Aimed at one focus, turned towards the other

The rung below showed the ellipse’s mirror property: a ray leaving one focus in any direction arrives at the other. The hyperbola has a property of the same kind and it is not the same property, and the difference is the difference between the two curves’ defining conditions.

An ellipse is the set of points whose two distances add to a constant. A hyperbola is the set whose two distances differ by a constant. One word changes, and the mirror behaves differently.

What the difference forces #

At a point $P$ of the near branch, write $r_1$ for the distance to the far focus and $r_2$ for the distance to the near one. The defining condition is $r_1 - r_2 = 2a$, the same $2a$ everywhere on the branch, and the figure checks it at every bounce point to nine decimal places.

Now the tangent. Moving $P$ along the curve keeps $r_1 - r_2$ constant, so the rates at which $r_1$ and $r_2$ change are equal — and each of those rates is the cosine of the angle between the tangent direction and the corresponding focal radius. Equal cosines means equal angles, so the tangent makes equal angles with the two focal radii.

That is word for word the ellipse’s condition. What differs is which side the two radii are on. For the ellipse the two foci are inside and the radii point the same way across the tangent, so the equal-angle condition means a ray from one focus reflects to the other. For the hyperbola the two foci are on opposite sides of the branch, and the same equal-angle condition means a ray heading for the far one leaves heading for the near one.

One condition, two configurations, two statements. The tangent of an ellipse is the external bisector of the angle between the focal radii and the tangent of a hyperbola is the internal one, and everything else follows.

That pair of words is worth unpacking once, since it is where the whole difference is stored. Two rays from a point divide the plane into two pairs of vertical angles, and there are two lines bisecting them: one splitting the angle between the rays, one splitting the angle between one ray and the other’s continuation. The first is internal and the second external, and they are perpendicular to each other.

So the tangent to an ellipse and the tangent to a hyperbola through the same point with the same two foci are at right angles. That is not a coincidence and it is not a remark: it is the next rung’s entire subject, arriving here as a consequence of the two bisectors being the two bisectors.

The same argument without calculus #

The rung below gave a shortest-path proof of the ellipse’s property, and the hyperbola has the corresponding one with a subtraction in it.

Take a straight line and two points on the same side of it. Among the paths from one point to the line and on to the other, the shortest is the one making equal angles — reflect one point across the line and take a straight segment. That argument proves the ellipse’s property.

Now take two points on opposite sides of the line and ask for the path whose two legs differ by as much as possible — an extremal question of the kind that also decides how much area a fixed boundary can hold. The answer is again an equal-angle configuration, found the same way: reflect one point, and the difference of the two legs is largest when the three points are collinear, by the triangle inequality.

The tangent to a hyperbola at $P$ is the line for which $P$ is that extremal point, because every other point of the tangent lies on the wrong side of the branch and has a smaller difference. So the equal angles follow from an extremal property, with no derivative anywhere — and the two curves’ proofs differ by whether a sum is minimised or a difference maximised.

What it is used for #

The property is not a curiosity; it is why hyperbolic mirrors are ground.

The Cassegrain telescope. A parabolic primary mirror gathers parallel light and sends it to a focus. Put a small hyperbolic secondary mirror in the way, positioned so that one of its foci coincides with the primary’s, and every ray heading for that focus is turned towards the hyperbola’s other focus — which can be placed behind a hole in the primary, where an instrument sits. The design uses two of this ladder’s reflection properties in sequence and would not work with either alone.

The arrangement is worth stating as an operation on light rather than as a piece of hardware: the parabola converts parallel to converging on a point, and the hyperbola converts converging on one point to converging on another. The second is a relocation of a focus, which is exactly what a secondary mirror is for, and no other curve does it.

The two mirrors’ division of labour deserves one more sentence, because it explains a design choice that looks arbitrary. A single parabolic mirror already brings light to a point, so the secondary is not doing the focusing; what it is doing is moving the focus from a place where an instrument would block the light to a place where it would not, and lengthening the effective focal length while it is at it. The hyperbola is the unique curve that relocates a focus without introducing a new kind of aberration, and its being unique is the same uniqueness the rung below established for the parabola: demand the property, write it as a condition on the slope, and the differential equation’s solutions are exactly this family.

