The Surprising Uses of Conic Sections

- Okay, so conic sections are the curves made by slicing through cones at different angles. There are three types, the ellipse, which is slicing at a kind of narrow angle or a low angle. As you steepen the angle of the slice, you get parabolas and they occur when the angle of the slice is exactly the same as the angle of the cone. And then if you get steeper still, the slices start to cut through both halves of the cone and that's the hyperbola. Now these three types of conic section as they're called, the ellipse, parabola and hyperbola have beautiful and intriguing properties and we're going to talk about these today, as well as some applications. We'll see how ellipses are used in medicine, how to make a parabola by folding paper and what hyperbolas have to do with economics. There are three of these conic sections, and for each one, I'm going to show you three quite varied applications. So let's get started and talk about the ellipse first. And I should mention there are three of these things, but the ellipse will take up approximately half the time we have, so don't panic if I'm still talking about ellipses halfway through, and you think you're never going to, this will never end, although it's exciting, of course. Right, let's talk about ellipses. This is one way to define an ellipse. If you think about a circle, which I guess is a special case of an ellipse where you slice horizontally through a cone, with a circle, you can make a circle by having a fixed point, the center, and then the circle is the collection of points that are all exactly the same distance from that center, a radius R. well, with an ellipse, it's a bit similar, but this time you have two important points. Each one is called a focus. So the plural of focus, foci. I've marked them on the diagram, they're F and F dash. And the definition with this is that it's the sum of the distances to the foci that is a constant. So the points on the ellipse have this property. Point P has a property that the distance to F plus the distance to F dash is a constant, and I'll call that R and I'll do that throughout the talk. So this is the definition of an ellipse. That gives us a way to draw ellipses. With a circle, we can just have, you know, a pair of compasses and we can draw like this. With an ellipse, you can get a piece of paper and you can put two pins in it, and then you get a piece of string. I'll make this whole piece of string here. You get a piece of string and loop it round, not an elastic band or something,'cause it needs to have a fixed length, and then you pull it taut and you get a pen or pencil and just draw around like that. Now, the reason this will give you an ellipse is because the total length of the string is constant, and then the distance between the two foci is always constant, and then the rest of the string is giving you the sum of the distances from the point to the two foci, F and F dash. So I've done a not very good video of me drawing an ellipse using this technique. So I'm about to- This is not an absolutely mathematically perfect ellipse. There's a little bit of a bump where I go over the knot in the string, but you can see that this is producing something that looks sort of elliptical. So that's that. And this gives us, this definition of an ellipse gives us the first application of ellipses. I should say, by the way, it isn't clear if you're thinking,"How does she know that that's the same curve that you get by slicing through a cone?" It's not obvious that that is the case. It is the case, but I'm not going to give the detailed proof here now. It takes a little bit of algebra. It's not too hard, but it can be done. But I want to focus on talking about the properties and the applications in this talk. So first application of ellipses is in gear systems. Now, usually when you have a system of gears, they are usually circular. So you might have an input gear that's say, a circle of some size or some radius, and then it'll have an output gear or a series of gears, and you usually would turn the gears. The input gear would be turning at a constant rate, and that would transfer to the next gear along, which will then start turning at a constant rate too. It'll not necessarily be the same constant rate, because if the gears are different sizes, that will affect the speed at which they rotate. So usually if you- Well, not usually, always, if you have a small input gear turning at a particular speed and it's next to a large output gear, then the larger one will turn at a slower rate. Because of course you need lots of revolutions of the small one to get one revolution of the big one. Whereas if you have a large input gear and a small output gear, then that small output gear will turn at a faster rate than the input gear. Sometimes in some applications, you don't want a constant rate of motion. One example is if you have a production line where things are moving along a conveyor, and they're having various things done to them at various stages, what you might want is for it to go very, very slowly while something is happening, some bit of the process is being done, move quickly to the next place where something happens. So in that case, what you would like to have is a variable rate where it's slow sometimes and faster other times. Now an elliptical gear system can do that for you, and I want to show you why. What we've got here in the picture is two ellipses that are the same size and shape. So what that means is we've got this constant R, right? For both of them, it's the same, where points on the edge of these ellipses, the sum of the distances to there to foci is this R, this value R. I've just marked one example of that on the green ellipse there. So I've got on both of these ellipses, I've got the foci are marked slightly differently, and I'll tell you why in a second. And on that green ellipse there, there's a point on the edge of the ellipse, which I'll just- For those in the room, I'm showing you, and for those at home, it's where the yellow line meets the dotted line. There's a point on the green ellipse. It looks like it's on the blue ellipse as well, but that's not guaranteed yet. I want to explain why it must always also be on the blue ellipse. So what is this setup? I've got my two gear ellipses, and they have to sort of turn around something. So the one focus of each ellipse has an axle through it, and that's what the ellipses are turning around. So that's marked with these orange circles and little red