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I foolishly said this on Twitter about a month ago:
David Butler | @DavidKButlerUoA | 14 Jun 2016
My PhD supervisor used to say 2 conics share 4 points - maybe coincident, maybe complex, maybe on the line at infinity
Tina Cardone 🏳️🌈 | @TinaCardone | 14 Jun 2016
on the line at infinity? Tell me about those!
At the time I declared this was a bit of a can of worms and I promised to write something and post it later. Well, here it is. But it might take a few blog posts to untangle it. In this post, I’m going to talk about how we can construct the line at infinity.
Let’s think about the geometry of the flat plane, and focus on just the straight lines and the points. Forget about the circles and the conics and the functions, just look at the straight lines. (I’m not even going to use the word straight any more, since straight lines are the only kinds of lines we’re going to look at.)
What can you say about points and lines? Well, if I had a single line, I could certainly locate a lot of points on the line. And if I started with a single point, there would be a lot of lines through that point.
If I had two points, then there would be exactly one line that I could draw through both of those points. And if I had two lines, then... well that’s not so easy. Either the two lines meet and there would be exactly one point they share, or they don’t meet at all, and we’d call them parallel.
But it’s not as if parallel lines have nothing in common. If you draw two lines parallel to each other, then they are obviously very very similar. Some might say they are the most similar kinds of lines. I find that rather interesting: somehow, by NOT meeting, two parallel lines are inextricably linked. What is it that they have in common?
Well, one thing they have in common is their slope. Two parallel lines have the same slope. If we’re thinking algebraically, you can rearrange their equations so that they have equations like y = mx+c and y = mx+d, for the same m but different c and d. Of course if they’re vertical, then the equations can’t be rearranged to look like this, but we can still say they have the same “slope” of “no slope”.
So, even if parallel lines don’t share a point, they DO share a slope. Interestingly, any two lines which do share a point, DON’T share a slope. So it seems that two lines must share one thing: either they share a point or a slope.
Now that we have this idea that two lines must share SOMETHING, they are a lot more like points.
Two points share a unique line; two lines share a unique point-or-slope.
This has a kind of pleasant symmetry about it. It would have even more symmetry if it was
Two points-or-slopes share a unique line; two lines share a unique point-or-slope.
Only it IS this! If I pick a point and a slope, there is a unique line through that point with that slope! (You may say that two slopes don’t share a line, and yes I realise this, but I’ll get back to it I promise.)
So it seems that slopes and points are the same kinds of thing, at least with respect to how they interact with lines. I think this is rather cool. It’s a bit philosophical, but often cool things are. The REALLY nice bit of philosophy is this: if points and slopes are so similar, why don’t I declare that a slope is a new kind of point? Then I can just say
Two points share a unique line; two lines share a unique point.
Can I just do that? Can I call something by a new name? It’s not as if the slopes really are points — I can’t see them or point to them (excuse the pun). But that sort of thing has never stopped us mathematicians before (think of the number 0). Plus, perhaps there is a way to see them after all...
If you look at a pair of parallel lines from above, then it actually does look like they ought to meet somewhere. Look out to the horizon and there they seem to meet. Of course if you walk towards it, that intersection point just keeps getting further away. However far you go towards it, it’s still just as far away. Of course, that’s because the point isn’t really there, it just LOOKS like it is. But it does look like it’s there, and you feel like if only you went far enough, you’d see it.
This is the extra bit of philosophy we need to legitimise calling our slopes points. It FEELS like two parallel lines actually do meet, but they meet infinitely far away. Philosophically, we know the two parallel lines DO share something that acts like a point acts — it’s their slope. So what we do is we declare that our slope-points we made earlier are called “points at infinity”.
It’s not just matching with our visual instincts either. If you were going to add a new point to a line, it would have to go SOMEWHERE, but there’s no space over here to put that extra point — our lines are already full. We’re going to have to add our extra point AT THE END of our line — that is, out there at infinity.
So now we have two kinds of points: we have our original common-or-garden points that we can see. And we have our new points at infinity that we can’t see. These points at infinity aren’t really there in the world, but they are really there in our minds — they are the slopes of the lines, and we know for sure that two parallel lines do indeed share a slope. We know that slopes are like points in the way they interact with lines, and so it’s ok to call them points. But we can’t see slopes either, so in a way it is only fitting to call them “points we can’t see”.
