposed to integrate around the curve exactly once in the counterclockwise direction. Well let’s see: for our deltoid we have this parameterization here. We’ve already seen that a full trace is accomplished by having t run from 0 to 2 pi. This means that in this special case our integral can be written like this. Now evaluating and simplifying the expression in the brackets gives this integral here, which can be broken up into two parts. Maths students won’t be surprised that the trig(onometric) integral on the right evaluates to 0 which then means that the area where after is equal to this baby integral which of course is equal to 2 pi.
Now the little rolling circle that is used to produce our deltoid is of radius 1 and is therefore of area pi. This means that the area of the deltoid is exactly double the area of the rolling circle. Neat isn’t it? Okay, up for a couple of challenges? Then explain in the comments what the number stands for that the shoeless formula or the integral formula produce in the case of self intersecting curves like these here. Another thing worth pondering is how the argument for our triangle formula has to be adapted to account for the blue points ending up in different quadrants, for example, like this. And that’s it for today. I hope you enjoyed this video and as usual let me know how well these explanations worked for you. Actually since I mentioned the Kakeya video and fish, I did end up turning my Kakeya fish into a t-shirt. What do you think? Well and that’s really it for today.