arrows there. And we will make it so that they are this distance R apart. So that distance between the two axles is R. Now, if you just had them like that, just fixed around the axles, there's no reason for the ellipses to sort of touch each other. They might just flop downwards to the bottom and be useless. So we have to do another thing, which is that we have to have some, a rigid thing like a rod joining the other two foci of those ellipses, and that's what that yellow line is signifying in the diagram there. Okay, and that also is going to be length R. Okay, so that's the setup. Now we imagine our point that's on the green ellipse, the point that's marked where those lines intersect. Now I've said, let's just say it's distance A from one focus and B from the other. We know because it lies on the ellipse that A plus B must equal R, the sum of the distances to the foci is constant R. So that point then, let's think about it in relation to the blue ellipse. We want to show that it also must lie on the blue ellipse, and that will mean that these cogwheels are always touching all points in the motion. Well, first of all, let's look at this dotted line that joins the two, the two foci where we have our axles. The bit that's in the green ellipse is B and we know that total distance between them is R so the missing part, the part that's in the blue ellipse is A, A plus B is R. And the same thing between the other two foci where we've got this rod joining them together, the bit that's in the green ellipse is A, the total distance is R. A plus B is R, so the missing measurement is B there. So that point that we know is on the green ellipse, the sum of its distances from the foci of the blue ellipse is also A plus B, which is R. So it must mathematically lie on the blue ellipse as well. So what this tells you is that these elliptical gears are always going to be touching each other, and that means that it'll be a smoothly operating gear system with never any gaps or jamming or overlapping or anything horrible. So these will always be touching. And so when you set the blue, let's say the blue one is the input gear. If you set that one rotating, it will make the green one rotate in the other direction, of course, but it will start moving as well, but here's the thing. If you turn the blue one at a constant rate, the green one will have a variable rate of motion. And the reason is that at parts of the motion, the distance to the edge of the blue gear wheel is large, and the distance to the edge of the green one is small. So it's like a large input gear leading to a small output gear, so then the green one will be moving faster. But then at other points of the motion, when we rotate a bit further around, the distance in the blue one is large, is small, and the distance to the green one is large, and then the green one will be moving slower. So if you rotate the blue one at a constant rate, the output gear will be rotating at a variable rate. And that's a really, really helpful thing, as I've said, in some applications. So already, just with the definition, we are able to think about these elliptical gears as an application. To tell you a bit more about applications, I want to tell you a property of ellipses, and I'm not going to give you a proof of what I'm about to tell you. Again, it's not too difficult. I have put the proof in the transcript if you're interested. Need a little bit of geometry, a little bit of- The sort of thing you might have done in school with similar triangles and things like this. But I'll tell you the property. Here's an ellipse. I've drawn a tangent to the ellipse at a point, a random point, P. So the tangent is a line that just touches a curve. So I've drawn a tangent at this point, P, and then I've drawn the lines from P to the foci, to F and to F dash. And the equal angle of property says that the angle between the tangent and each of these lines is equal. This is very useful because, if you remember for the properties of light, says that when you have a light ray and it reflects, the angle of incidence equals the angle of reflection rate. You might remember that phrase. Well, what this tells you is that if you were to have an elliptical mirror and you send a beam of light from one focus, it will bounce off the mirror and pass through the other focus. Or, a bit more frivolously, by the same token, if you had an elliptical pool table and you had a pocket at one focus, and you put the ball at the other focus and you hit it completely randomly in any direction at all, it's guaranteed to bounce off the cushion and go through the pocket, so go into the pocket so you can always win. You can always win at pool on an elliptical pool table. That isn't my second application. I have seen one of these things in real life. Someone did decide to make one, but yet that's not a very useful application. However, here is a useful application and it's an application to medicine, and it's to do with kidney stones. So kidney stones are sort of hard crystalline growths that can appear in your kidney, and they can be very painful and to get rid of them, you know, one option might be to do surgery, but of course that it's invasive, there are risks, it's not ideal. So there's a treatment for kidney stones called lithotripsy, which bombards the kidney stones with high energy sound waves, and literally sort of shatters them apart, breaks them apart, they are shaken apart into tiny little pieces, which can then pass harmlessly out of the body. But of course, you don't want just fire randomly high energy sound waves at your body. You know, you don't want to damage the healthy tissue. So what you'd like to do is just focus those waves only on the bit that you want to break up, the kidney stone. So we use this property of the ellipse, the equal angle property in a machine called a lithotripter And here's the setup. So what you do is you have your emitter of high energy sound waves at one focus of an ellipse. And then you have, I mean, not the whole ellipse,'cause that would hit your body, but you have part of the ellipse as a reflector. And so the emitter beams out these sound waves, they reflect off the elliptical reflector, and they are focused at the other focus of the ellipse, which you have carefully ensured is where the kidney stone is. So it's this non-invasive treatment for kidney stones, which uses the mathematical properties in the ellipse. That's really quite nice. Okay, so there's another way we can think about ellipses. I've given you the