We can now start filling out our new terminology with some imagination. Say I have a point and I want to draw a line through it of a certain slope. Well, that slope is a “point at infinity”, so drawing the line we want would be joining this point to the appropriate “point at infinity”. I imagine it as standing at this point and looking out to infinity in the direction I want the line to go, and drawing the line in that direction. I am still joining it to that point at infinity, I just won’t ever get there.
You might be thinking that surely I could have looked in the opposite direction and drawn my line that way. Well, yes I could. And that’s where the analogy needs a bit of tweaking. Remember that these points at infinity are not really points that live in a real place: they are slopes that don’t live anywhere but in our minds. It looks like if we want them to live in the real world, we’re going to need them to be both in front of us and behind us. Needs must.
So there’s only one thing left to do: we have to deal with the niggling little detail that two points don’t quite share a unique line. We know that two ordinary points share a unique line (that was true before), and we know that an ordinary point and a point at infinity define a unique line (that’s an ordinary point and a slope, really).
But what about two points at infinity? That’s two slopes, and we know there’s no lines with two slopes. Well that’s ok, we’ll just collect all the slopes together to make one new line. Since this line is made up entirely of points at infinity, we’ll call it the line at infinity.
From now on, we will start talking about the points at infinity as if they were really real points, that really were really at infinity. If we try hard we can remember that they’re really just slopes, but we don’t have to do that to make the maths work. Our imagination will tie in fine with how the maths works, because we’ve carefully made it do so. We’ve had to bend our imagination a little to make the concept of a really real line at infinity fit with our words and pictures, but that’s ok, we can always check back with the slopes if we need to make sure. Plus, really cool maths often requires a change of perspective.
I made a promise to someone on Twitter to talk about how conics relate to the line at infinity, but when I came to do that, I realised it’s a can of worms that will take a few blog posts to untangle. Last time, I talked about how I construct and think about the line at infinity itself.
Back in that post, we added a line to our plane called the “line at infinity” made up of “points at infinity”. These points were really the slopes of the straight lines in our original plane, but we extended our imagination to give them a sort of physical reality. This produced a pleasant symmetry in the way points and lines interact. Two points share a unique line; two lines share a unique point. Now I’ll talk about how we might add coordinates to these new points at infinity.
All our ordinary points already have coordinates like this: (0,0), (1,3), (1/2, 2004), (-24, π). That is, they have coordinates that are two real numbers in pairs. We can’t make any more new coordinates like this, because we’ve already used up all the real numbers. Our new points are slopes, so we could just give them one coordinate, which is the slope that they really are. For example, the lines with equations y = 2x+1, y = 2x and y = 2x-7 could all share the point at infinity with coordinate (2). This isn’t such a bad solution, but it will require us to deal with lines of the form x = 4, x = 0, x = -17. We could give their point at infinity a coordinate of (∞), I suppose.
This is actually one of the ways that people do go about giving coordinates to the points at infinity. It preserves their “true” nature as the slopes of lines, and makes some rather useful connections to some other ideas in this corner of geometry. But I find it a bit unsatisfying. The whole idea of this was to elevate slopes to the status of proper points and here we are reminding ourselves of their origins as slopes. It feels like a sort of discrimination. Plus, I’d really love a system of coordinates that highlights the symmetry with lines in some way, which was the other reasons to introduce these points at infinity in the first place.
So let’s seek out a new way of doing coordinates. Since one of the goals is to emphasise the point-line symmetry, maybe the best place to start is to look at the lines.
Lines have equations. Most lines have equations of the form y = mx+c, for some numbers m and c, but this misses out the vertical lines of the form x = k, for some number k. To include all lines in one equation format, you can write them as ax + by = c for some numbers a, b and c. For reasons that will hopefully be clear later, I prefer to write them like this: ax + by + c = 0.
