Introduction

[♫ opening music ♫] Hi, I'm Kevin, and I'm a geek. I really like working on mathematical problems, and explaining them to people. I've retired from a job at a big industrial research lab where a lot of my work consisted of just that. Now that I'm on my own for a while, I'm looking for whether I can get the same sort of fun out of explaining things in videos. A lot of what I've done involves various aspects of geometry, but I'm likely to stray far and wide because the field of mathematics is all connected. In this video, I'm going to continue exploring the cardioid, which is one of my favorite geometric figures. I'll look at some of its analytic geometry.

Review

Last time, we defined the cardioid with a spirograph, and in Euclidean fashion with straightedge and compass. We defined the cusp, the vertex and the diameter, and proved that all our constructions of the cardioid so far are equivalent.

Analytic formula - polar coordinates

Now let's shift our focus to analytic geometry and define an analytic formula for the cardioid. We'll work in polar coordinates. It turns out that we get the simplest formulae if we start with our coordinate system centered on the cusp. Let the radius of the wheels be a. If we take the line joining the centers of the wheels, O and W, and drop perpendiculars onto that line (the points D and E) we get two new line segments OD and EW, that are both of length a cos theta. Since the total length OW is 2a, we see that DE = 2a - 2a cos theta which is 2a times the quantity (1 - cos theta) And CPDE is a rectangle, so CP, the distance from cusp to cardioid, is also 2a times the quantity (1 - cos theta) That's the formula we are after.

The diameter

This formula now gives us some insight into why we chose to call 4a the diameter. Remember that every chord through the center of a circle is a diameter of the circle and all the chords are of equal length? Let's look at the chords through the cusp of the cardioid. They certainly look equal... And it's easy to prove it. One part OA of the chord is defined by the angle theta and its length is 2a times the quantity (1 - cos theta). The other part OB of the cardioid is defined by the angle theta plus a straight angle. Its length is 2a times the quantity (1 + cos theta), and the two parts add up simply to 4a. Since we said that 4a is the diameter, we can say that the length of every chord through the cusp of a cardioid is equal to the diameter, just as the length of every chord through the center of a circle is equal to the diameter.

Midpoints of diameters

There's also an interesting fact about the midpoints of the diameters. They're all on the circumference of the fixed wheel, by contrast with the midpoints of a circle's diameters, which are all on the circle's center. This observation about the midpoints of the chords is not terribly difficult to prove, and I'm going to skip over the proof.

Area of a cardioid

The real meat of having an analytic description is that it lets us find the area and perimeter of a cardioid. If you're learning calculus, you might find it a useful exercise to try to compute them from what we know. If you know no calculus at all, you might want to just doze through the proofs and wake up for the conclusions. We'll start with the area. I've repeated the formula for the cardioid in polar coordinates, and I'm giving the formula for area within a curve in polar coordinates to refresh your memory. I'll give you a few seconds to pause the video if you want to try to work this one out for yourself. [♫ music ♫] And here's the working through the problem on 'algebra autopilot'. [♫ music ♫] If you got stuck part way through, I'll bet this is where it happened. You either hadn't learnt or had forgotten what to do about an integral like this one. Who remembers an integral like this off the top of the head? We could look it up in a table of integrals. I'm an old man. There's still one on my bookshelf. Or a younger person would hand it to a computer algebra system But it isn't actually that hard to re-derive from trig identities. We can get this one, if we're clever, by using the double angle formula cos 2 theta = 2 cos2 theta - 1. Rearrange that to put cos2 theta on the left hand side, and we have a sum that we can integrate. [♫ music ♫] And once we have that insight, a little bit of algebra finishes the job. Let's go back and substitute this in. And we get that a cardioid's area is 3/2 of the area of a circle of the same diameter.

Perimeter of a cardioid

Now what about the perimeter? The perimeter formula is easiest to remember if you think of it as being like the Pythagorean theorem, where the legs are the radius and its derivative. For reasons that I'll get to in a moment, I'm actually going to integrate only half the perimeter, and get the other half by symmetry. Of course, we need the derivative to plug into the formula, so we start by differentiating the radius function. And now I'm going to give you a few seconds to pause the video and continue the calculation for yourself. Then I'll engage my handy autopilot once again. [♫ music ♫] If you tried to work through it for yourself and got stuck, I bet this is where it happened. Once again, a trig substitution comes to the rescue. This time it's a half-angle formula. You have to be careful with the half-angle formula because of that annoying "plus or minus". If we restrict ourselves to angles between 0 and pi, the plus-or-minus goes away because the sine of half the angle is always positive. We can fill in the other half of the cardioid by symmetry - and that's why I set up the problem as I did. From here, we need to tweak the form of the integrand to match the half-angle formula, and we can go back on autopilot. And we get the somewhat surprising result that the perimeter is 16 times a, or four times the diameter of the cardioid. I really don't like this result. To have to do all this calculation and come out at the other end with a small integer seems like a lot of work for nothing. But I don't know an easier way to get there. If you do, please let me know in the comments!

Summary

So, let me do a quick review. We generated a cardioid by rolling a wheel around a another wheel of equal size that's held stationary, and we called four times the wheel radius the diameter. Every chord through the cusp is equal to the diameter. The midpoint of the chords through the cusp lie on the circumference of the stationary wheel. The area of the cardioid is three-eighths pi times the diameter squared, or three-halves the area of a circle of the same diameter.

Coming attractions

The perimeter is four times the diameter. And that puts us just about out of time for this video. Next time, we'll start looking at a couple of real-life cardioids, so stay tuned for that! Thanks for watching, stay healthy, and keep calculating! [♫ closing music ♫]