Hyperbolic navigation. The constant-difference definition is a statement about timing. Two transmitters send a synchronised pulse; a receiver measures the difference in arrival times, which is a difference of distances, and therefore knows it lies on one branch of one hyperbola. A second pair gives a second hyperbola, and the crossing is the position. That was the LORAN system, and the mathematics is the definition read backwards — with the crossing of two branches playing the part that the intersection of two circles plays in a construction.

Where the branches come from #

A hyperbola has two branches and the definition explains why, which the drawing does not.

The condition $|r_1 - r_2| = 2a$ has an absolute value in it. Dropping it gives $r_1 - r_2 = 2a$, which is one branch — the points nearer the second focus — and $r_2 - r_1 = 2a$ gives the other. The two branches are the two signs, and they are two separate curves that the standard equation happens to describe together.

That matters for the reflection. Each branch has its own near focus and its own far one, and the property is stated per branch. A ray aimed at $F_1$ striking the branch nearer $F_2$ is turned towards $F_2$; the same ray striking the other branch does something else entirely, and the figures draw the far branch dashed for exactly that reason.

The two-branch structure is also where the curve’s algebra differs from the ellipse’s. $x^2/a^2 + y^2/b^2 = 1$ has bounded solutions; $x^2/a^2 - y^2/b^2 = 1$ requires $|x| \ge a$ and has two unbounded pieces, and the sign in the equation is the same sign as the one in the definition.

There is a reading of the two branches that makes them one object again, and it is worth having because it is the one the next rungs use. On the cone the section is a single connected curve; it appears as two pieces on the plane because the plane misses the cone’s apex, and the two pieces are joined through it. Add the points at infinity — the single point that turns a plane into a sphere is the same manoeuvre one dimension down — and the two branches close up into one closed curve, meeting at the two points where the asymptotes run off.

On that reading a hyperbola is an ellipse that happens to cross the line at infinity, and the four conics stop being four. Nothing in the reflection property survives the identification, because reflection needs distances and the points at infinity have none; but the classification does, and it becomes much simpler.

What the constant difference is, in the other definitions #

The hyperbola has the same three descriptions the ellipse has, and the constant difference is what each of them produces.

From the cone. The cut is steep enough to meet both nappes of the double cone. Dandelin’s argument runs with one change: the two inscribed spheres now touch the plane on opposite branches, and the tangent-length equality that gave a sum for the ellipse gives a difference here — because the two tangent lengths are measured along the same generator in opposite directions rather than the same one.

From the focus and the directrix. The distance to the focus is a fixed multiple of the distance to a line, and the multiple exceeds one. That single number, the eccentricity, is what the constant difference amounts to: $2a$ and the focal separation $2c$ satisfy $e = c/a$, and $e > 1$ is exactly the condition that the distances differ rather than adding.

And from the algebra. $x^2/a^2 - y^2/b^2 = 1$ with $c^2 = a^2 + b^2$. The plus sign in that relation is the one that separates the two curves — the ellipse has $c^2 = a^2 - b^2$ — and it is why a hyperbola’s foci lie outside its vertices while an ellipse’s lie inside.

Three descriptions and one curve, exactly as the rung below records for the ellipse. What is worth noticing is that the reflection property is easy in only one of them: the focal one. From the cone it is invisible, and from the algebra it is a calculation.

The asymptotes, which the ellipse has nothing like #

A hyperbola has two straight lines it approaches and never meets, and they are the one feature with no counterpart on the closed curve.

They come out of the equation directly: for large $|x|$ the constant $1$ on the right is negligible beside the two large terms, so $x^2/a^2 - y^2/b^2 \approx 0$, which factors into the two lines $y = \pm (b/a)x$. The curve approaches them and the gap goes to nought without ever closing.