definition. We've got this kind of way of drawing it with two pins and a piece of paper, or I've seen people do this with- You can put pegs and then rope, and then it'll help you to make an elliptical lawn for your garden, very nice. But there's another way that you can perhaps not draw, but construct an ellipse, and to tell you about it, I'm going to show you. Here's our diagram of the ellipse again, with a, you know, generic point, distance to the foci, the sum of those distances, there's a constant R. So what I'm going to do is I'm going to take the line that goes from one focus, F dash on the diagram to P, and I'm just going to extend it out until we reach a point X that's the same distance from P as P is from F. So PX equals PF. That's how I'm going to create this point X. So what that tells you, this line we've now got from the focus F dash up to X, the total length of that line must be R, because you've got the bit from F dash to P plus the bit from P to X, but we've constructed that on purpose so that that length is equal to the length from P to F. So the total length, the distance from F dash to X is R, and that did not depend on our choice of the point P. We get R every time. So these points X actually lie on a circle whose center is F dash and whose radius is R. And you can do this for all the points P on the ellipse and get all these points X. So you can make a circle in theory out of these points X. That gives us a way to construct an ellipse. It's really quite cute. Here's what you do. Imagine this was just, you know, we printed this picture onto a piece of paper, and you folded, you physically folded the paper so that the point X goes down to the point F and you make a fold line. I've drawn it dotted on the diagram. Make a nice fold line and unfold, okay? Now that fold line, if you think about the symmetry of the situation, when you fold things, it's like a mirror line. So everything on each side is going to match up, going to be equal. And so for example, everything that's on that fold line must be an equal distance from the point X and the point F. So in geometry we call that line the perpendicular bisector of the line XF. It's all the points that are equidistant from X and F. And we know that P, our point in the ellipse, is exactly one of those points, right? It's the same to distance from X as it is from F. So what we're going to do is we're going to get a circle, get a circular piece of paper, cut out a circle, we're going to mark a point, so that circle, the center of it is going to be one focus of our ellipse, that F dash. The radius will be our R and we're going to need to have another focus. So we'll mark a little point on our piece of paper, and then we're going to take points on the circumference. I'm going to show you this now, me starting to do this. So you're going to take points on the circumference and we're going to fold them in so that they just touch that point that's going to be one of our foci. I called it S. Dunno why, it was F before, but now it's S. So we're going to do this. Now, I'm not going to do infinitely many folds, but I'm just showing you the first two very slowly so you can see what I'm doing. You fold in, make a nice fold line, fold back out again. All right, and then if we can go onto the next video, I've sped up the process a bit, so now I'm doing it quickly. Look how quickly my hands are moving. So I'm going all the way around the circle then. That, okay, is what you end up with. And the person who taught me to do this was a teacher at some master classes I went to when I was a kid and he was called Terry Herd. He was a great inspiration. And he said, if you this for yourself, it's better to do it when your hands are slightly dirty, because then you sort of leave a better impression of the folds that you're making. But I think my hands were maybe a bit too clean, but I hope you can see there that we are creating something that looks like an ellipse there out of these. So I think I really love that way of folding an ellipse out of a circle. Okay, the reason in my mind when I did this, I called that focus S is because I had in mind the third application that I want to mention, which we probably all know that the planets move in elliptical orbits around the sun. And again, that the proof of that, the algebra behind that is a little time consuming for one lecture, so I won't do that for you, but Isaac Newton did it, Isaac Newton did it, and he is commemorated, or was in the olden days before we had pound coins, we had pound notes. And it has to say specimen on it, because there are laws about showing pictures of bank notes, even old ones, so that's why, but this is a picture of the old pound note, and it commemorates Isaac Newton's work on gravitation, on the astronomy, and it shows the elliptical orbits some of the elliptical orbits, but there is a slight error here in this. I dunno if you are spotting this right now, but the sun is at the center of some of those ellipses, not at a focus. That's slightly embarrassing moment for mathematics on bank notes, but still anyway, let's just be pleased that there was a mathematic on a bank note, and that has happened a few times since then. Florence Nightingale and Alan Turing are some others, mathematicians on bank notes. This is a place where ellipses, of course, are absolutely vital and crucial to our understanding of astronomy. You can derive this fact from the inverse square law of gravitational attraction. As I say, we won't do that right now, but it can be done. But I thought I should at least show you the equation of an ellipse, or at least a form of the equation of an ellipse. So if it's been a while since you have done any algebra, you're allowed to let your eyes glaze over for a couple of minutes, and then as long as they unglaze again, we're all good. So just wanted to show you, if you take the definition of an ellipse and just, you know, convert that into algebra, so the sum of the distance is, you can work out there's an algebraic expression for the distance between two points, and you want the sum of the two distances to be some constant thing. If you sort of put that into a bit of algebra, crank the handle, you can come out and you get an equation that looks a bit like this. So it's a quadratic expression, otherwise it's got X squares and Y squares in it. I just wanted you to see that. And the A and the B, depending on your choice of A and B, you can get different shape ellipses, different shapes and different sizes of ellipses. I've drawn