These numbers a, b and c are likely candidates to be coordinates for our lines. They describe very clearly which line we have in a way that relates nicely to our existing understanding of lines. So we could make the coordinates of our line [a,b,c]. The problem is that it’s not really “the” coordinates is it? In this format, every line has multiple equations. The line with equation 2x + 3y + 7 = 0 also has equation 4x + 6y + 14 = 0 and x + 3/2y + 7/2 = 0 and -2/3x – y – 7/3 = 0 and 200x + 300y + 700 = 0 and so on. This gives us coordinates of [2,3,7], [4,6,14], [1,3/2,7/2],[-2/3,-1,-7/3], [200,300,700] and so on. I suppose that’s not so bad, though — we can just say that any multiple of our coordinates for a line represents the same line.
So, can we do this for points? To make their coordinates match with the ones we have for lines, we should extend them to three coordinates. I suppose this makes sense because we’d already used up all the possibilities when we used two coordinates; it seems natural to just add an extra coordinate to our existing two in order to accommodate the extra points. But what should our third coordinate be? Maybe the way to solve this is to look at how points interact with lines.
The point (-5,1) is on the line with equation 2x + 3y + 7 = 0, because when I substitute the coordinates of the point into the equation, it works: 2*(-5) + 3*1 + 7 is indeed 0. Let’s look at how that works when you use our new “coordinates”. We have a point with an extra coordinate (-5,1,?) — we don’t know what the extra coordinate is yet — and a line with three coordinates [2,3,7] and we combined them together like this 2*(-5) + 3*1 + 7. Hmm. The first coordinate of the point matched up with the first coordinate of the line, the second coordinate of the point matched up with the second coordinate of the line. It seems reasonable that the third coordinate of the point ought to match up with the third coordinate of the line in the same way. That is, our calculation ought to be 2*(-5) + 3*1 + 7*(?). If this is going to work, that ? has to be a 1.
So, our ordinary point (-5,1) has new coordinates (-5,1,1), and we’ll say any point (x,y) has new coordinates (x,y,1). We’ve tacked on an extra coordinate, which is designed to combine with the constant term in our line equation. I’ve used the traditional x and y for the first two coordinates, since that’s what I was using in my line equations above anyway. We need a name for this last coordinate too. I don’t want to call it z, because we’re not really in 3D space. How about w? (Really I’d prefer to use the Greek letter omega ω, so that it means “the very end”, with reference to “alpha and omega”, but it looks like a w anyway and I can’t be bothered doing the symbol all the time.)
Our line equations turn from ax + by + c = 0 to ax + by + cw = 0, and it’s still true that you can tell if a point is on a line by substituting its three coordinates into the line equation. Alternatively, you could do the sumproduct (official name: dot product) of the line’s coordinates and the point’s coordinates: [a,b,c].(x,y,w) = ax + by + cw = 0. Lovely and symmetrical.
But it’s a bit bizarre to only be able to have a 1 in that last coordinate. What would happen if I had a 2 there? Well, with line coordinates any multiple of the coordinates represented the same line, why not do that for points too? Let’s check if it works: the point (-5,1,1) was on the line with equation 2x + 3y + 7w = 0. Does (-10,2,2) work? Well, 2*(-10) + 3*2 + 7*2 = -20+6+14 = 0. Yes it does. And I can see how that would work for any other multiple — we just multiplied each term by the same number, which is the same as multiplying the whole thing by that number, so the answer would have to still be 0. Now the symmetry is even more lovely — lines have three coordinates and any multiple of them represents the same line, and points have three coordinates and any multiple of them represents the same point.
Just one more thing to wrap up: if we DO have a random set of three coordinates like (6, -9, 3), how do we tell what its original set of two coordinates are? Well, you know that a point like (x,y) has coordinates (x,y,1) so all we have to do is make that third coordinate into a 1 and we’ll be good. We can divide by 3 to get (6,-9,3) = (2,-3,1) so the original point was (2,-3). In general we’ve got that a random point with coordinates (x,y,w) has its original coordinates (x/w,y/w). So we can now convert back and forth between our old coordinates and our new coordinates.
This is all well and good, but the whole purpose here was to have coordinates for our points at infinity! So far we just have coordinates for our ordinary points and our ordinary lines! Admittedly they are wonderfully symmetrical, but was that really worth the effort?
Well yes it was, because we haven’t covered all the possibilities for three coordinates, and those missing possibilities ought to cover our new points. All the ordinary points have a 1 in their third coordinate when you first construct them. If you multiply all three coordinates by a number, the third coordinate will be whatever that number is. But importantly, it won’t be zero. (It doesn’t make any sense to multiply them all by zero, or (0,0,0) could be any of our points and that’s just confusing.) So all of our ordinary points have a nonzero third coordinate — we’re missing all the points with 0 in their third coordinate. These ought to be our points at infinity.
