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.