in, so I've made the choice here for this equation to have kind of the fattest, widest bit of the ellipse, which we call the major axis. That's going to be X axis, and the thinnest bit of the ellipse, the minor axis is the Y axis here. So A is at least as big as B right there. And I've drawn in the foci. I've called them C and minus C, just because I want to talk for a second about, you know, the R that we've talked about, this constant distance which is the sum of the distances to the foci, that's nowhere to be seen on this diagram it looks like. So you might be thinking,"Well, where is it?" Well, it's there to be found. If you think about the total width of the ellipse, which be twice A, right? The total width of the ellipse at its widest. Now at that point A on the X axis there, that is a point on the ellipse. So the sum of its distances to the foci must be our R. So we can actually work that out, the distance to the ellipse and to the focus on the far left will be A plus C and the distance to the nearest focus will be A minus C. So the sum of the distances will be A plus C added to A minus C, so those C's cancel out and you just get two A. So that is our R. So we can find that from the graph here. Our R is two A. Now I've said that by varying the A and the B, you can get different shapes and sizes of ellipse, and I want to explore that a little bit. If you think about a circle, all circles are the same shape, right? They can be different sizes, but they're all the same shape. We know what a circle is like. You can have big circles and small circles. But ellipses aren't like that. You can shrink and grow them, sure, but also you can change the shape. You can have sort of long, slim ones, or you can have slightly fatter ones and you can have ones that look almost circular. So there's something more. There has to be kind of another parameter, I guess, that we need in order to understand what our ellipse looks like. With a circle, we can just get the radius, but with an ellipse, you need to actually specify which ellipse we're talking about. We clearly need a little bit more information, some other parameter. Now that parameter that we use is known as the eccentricity of an ellipse. I want to just show you how varying it- I'll tell you what it is, and then show you how varying it can vary the kind of shape we get. So the eccentricity, if you think about what that word might mean, it sort of means a bit how far away from the center, in some sense, that you are. And the definition of it is, it's kind of how near the foci are to the edge of the ellipse. So, you know, to be exact, it's the distance between the foci as a proportion of the full width of the ellipse. So it's the distance between the foci divided by our R, okay? I will show you. So here's one where I think I made it 4/5 here, the eccentricity. So the distance between the foci is 4/5 of the total width. You can see the eccentricity is always less than one,'cause you can never- The foci are always inside the ellipse and they're not ever on the ellipse. So the distance between the foci is a proportion of the total width is always less than one. As we move those foci closer to the center, so their distance is a smaller proportion, I think this might have been 3/4. That's eccentricity of 3/4, looking a bit fatter there. This is eccentricity of half, looking really quite circular. And then, you know, they can actually- You know, the endpoint of this is where they actually coincide, which is allowed, that can happen. If the foci coincide, that means the eccentricity is zero because the distance between them divided by the total width is zero. And that is a circle, so we recover a circle there, a special case of an ellipse. So I've said the eccentricity is always less than one. I want to show you what happens as the eccentricity gets larger and larger. And we'd sort of seen a bit of a, you know, perhaps theme ellipses somehow, but a better way or a good way of looking at what happens as that eccentricity gets higher is to go to our kind of circle idea that we had before. So remember here, we've got our ellipse and we've extended the line from one focus, F dash, out to a point X, which is so that PX equals PF, and then those points X lie on a circle whose center is the focus F dash and whose radius is R. And when we've defined the eccentricity of an ellipse as the distance between the foci, so the distance between the foci divided by our R, which is the radius of this circle. So what I'm going to do is show you a series of ellipses where the eccentricity is getting higher and higher and higher and approaching one, never quite reaching it. So this is what I'm going to do, the way I'm going to do it. So I'm going to get a radius R of this big circle, and I'm going to make the distance between the foci be just one less than that. So in other words, that focus F is just distance one from the circumference of the circle. And then if we, if we do that, the eccentricity then will be R minus one over R, which if you cancel the R's from the top and the bottom is one minus one over R. So as we increase R and R gets closer and closer to infinity, getting larger and larger and larger, this eccentricity will get closer and closer to one and in the limit, so in the limiting case, there won't be an ellipse anymore because the foci cannot lie actually on the circle, but we'll see what we approach in the limit. So as R tends to infinity, in the limit, we'll get a circle of infinite radius, which, you know, think what might be, and somehow eccentricity one. So that won't be an ellipse anymore, but what will it be? Well, let's have a look. So I've drawn a few of these things in pretty rainbow colors'cause I like them, and the big white blob is that focus F, the one that isn't the center of the circle, the one that is the other one, the left-hand side. And you can see, we start off with something that's a circle when R equals two, I guess, and then we just keep going and going, and in the picture, you know, the right-hand edge of the ellipse vanishes off the edge of the screen as we increase our value of R and increase the eccentricity. But there is it seems to be a sort of limiting curve, which you can see, we're going up incrementally, but by the time we get to this purple curve, which is the kind of the extreme more case, it's starting to look like a curve which we have already seen in this talk, and that was the parabola, and that is what the limiting case is. So we have these ellipses and in the limit, I