If they really are points at infinity, then they ought to be the points where lines of the same slope meet. Let’s give it a go with an example. Take two lines with the same slope: 2x + 3y + 1 = 0 and 2x + 3y + 7 = 0. In our new coordinate system, there’s a third coordinate which attaches itself to the constant term, so we have 2x + 3y + w = 0 and 2x + 3y + 7w = 0. Subtracting these two gives us 6w = 0, so w = 0. That’s promising. Whatever point is on both those lines has 0 as its third coordinate. I wonder what the other coordinates are? Let’s sub w = 0 back into the first equation: 2x + 3y = 0=> 3y = -2x=> y = -2/3x. Hmm. I can’t figure out what the other coordinates are from here. Oh, well, the point will be something like (x, -2/3x, 0). But wait a minute! Any multiple of this set of coordinates ought to be the same point, so I could just divide by x! So my point is (1,-2/3, 0). Brilliant. Interestingly, that number -2/3 is exactly the slope of our line, so it turns out we do have the slope as part of our coordinates after all!
Let’s try this in general with the lines ax + by + cw = 0 and ax + by + dw = 0, with c not equal to d. Subtracting these gives (c-d)w = 0, so w = 0. So definitely whatever point is on both those lines has its third coordinate equal to zero. And subbing this back into either gives ax + by = 0. Hmm. I can’t do what I did last time and just divide by a or b because one of them might be zero. But I can guess a point satisfying the equation. It’s (-b,a,0) and any multiple of that will do. If b isn’t zero I can get (1,-a/b,0), which has the slope of the line as its second coordinate. Excellent.
This just leaves the line at infinity as a line. We already know that all the points on the line at infinity have their third coordinate equal to zero, so the equation of the line at infinity ought to be w = 0. Let’s see if this matches up with our missing coordinates.
All the ordinary lines have equations like ax + by + c = 0, based on the old coordinates. You can’t have both a and b being zero or there would be no points (x,y) satisfying the equation. But any other combination of a and b will be fine. In our new coordinates, this means we have covered all equations ax + by + cw = 0 where a and b aren’t both zero; that is, all line coordinates [a,b,c] where a and b aren’t both zero.
What happens if a and b ARE both zero? Well then we have cw = 0, which is w = 0. In line coordinates this is [0,0,c] which is the same as [0,0,1] when I divide by c. I know I can divide by c, because it doesn’t make sense to have ALL the coordinates being zero — 0x + 0y + 0w = 0 is always true, so this wouldn’t define a line but the entire plane.
So now we have a complete system of coordinates for both points and lines, whether ordinary or at infinity. They’re usually called “homogeneous coordinates” because the points and lines have the same set of coordinates, and because all the parts of the new line equations are the same – there’s no constant term anymore.
So what’s the benefit of all this? Well, the goal was to have coordinates for our points that allowed us to have coordinates for the points at infinity, and we’ve done that. What more do you want?
The real question is, why do we need coordinates at all? Well, one use for coordinates is locating things, such as the point where two lines meet, or the line that two points define. Our new coordinates unify this approach quite a lot. I’m not going to prove that it works now (maybe another day), but if I want to find the point where two lines meet, I just need to do the cross product of the two line coordinates. And if I want to find the line joining two points, I just need to do the cross product of the two point coordinates. This is fabulous!
For example, where do the two lines with old equations 2x + y + 2 = 0 and -x + 3y – 8 = 0 meet? Well, they have coordinates [2,1,2] and [-1,3,8], whose cross product is (-14,14,7) = (-2,2,1). And yes this point does actually satisfy both equations.
What is the equation of the line that joins the points with old coordinates (1,5) and (2,-7)? Well they have new coordinates (1,5,1) and (2,-7,1) whose cross product is [12,1,-17], so the line has equation 12x + y - 17 = 0.
And what is the equation of the line with slope 2 through the point with old coordinates (3,-4)? Well the point at infinity for lines with slope 2 is (1,2,0) so we need the cross product of (1,2,0) and (3,-4,1). This is [2,-1,-10] so the line has equation 2x – y -10 = 0.
