My Little 2021 Mathematics A-to-Z: Hyperbola

John Golden, author of the Math Hombre blog, had several great ideas for the letter H in this little A-to-Z for the year. Here’s one of them.

Hyperbola.

The hyperbola is where advanced mathematics begins. It’s a family of shapes, some of the pieces you get by slicing a cone. You can make an approximate one shining a flashlight on a wall. Other conic sections are familiar, everyday things, though. Circles we see everywhere. Ellipses we see everywhere we look at a circle in perspective. Parabolas we learn, in approximation, watching something tossed, or squirting water into the air. The hyperbola should be as accessible. Hold your flashlight parallel to the wall and look at the outline of light it casts. But the difference between this and a parabola isn’t obvious. And it’s harder to see parabolas in nature. It’s the path a space probe swinging past a planet makes? Great guide for all us who’ve launched space probes past Jupiter.

When we learn of hyperbolas, somewhere in high school algebra or in precalculus, they seem designed to break the rules we had inferred. We’ve learned functions like lines and quadradics (parabolas) and cubics. They’re nice, simple, connected shapes. The hyperbola comes in two pieces. We’ve learned that the graph of a function crosses any given vertical line at most once. Now, we can expect to see it twice. We learn to sketch functions by finding a few interesting points — roots, y-intercepts, things like that. Hyperbolas, we’re taught to draw this little central box and then two asymptotes. Also, we have asymptotes, a simpler curve that the actual curve almost equals.

We’re trained to see functions having the couple odd points where they’re not defined. Nobody expects to mean anything when is zero. But we learn these as weird, isolated points. Now there’s this interval of x-values that don’t fit anything on the graph. Half the time, anyway, because we see two classes of hyperbolas. There’s ones that open like cups, pointing up and down. Those have definitions for every value of x. There’s ones that open like ears, pointing left and right. Those have a box in the center where no y satisfies the x’s. They seem like they’re taught just to be mean.

They’re not, of course. The only mathematical thing we teach just to be mean is integration by trigonometric substitution. The things which seem weird or new in hyperbolas are, largely, things we didn’t notice before. A vertical line put across a circle or ellipse crosses the curve twice, most points. There are two huge intervals, to the left and to the right of the circle, where no value of y makes the equation true. Circles are familiar, though. Ellipses don’t seem intimidating. We know we can’t turn (a typical circle) into a function without some work. We have to write either or , breaking the circle into two halves. The same happens for hyperbolas, though, with (a typical hyperbola) turning into or .

Even the definitions seem weird. The ellipse we can draw by taking a set distance and two focus points. If the distance from the first focus to a point plus the distance from the point to the second focus is that set distance, the point’s on the ellipse. We can use two thumbtacks and a piece of string to draw the ellipse. The hyperbola has a simliar rule, but weirder. You have your two focus points, yes. And a set distance. But the locus of points of the hyperbola is everything where the distance from the point to one focus minus the distance from the point to the other focus is that set distance. Good luck doing that with thumbtacks and string.

Yet hyperbolas are ready for us. Consider playing with a decent calculator, hitting the reciprocal button for different numbers. 1 turns to 1, yes. 2 turns into 0.5. -0.125 turns into -8. It’s the simplest iterative game to do on the calculator. If you sketch this, though, all the points (x, y) where one coordinate is the reciprocal of the other? It’s two curves. They approach without ever touching the x- and y-axes. Get far enough from the origin and there’s no telling this curve from the axes. It’s a hyperbola, one that obeys that vertical-line rule again. It has only the one value of x that can’t be allowed. We write it as or even . But it’s the shape we see when we draw , rotated. Or a rotation of one we see when we draw . The equations of rotated shapes are annoying. We do enough of them for ellipses and parabolas and hyperbolas to meet the course requirement. But they point out how the hyperbola is a more normal construct than we fear.

And let me look at that construct again. An equation describing a hyperbola that opens horizontally or vertically looks like for some constant numbers a, b, and c. (If a, b, and c are all positive, this is a hyperbola opening horizontally. If a and b are positive and c negative, this is a hyperbola opening vertically.) An equation describing an ellipse, similarly with its axes horizontal or vertical looks like . (These are shapes centered on the origin. They can have other centers, which make the equations harder but not more enlightening.) The equations have very similar shapes. Mathematics trains us to suspect things with similar shapes have similar properties. That change from a plus to a minus seems too important to ignore, and yet …

I bet you assumed x and y are real numbers. This is convention, the safe bet. If someone wants complex-valued numbers they usually say so. If they don’t want to be explicit, they use z and w as variables instead of x and y. But what if y is an imaginary number? Suppose , for some real number t, where . You haven’t missed a step; I’m summoning this from nowhere. (Let’s not think about how to draw a point with an imaginary coordinate.) Then is which is . And despite the weird letters, that’s a circle. By the same supposition we could go from , which we’d taken to be a circle, and get , a hyperbola.

Fine stuff inspiring the question “so?” I made up a case and showed how that made two dissimilar things look alike. All right. But consider trigonometry, built on the cosine and sine functions. One good way to see the cosine and sine of an angle is as the x- and y-coordinates of a point on the unit circle, where . (The angle is the one from the point to the origin to the point (1, 0).)

There exists, in parallel to the familiar trig functions, the “hyperbolic trigonometric functions”. These have imaginative names like the hyperbolic sine and hyperbolic cosine. (And onward. We can speak of the “inverse hyperbolic cosecant”, if we wish no one to speak to us again.) Usually these get introduced in calculus, to give the instructor a tiny break. Their derivatives, and integrals, look much like those of the normal trigonometric functions, but aren’t the exact same problems over and over. And these functions, too, have a compelling meaning. The hyperbolic cosine of an angle and hyperbolic sine of an angle have something to do with points on a unit hyperbola, .

Thinking back on the flashlight. We get a circle by holding the light perpendicular to the wall. We get a hyperbola holding the light parallel. We get a circle by drawing with x and y real numbers. We get a hyperbola by (somehow) drawing with x real and y imaginary. We remember something about representing complex-valued numbers with a real axis and an orthogonal imaginary axis.

One almost feels the connection. I can’t promise that pondering this will make hyperbolas be as familiar as circles or at least ellipses. But often a problem that brings us to hyperbolas has an alternate phrasing that’s ellipses, a nd vice-versa. But the common traits of these conic slices can guide you into a new understanding of mathematics.

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Thank you for reading. I hope to have another piece next week at this time. This and all of this year’s Little Mathematics A to Z essays should be at this link. And the A-to-Z essays for every year should be at this link.

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