mean, you can sort of think of a parabola as an ellipse with eccentricity one. It isn't an ellipse, but that's the limit and you can perhaps see why we get this happening. So just to remind us how we got to this point, we are saying that in this limiting case, we actually are looking at that green circle as a circle of infinite radius. And, you know, you can't have a circle of infinite radius, but if you could, it would be a straight line. The curve is getting less and less and less and less, and so you end up in the limit with a straight line. But you still have that property of this PX equals PF. You've kept that all along and you still have that in the limit, and that is the way we define a parabola. There's a straight line, we call it the directrix and there's a focus, there's just one focus'cause the other one zoomed off to infinity and the distance, the parabola is the center point whose distance from that straight line is equal to the distance from the focus. So that's the definition of a parabola. I'm going to show you how we can fold a parabola now. This time we're folding a straight line, not a circle. So we get just a normal piece of paper and we fold from the edge and you'll see it happen. We just fold points from the edge to the center, not to the center, to the focus that I've just drawn on there. I think we might have had a sneak peek of this one earlier. There it is. So you can see that's, that shape is a parabola and we've made it by folding this time from the straight line that's the edge of our paper. So that's kind of a cool way to make a parabola. I won't go through all again. I won't go through some algebra, but you can do some algebra here. Again, it's just taking a point and using the fact that its distance from a straight line is equal to its distance from a particular given point. I have chosen to put my axes in such a way that the focus lies on the X axis, it's point A, and the directrix is on the other side of the Y axis and halfway between, right, is the Y axis. So the directrix is the line X equals minus A and the focus is A. If you do that, you get the equation Y squared equals four AX, and by varying A, you can shrink or grow the parabola. Now you may have drawn a parabola yourself at school. Perhaps if you ever drew graphs and you plotted points, then probably the first curve you drew using that was Y equals X squared. That is also a parabola, but it's just, it's flipped over, so we've interchanged the roles of X and Y, and in that special case, you know, A is a quarter. But it's exactly the same reasoning, really, and so that Y equals X squared, that is a parabola, and this is kind of why. If you want to try and do those few lines of algebra, it's a fun exercise to do. Now, parabolas have an equal angle property just like ellipses, and it's the one we might expect because we've done this sort of limiting process. Remember that somehow the parabola has this focus way over infinity. So rays of light coming from the other focus, which, I mean, isn't really there, it's infinity, will be coming in and they'll be kind of parallel lines like this, and they will be reflected off the parabola and pass through its focus there. So this gives us an application of parabolas, or I guess usually it's paraboloids, right? Which let me just show you the picture. You take a parabola and you just rotate it around its axis so you get this surface. Light comes in from a distant star, say for a telescope, so far away that we can essentially assume it's coming in from infinity and the rays are parallel, and it bounces off the parabolic reflector and passes through the focus. So that focuses a line gives us a really good, clear image, same technology in satellite dishes as well. So that's the first application. I promised you three applications of each thing. So what might the next one be? Well, is it the Archimedes death ray? I apologize, I was being very frivolous. Archimedes did not fire laser beams out of his eyes, let's just get that clear. But there's a story about Archimedes, a great mathematician and scientist. He was from Syracuse and Syracuse was under threat of invasion by the Romans. And story goes that one night, the Roman fleet sailed into the bay at Syracuse and you know, "Oh no, what are we going to do? Let's fetch our brainiest person." And what Archimedes did, he had a solution to this, a very powerful weapon. He got parabolic reflectors and reflected the rays of the sun so that they focused onto the ships and the ships burst into flames and problem solved, right? Now, one or two teeny tiny weeny issues with that. Number one, the technology at the time was not absolutely amazing at making incredibly reflective paraboloid shapes. That's really quite hard to do. It's not at all clear that it would've been possible to do it in such a way that you could generate enough heat to set things on fire at a distance. That's number one. Number two, contemporary accounts did not bother to mention this amazing thing that would've been absolutely astonishing. The first account that mentions it is 350 years later. Galen writes about it. So that's a bit suspicious. Number three, if this was so fantastic, why has no one else ever done it in the history of warfare? That's number three. And finally, how did my story begin? One night.(crowd laughing) There you go. The clue was in the first word. So this is not the second application of parabolas as far as we know. However, there is another application and it's done by reversing the process essentially that we have talked about. If instead of light coming into the parabola and being focused, if you have a light source, a bulb at the focus, then rays from it will come out and they'll be reflected into a parallel beam of light, a nice strong beam of light. So you see this in torches, you see it in car headlamps or motorbike headlamps. If you've got a really old vintage car, sometimes the headlamps are actually external to the body work of the car and you can actually really see that nice parabola shape. So always good to go out parabola spotting. So that's the second application. The third one really as it relates to what we said about gravity causing the planets to orbit the sun with ecliptic orbits. Now, if you have a situation where the two bodies that you're studying are vastly different sizes, for example, a whole planet Earth and a petal, then actually, although the motion is strictly speaking, still in ellipse, it's sufficiently an extreme one that a parabola is a very, very good