I find this rather awesome, but not as awesome as the fact that lines and points now have coordinates that are not different in any essential way. That is really wild!
The other thing that coordinates can do is allow us to define objects using equations. We’ve already defined our lines using equations. The next kind of object is a conic, which is very very cool...
I promised Tina on Twitter that I would write about how the line at infinity relates to conics, and I’ve been doing it in the last two blog posts.
First, I talked about what the line at infinity is. We noticed that a set of parallel lines all share a slope, so we elevated all the slopes to the status of points and gave them a more physical reality as “points at infinity”. Now lines we thought were “parallel” aren’t anymore because they actually meet. ANY two lines meet, though possibly at a point at infinity.
Second, I talked about a system of coordinates that includes the line at infinity. It turned out we could do this by giving both points and lines three coordinates, with the understanding that any nonzero multiple of a set of coordinates represents the same point/line.
These coordinates are called homogeneous coordinates and they go like this: Points have coordinates of the form (x,y,w). Ordinary points have w not equal to zero, and their original coordinates can be calculated as (x/w, y/w). Points at infinity have w equal to zero, and a point of the form (a,b,0) is the point at infinity belonging to all the lines parallel to bx – ay = 0. Line equations go from ax + by + c = 0 to ax + by + cw = 0, which can be shortened to coordinates [a,b,c]. The line at infinity has equation w = 0.
Last time I noted how awesome it is that with these new coordinates, lines and points are not really distinguishable from each other, if we just look at their coordinates. There is another thing I think is very cool too: with these new coordinates, the points at infinity don’t look fundamentally different from ordinary points. So what if one of their coordinates is zero? Other points have a zero coordinate, such as all the points on the x-axis — they have coordinates of the form (x,0,w). The line at infinity has equation w = 0, but that’s not really any different from y = 0. If I try not to think about WHERE it is, and just focus on the coordinates, then the line at infinity is not different from any other line.
If we want to, we can think of this like magically travelling to stand on the line at infinity, and from that perspective it looks just like any other line. Interesting things happen when you shift perspective this way. For example, consider two parallel lines and a transversal: if you stand on the line at infinity, you can see the point at infinity involved, and the three lines become just an ordinary triangle. We’re going to talk about what happens when you do this to conics.
Conics began life millenia ago as literal sections of a cone. You took a circular cone, including both a top nappe and a bottom nappe, and you sliced it with a plane at various angles to produce various results – parabolas, ellipses, hyperbolas. If you arranged it just right, you got a point, or a single line or a pair of lines. These are often called the “degenerate conic sections”.
Later, when coordinate geometry became a thing, it turned out that the conic sections could be described as the set of points whose coordinates satisfy a quadratic equation. Take a quadratic equation in two variables like x2 – 3xy + y^2 – x + 4y – 7 = 0, and find all the points that work when you sub their coordinates into the equation and this give you a conic section. Very nice.
There are a few extra possibilities that appear from our equations that aren’t covered by our cone: a pair of parallel lines, and an empty set where no points at all satisfy the equation. To include these, we’ll drop the word “section” from the name, so that they’re still objects related to cones in some roundabout way, but not necessarily a section of a cone.
When we introduced a third coordinate to our points, and so introduced a third variable to our lines, we simply attached our new variable to the constant term in our line equation. Like this: 3x – 7y + 5 = 0 becomes 3x – 7y + 5w = 0. But it’s not so simple with quadratic equations. In fact, it’s just the slightest bit dodgy to simply attach a w to the constant term, because the x and y in the first equation aren’t really the same as the x and y in the second equation if w isn’t 1. A more rigorous approach would be to remember how to convert from three coordinates back to two like this: (x,y,w) becomes (x/w, y/w). So that “x” in the first equation is actually x/w, and the “y” is actually y/w. Then 3x – 7y + 5 = 0 becomes 3(x/w) – 7(y/w) + 5 = 0, and multiplying by w gives 3x – 7y + 5w = 0.
(Someone is sure to notice the slight dodginess here too: I’ve still used x and y to mean different things before and after the change. My geometry lecturer back in the distant past switched to capital letters to mark the change, but I think we’ll cope with just a little bit of notational sloppiness for the ease of explanation. I hope so anyway.)