In the reflection picture they say what happens to a ray striking the branch far from its vertex. Out there the curve is nearly straight, the normal is nearly perpendicular to the asymptote, and a ray aimed at the far focus is nearly parallel to the asymptote — so the reflected ray is nearly parallel to it too, and the “convergence at the near focus” happens at a very shallow angle over a very long distance.

A property that holds exactly everywhere can be useless where the geometry is nearly degenerate, and that is the practical limit on how much of a hyperbolic mirror is worth grinding. The figures draw the part near the vertex for that reason, and the sample is honest about being a sample rather than the whole branch.

Where the account needs care #

The rays are aimed, not emitted. For an ellipse a ray leaves a focus; for a hyperbola a ray arrives from far away travelling towards a focus and never gets there. That distinction is invisible in a still picture — the drawn segment looks the same either way — and it is the whole difference between an emitter and a mirror in an optical system.

The near focus is not on the concave side. Both foci of a hyperbola are inside the two branches’ opening, and each branch’s near focus is on its convex side. An ellipse’s foci are both inside the closed curve. That is why the ellipse concentrates and the hyperbola relocates.

The property is about one branch. A statement about “the hyperbola” reflecting is ambiguous until the branch is named, and the ambiguity is not harmless: the two branches turn rays towards different points.

And the mirror has to be shaped exactly. As with the parabola, a spherical approximation to a hyperbolic secondary introduces aberration, and the tolerance in a telescope’s secondary is as tight as in its primary because a small mirror close to the focus magnifies the error.

The third case, and the one with no second focus #

Putting the three mirrors side by side says what the family is doing.

The ellipse takes rays from one point to another point. The hyperbola takes rays aimed at one point towards another point. The parabola takes rays from one point to a direction, and rays from a direction to a point.

The parabola is the limit of both. Push an ellipse’s second focus away and the rays arriving there arrive more and more nearly parallel; push a hyperbola’s far focus away and the rays aimed at it arrive more and more nearly parallel. The parabola is where the two families meet, which is the same statement as its eccentricity being exactly one and its cut being exactly parallel to the cone’s side.

What the pictures cannot show #

Thirteen rays are drawn and the property holds for every direction. The figures are evidence that the reflection has been implemented correctly, and the argument in the second section is what covers the infinitely many rays that are not drawn.

The pictures also cannot show the ray that misses. A ray aimed at the far focus but arriving on the wrong side of the branch does not strike the mirror at all, and the drawn sample is chosen to strike it — so the figure shows the property where it applies and gives no sense of the aperture it applies over, which is the first thing an optical designer would ask.

And nothing here is a wave. Every segment is a geometric ray, and the concentration a real mirror achieves is limited by the wavelength in a way that no ray diagram contains.

The ladder from here #

Rungs above: confocal conics, where the ellipse and the hyperbola with the same foci turn out to meet at right angles and so form a coordinate system. The single sign that names the curve, where the general quadratic is classified. The caustic, which is what a mirror of the wrong shape produces instead of a focus. Conics as projective objects, where the distinction between the bounded and unbounded cases dissolves. And the reflection property of a general curve, where the equal-angle condition becomes a differential equation whose solutions are exactly the conics.

A word changed, and a proof that survived it #

The habit is worth naming because the saving is large and the move looks like cheating.

The ellipse’s proof is: the defining quantity is constant along the curve, so its rate of change vanishes, so the two focal radii change at rates that cancel, so the angles are equal. The hyperbola’s proof is the same four steps with add replaced by subtract, and the cancellation becomes an agreement.

Nothing in the argument had to be redone, because nothing in it used which of the two operations was involved — only that the combination was constant. A proof written to depend on the sum would have needed a second proof; a proof written to depend on constancy covers both.

The general form is to notice which feature of a hypothesis an argument actually consumes. Here it consumes a function of the two distances is constant, which is satisfied by the sum, the difference, and — as the constant-ratio definition shows — by other combinations too. The class of objects a proof covers is decided by what the proof reads, not by what the statement says, and reading the proof for that is usually cheaper than generalising the statement and starting again.