approximation to it. And that's what we tend to use as our approximation when we are studying, you know, I could talk about ballistics or cannon balls and warfare and things, which I don't like that, so instead, I'm going to use the example of lovely fountains, where you have water coming out and it's fired out of a pipe or something like this, and it follows a parabolic trajectory to an extremely good level of approximation. And you can see some of these beautiful things. You can see them in gardens or all over the place and lovely parabolas there to be seen. And if you had a waterfall, you probably get kind of half a parabola, you know? So you can see those, and I mean, they're very, very beautiful, but if you know the mathematics, you can go a bit further and you can know exactly where the other end of the parabola will be, and that means, I dunno if you've seen these, but you can sometimes create these effects where you have a series of kind of holes through which water is pumped out in little jets and it'll go up and it'll land on the next one, and then a jet will come out of that one, a little pulse of water, so it looks like the water's kind of bouncing along the ground, and it's a very, very beautiful effect. So this is an application then of parabolas that these trajectories of essentially very small things in relation to planet Earth, to a good approximation follow a parabolic trajectory. So we talked about two of the three conic sections and the final one that's left is the hyperbola. And hyperbolas, if you remember, we're imagining our cone and we're slicing through it and sort of shallow angles, we got ellipses. Then at this critical point where the slope was equal to the slope of the cone, we got parabolas, and if you then go a little bit further, we cut both halves of the cone and you start to get hyperbolas. And so here's a picture of this. And on the right you see somewhere where we see hyperbolas in the home. I'm all in favor of spotting things in everyday life. What you've got on the right there is a perfectly normal lampshade, a cylindrical lampshade. We think what's happening here with the light, you've got a bulb in the middle, and of course, the light's going everywhere, but the bit that can actually escape from the top and the bottom of the lampshade, you've got the top a circle there, so what's escaping is a cone of light, right? So you get a cone of light and then that cone of light, what will the shadow look like on the vertical wall? Well, it's slicing that cone of light vertically. So you exactly get a hyperbola in the shadow there. That's lovely. So we do see hyperbolas all around us. The equation of a hyperbola, I'm not going to go into too much detail, but we can define a hyperbola, and it's sort of a "Through the Looking Glass" thing. With an ellipse, there are two foci, and the points on the ellipse, the sum of the distances to the foci is constant. With a hyperbola, it's the difference of the distances that is constant. And that's why you get two branches because on one branch, you know, one focus is closer and on the other branch, the other focus is closer, but we only care about the size of that difference. So if the distance to one of them is one and the other one is seven, then the difference is six. We don't care if it's plus six or minus six, right? So it's the difference of the distances with a hyperbola that is constant and that sort of reflect- You've gone out the other side through parabola and out the other side. Hyperbolas have an equal angle property as well, and I'll show you that here. If you've got light that is directed towards one focus of the hyperbola, and then you make that be a mirror, so it's coming in from the other side of the focus, there's the focus coming from the other side, it'll be reflected off towards the other focus. You want to throw away the other half of the parabola, of the hyperbola, if you want it to reach that point, but that's where we go. This can be used, this property, to make telescopes that are slightly more compact than the Newtonian telescopes that we talked about before that have just a parabolic reflector. I'm going to show you this slightly complicated diagram, but let's talk our way through it. So this is a kind of telescope where you have a parabolic mirror, and so light comes in from a distant star and it comes, and the first thing that happens is it hits a parabolic mirror just like in a normal telescope. By the way, I've shown you kind of the bit where the action is happening, but you should imagine the parabolic mirror kind of carrying on significantly further in both directions. So it hits the parabolic mirror first. Then it bounces off that, and of course, as we know, it's going to head towards the focus of the parabolic mirror, but before it gets there, it hits another mirror. And the second mirror that it hits is part of a hyperbola. And remember, the property we've just found out is that light that's heading towards one focus of a hyperbola will bounce off and be directed towards the other focus. So what you do, you set things up so that the focus of the parabola is also one focus of a hyperbola. So that light hits the hyperbolic mirror, it bounces off it towards the second focus, and that's where the viewer is. And this makes for a much more compact design. So it's useful for telescopes that you might be moving around maybe, and you also do sometimes see in satellite dishes. So it combines, this is great, two conic sections instead of one, so obviously it's my favorite kind of telescope. Okay, so that's one use of hyperbolas. Now I haven't yet given you or told you what the equation of a hyperbola might be. Now, there is quite a general form for the equation, but I wanted to give you one particular example, because again, you know, if you've drawn curves, at school, some moments ago and you were drawing the curves, you might have done Y equals X squared parabola, and then you might have done Y equals one over X. That's another curve that we commonly learn to draw or practice drawing when we learn how to draw graphs. So Y equal one over X, that's an example of hyperbola. But also Y equals two over X, Y equals three over X. You can scale it up, and in general, if M is any number, Y equals M over X will be a hyperbola. Now the heading of this slide might be doubling the cube. What's that to do with anything? I wanted to just mention conics have been studied, were first studied by the ancient Greeks, people like Apollonius and others, but it's believed