This is how we’re going to deal with quadratic equations. That is, we’ll replace the x’s with x/w and the y’s with y/w:
x^2 – 3xy + y^2 – x + 4y – 7 = 0 (x/w)^2 – 3(x/w)(y/w) + (y/w)^2 – (x/w) + 4(y/w) – 7 = 0 x^2/w^2 – 3xy/w^2 + y^2/w^2 – x/w + 4y/w – 7 = 0 x^2 – 3xy + y^2 – xw + 4yw – 7w^2 = 0
(To get to this last line, we multiplied by w^2.)
So the linear terms get a w attached to them, and the constant term gets a w^2. This has the effect of meaning that in every term there are two variables multiplied together, either two different ones like xy or yw, or two of the same like x^2 or w^2. All the terms are the same in some sense now — homogeneous, as it were.
We’re now ready to figure out how our conics meet the line at infinity. Well, nearly. First it’s probably a good idea to know how conics meet ordinary lines! Drawing some pictures, I can convince myself that a line either misses a conic entirely, meets it in exactly one point, meets it in exactly two points, or (in the case of some degenerate conics) is actually part of the conic itself.
The algebra matches the drawings (as it ought to). For example, take a line equation in our old coordinates like 2x + 3y – 7 = 0 and a quadratic equation in our old coordinates like x^2 – 3xy + y^2 – x + 4y – 7 = 0. You can rearrange the line equation to get y in terms of x, and sub that into the quadratic equation to get a quadratic equation in terms of just x. And a quadratic equation has zero, one or two solutions, corresponding to zero, one or two points where the line meets the conic. The only other possibility is when everything cancels out in which case every x is a solution and the whole line is part of the conic.
There’s some terminology to name the different kinds of line with respect to a conic. If the line misses the conic entirely, we’ll say “the line is EXTERNAL to the conic”, or say “the line is an external line”. If the line meets the conic in exactly one point, we’ll say “the line is TANGENT to the conic”, or say “the line is a tangent”. And if the line meets the conic in two points, we’ll say “the line is SECANT to the conic”, or say “the line is a secant”. (And if the line is actually part of the conic we’ll say “the line is a GENERATOR LINE of the conic”, but that’s not going to come up for our line at infinity.)
A story about me: My PhD thesis was mostly concerned with tangents, secants, generators and external lines of conics and their higher-dimensional friends and relatives, so these words bring back all sorts of memories for me. In particular, I remember giving my second-year thesis seminar the title “David Butler and the Chamber of Secants”. (The first-year seminar was called “David Butler and the Philosopher’s Cone”, by the way.)
Ok, now we’re ready to find how our conics meet the line at infinity. Let’s try the conic I’ve been using as an example so far. Its equation is x^2 – 3xy + y^2 – x + 4y – 7 = 0, and in homogeneous coordinates this is x^2 – 3xy + y^2 – xw + 4yw – 7w^2 = 0. The line at infinity has equation w = 0, so let’s sub this in to get x^2 – 3xy + y^2 = 0. If we treat x as a constant, this is a quadratic equation in y, so we could complete the square/use the quadratic formula to get y = [3x ± √(9x^2 – 4x^2)]/2. Note the x^2’s under the root sign which mean we can pull it out, and indeed pull the x’s right out to get y = x [3 ± √5]/2. So we have two points at infinity of the form (x,x[3 ± √5]/2,0). We can divide by the x to get (1,[3 ± √5]/2,0), which looks a bit nicer. And actually, this means that the points at infinity on this conic are the same as the points at infinity on the lines with slope [3 ± √5]/2.
This is all well and good, but it’s not particularly edifying is it? No, not really. Perhaps it might be best to do this trick with our standard versions of the three non-degenerate conics. That might be a bit more illustrative. I’ll do them in this order: hyperbola, parabola, ellipse.
Your standard form of a hyperbola equation looks like this (x/a)^2 – (y/b)^2 = 1. In homogeneous coordinates this is (x/a)^2 – (y/b)^2 = w^2. When we sub in w = 0 we get
(x/a)^2 – (y/b)^2 = 0 (x/a)^2 = (y/b)^2 ±x/a = y/b ±xb/a = y
So our hyperbola meets the line at infinity at (x, ±xb/a,0) = (1, ±b/a,0). That’s two points, so the line at infinity is secant to our hyperbola. Interestingly, those are the two points at infinity on the lines with slope ±b/a. If you happen to know a lot about hyperbolas, these happen to be the slopes of its two asymptotes!