that the first person to study conics, write about them was an ancient Greek mathematician, Menaechmus, and he was interested in the problem of doubling the cube, which is, you may recall, there were these three kind of intractable problems of ancient Greek geometry, which is all about straight edge and compass constructions. So that there are only a very small number of things you're allowed to do in order to construct your geometrical figures. You can draw lines between points, you can draw circles, and you can make the intersections of those and get new points, but that's about it. So with this very strict set of rules of what you can do with your constructions, there were three problems that were difficult and no one could do. So trisecting the angle is one. If you've got an angle, can you do some construction to cut exactly into three equal angles? You can bisect any angle, but can you trisect any angle? Squaring the circles, probably the most famous. If you've got a circle, can you construct a square which has the same area? And doubling the cube is the third. If you've got a cube, can you construct a cube which has exactly doubled the volume of your original cube? Now, all of these are impossible it turns out. Many, many centuries elapsed before it was finally proved. I mean, almost 2000 years before it was finally proved that all three of these could not be done with the standard straight edge and compass construction. But Menaechmus was investigating conics and right, anachronism alert,'cause everything I'm about to say is absolutely not how Menaechmus would've expressed anything. There was no algebra at that time, there was no Y equals one over X, but I'm going to explain to you kind of the idea of this,'cause I think it's really cool. If you slightly expand what you allow, so you allow not just circles but conics, let's see what we can see. So take a cube of volume V. I'm going to find the intersection between two conics carefully chosen. So one of them is the hyperbola, Y equals two V over X and the other one is our favorite parabola, Y equals X squared. They will meet each other, they will intersect when those two things are equal, so when two V over X equals X squared. So you find that intersection point, you just draw your two conics, find the intersection point, and at that point X, the point X will satisfy two V over X equals X squared. So if you multiply it by X, you get X cubed equals two V. So now if you draw a cube, you've got that X now. If you draw a cube of side X, then its volume will be X cubed, which is two V. So we've doubled the cube. Now I'm not saying that that is- We haven't done it legally, right? Because we've used conics. But it's a really interesting thing that you can just, if you expand a little bit what you say you are allowed to use, you can then double the cube. Okay, I want to show you a way, a way that hyperbolas are used again, in buildings that we might see. So the iconic cooling tower, there's three of them on the left there that we all see. That is a hyperboloid. So again, take a hyperbola, rotate it around its axis, and you get this surface called a hyperboloid. Cooling towers, it's interesting 'cause you know, we've all grown up just seeing them as iconic things. You sort of imagine they've been around forever, but actually the design, the hyperbola design was first suggested and built in 1918. So it's a century old, but you sort of feel like it's as old as time somehow. Well, maybe that is. But that's a cooling tower. Now why is this shape used? Well, in a cooling tower, in industry water is used as a coolant, but of course, it then warms up and you have to cool it down itself before you can reuse it, right? So water comes in part of the way up the tower, it goes onto something called an exchange surface, a little bit of the water evaporates, and that creates a vacuum at the base of the tower, which kind of sucks the rest of the water down through the exchange surface, cooling it and it goes into a reservoir and then it can be reused. Only about 2% of the water is lost as water vapor. So that stuff, that gas coming out of the top of the cooling tower is just water vapor. It's nothing horrible. Why this shape? Well, it's really strong. The shape helps the flow of the vapor in the tower, and also it uses less material. So it's like a cylinder that's been on a diet, right? It uses less material'cause it kind of goes in in the middle. So it's very efficient in that way. The hyperbola shape also has a really brilliant property in terms of, you know, building costs, I guess. It looks great, but it's what's called a ruled surface and that means you can actually make it with straight lines. And you can see that in the building on the right, which is the air traffic control tower at Newcastle Airport, actually. And you could see that it's made- The lattice of straight lines are defining this curved shape, which is really great, and in terms of building costs, it is cheaper to have straight line beams than to make curved ones. So you see these shapes and there is actually, there's another kind of shape which I haven't got a picture of, but if you remember the London Olympics, the velodrome there got the nickname the Pringle, right? Because of a well known snack food that it looks like. That has a cross section. So in one direction its cross section are hyperbolas, and in the other direction its cross are parabolas. So it is a hyperbolic paraboloid. So impress your friends and family when they are eating certain snack foods by telling them that they are eating hyperbolic paraboloids. We must never say Pringles again. So that's how we see hyperbolas in building. In the last few minutes, I want to show you a way- It's not really a practical way of making hyperbolas, but I quite like it anyway. I call it tidying rectangles. And the inspiration for this is tidying up after small children who have building blocks all over the floor, and just imagine that you happen to have a rather strange set of building blocks where they all have the same area, all right? So I picked 12, 'cause it has lots of factors, but you could pick any number and then you can have these various blocks, 12 times what? 12 by one or three by four, six by two. Let's not be prejudiced against non whole numbers, eight by one and a half. Anything you like as long as the area is 12 and we're going to tidy them away. We're just going to put them all in the corner nicely, not all over the floor. And so here here's my corner, this is the axes