Actually, let’s just check that shall we? The two asymptotes of the hyperbola with equation (x/a)^2 – (y/b)^2 = 1 have equations y = (b/a)x and y = -(b/a)x. In homogeneous coordinates, we have (x/a)^2 – (y/b)^2 = w2, y = (b/a)x and y = -(b/a)x. Subbing y = (b/a)x into the conic’s equation gives
(x/a)^2 – (y/b)^2 = w^2 (x/a)^2 – ((b/a)x/b)^2 = w^2 (x/a)^2 – (x/a)^2 = w^2 0 = w^2 w = 0
So this asymptote meets the hyperbola at a point ON THE LINE AT INFINITY. I suppose this makes sense — the conic gets closer and closer to the asymptote, so it only makes sense that they actually meet AT INFINITY.
It’s interesting to note that the asymptote meets the conic in exactly one point (x, (b/a)x,0) = (1, b/a,0). That makes it a tangent. Also, even though the two branches of the hyperbola both approach the asymptote, there is only one point where they meet. That does make sense — way back in the first post, we figured out that the point at infinity on any line is both behind and in front of us. If we imagine standing on the line at infinity, you’ll be able to see this point where the conic meets the asymptote. One branch of the hyperbola meets this point on one side, and the other branch meets it on the other side. So here at infinity, the hyperbola links up to itself, like an ellipse.
Your standard form of a parabola equation looks like this: y = ax^2. In homogeneous coordinates, this is yw = ax^2. When we sub in w = 0 we get 0 = ax^2, so x = 0. This gives us one point (0,y,0) = (0,1,0). So the line at infinity meets our parabola in one point — it’s a tangent. Interestingly, this is the point at infinity on the y-axis, which is the axis of symmetry of the parabola.
Let’s check this with another parabola. What about x = ay^2? In homogeneous coordinates, this is xy = ay^2. When we sub in w = 0 we get 0 = ay^2, so y = 0. This gives us one point (x,0,0) = (1,0,0). So again we get one point, this time the point at infinity on the x-axis, which is again the axis of symmetry of the parabola.
Admittedly, this is only two examples, but I’ll trust for a moment that it works. If I do trust it, it seems if I stand on the line at infinity, it’s tangent to the parabola, and the parabola links up to itself at that spot, like an ellipse.
Your standard form of an ellipse equation looks like this: (x/a)^2 + (y/b)^2 = 1. In homogeneous coordinates this is (x/a)^2 + (y/b)^2 = w^2. When we sub in w = 0 we get (x/a)^2 + (y/b)^2 = 0. Since we can’t have both x and y being zero, this means (x/a)^2 + (y/b)^2 has to be a strictly positive number, so we have no solutions to this equation. In short, our ellipse doesn’t meet the line at infinity at all. If you like, the line at infinity is external to our ellipse. I suppose this makes sense — we can see the whole ellipse just fine so it shouldn’t meet the line at infinity.
So it turns out that there is actually only one kind of nondegenerate conic. Hyperbolas and parabolas are really just ellipses that meet the line at infinity in different ways. A hyperbola is an ellipse that has the line at infinity as a secant, and a parabola is an ellipse that has the line at infinity as a tangent. I often imagine it as an ellipse that is stretched out further and further, until it meets the line at infinity to become a parabola, and then crosses the line at infinity to come in on the other side as the opposite branch of the hyperbola.
And I think that might be enough. Except for one parting shot — if we have a random quadratic equation, we have a good chance of knowing what kind of conic it is by seeing how it meets the line at infinity. Our original conic with equation x^2 – 3xy + y^2 – x + 4y – 7 = 0 met the line at infinity in two points, so I can be pretty sure it’s a hyperbola. It is possible it’s actually a pair of lines, I suppose — a pair of lines also meets the line at infinity in two points. There is actually a way of telling the difference, which involves making a matrix and calculating its determinant. But that’s another story and shall be told at another time.