now graphed really, and I'm going to tidy up my rectangles. So I'm just going to put them all nicely into the corner. There they all go, very nice. And if you do this, you start to see that looks like some sort of curve maybe. Zero points at this stage, given the context of what I'm talking about. I mean, here's a hyperbola, right? But let's see why it's a hyperbola. So if we think about any old rectangle that we've got here, the top right-hand corner of the rectangle, that's the bit where, these are sort of making all the curve there. So if you've got a rectangle of width X and height Y, then the top right-hand corner will be the point XY, right? And the area of that rectangle will be width times height, which is X times Y. And we've said in this case, that will be 12. So all the points that are making up this curve have the property that X times Y is 12. So Y equals 12 over X. Well, that's exactly an example of one of the hyperbolas I mentioned earlier, right? So tidying rectangles in this way results in hyperbolas. I mean, okay, it's half a hyperbola. You'd have to add in rectangles of dimension minus four by minus three if you want the other half. That might be a little advanced for child's building blocks. But anyway, this is how you get this hyperbola. Okay. I've promised three examples or three applications of each conic and here is the final, the third application of the hyperbola and it's in economics. Now in economics, you have something called the law of demand, which says that usually for goods or products, when the price rises, the demand falls. If the price of cars suddenly doubles, then fewer cars will be sold. There's one or two little exceptions to this. For example, super high end luxury goods. Sometimes part of the appeal is that they're very expensive and almost nobody can afford them. So all the billionaires will want one. If you increase the price of those things, then the cache goes up and so maybe the demand goes up. Then all the trillionaires will definitely want one, but all the billionaires will as well. So that's an extreme example. At the other end of things, if the price of something is quite low, that can sometimes signal that the quality is perhaps low. So if you lower the price even more, that can be a bad signal and perhaps the demand goes down. But outside of those two special cases, in the vast majority of cases, there's an inverse relationship between the price and the demand, and this is plotted on what's called the demand curve. So it plots the price per item against the quantity demand. In other words, how many you can sell at that price point. And different goods have different demand curves, but one nice example is what, again, you might call the kids' sweet shop budget, which is if I give a kid 50 P to spend on sweets, they will be spending that 50 P, right? Doesn't matter what the sweets cost. If sweets are five P, they buy 10, if sweets are two P, they're buying 25 sweets, right? So they will spend their whole budget and it doesn't matter what the price is, they're just going to spend all of the money. So if you take any point on the demand curve, so at this point here, imagine drawing a rectangle underneath it, so we were doing our tidying rectangles. A rectangle underneath there, the kind of the width of that rectangle will be the price per sweet, and the height of the rectangle will be how many sweets get sold. So the total spend on sweets, it will be the price per sweet multiplied by the number that gets sold. And that gives you the area of the rectangle, right?'Cause it's the width times the height. So in this case where you're going to spend your whole budget, the area of the rectangle is going to be that whole budget, in our case, the 50 P that we had to spend on sweets. And that's true for any point on this curve, right? You're spending your whole budget. So the area of any rectangle we draw under this curve will be the total budget, our 50 P. And we've already seen that if you do that, if you pile up all your rectangles like that, you exactly get a hyperbola. So in that case where you have a fixed budget, the demand curve is exactly a hyperbola. And of course, you know, this is a toy example, but there are real life situations where we would do this. I mean, even as adults we don't have a sweets budget, well, maybe we do, but we might have something like a budget for eating out or a budget for holidays or something like this and we will spend that budget, and if the cost of eating out goes up, we eat out a bit less, but we have a fixed budget. So this is a really important example in economics, and it's a hyperbola, which is great. So I hope you've enjoyed this guided tour of conics. And, you know, it really highlights for me the issue with any kind of pressure to only study useful things. I mean, of course, I'm sure we all agree in this room that knowledge itself is worth having. You know, studying things because they're interesting and wonderful and beautiful, that's already enough. But imagine if Menaechmus had to say to a grant committee,"What are the applications of these conics of yours?" You know, you can't say,"Well, 2000 years from now, they'll be for treating kidney stones, seeing distant stars." You just never know, you never know. And so this is a plea, let's all study things for the love of them. Okay, and I will finish there with a little plug for my next lecture, the final one of this academic here, more coming next year, The Incredible Sine Wave and Its Uses and that's on the 23rd of May, so I hope to see you there. Thank you.(crowd applauds)- Are there parallel conic sections for fourth and higher dimensions?- Yes. Yes, but I'm not aware of any applications of them as yet, although let's wait 2000 years and maybe we will.- Great. And the second one is, is it true that there is no analytical formula for the length of an ellipse, its circumference?- Yeah, this is so weird. There's a really- Right, so with a circle, The area of the circle is Pi R squared, right? And the circumference two Pi R, easy. Ellipse, area of the ellipse in terms of the A and B, I said, it's Pi A plus B, right, squared? And circumference, no formula. Well, there is a formula, but it's got like integrals and calculus stuff and it's really hard. So that's a very, very odd and curious thing about ellipses, but yeah, that's true.- Thank you so much, Professor Hart. I'm sorry to bring these questions to a close. It's just we've overrun a bit. Thank you very much for the fascinating lecture.(crowd applauds)