Every ray comes back to the other focus

The four conic sections come from one cone cut at four angles, and the same four curves come from a definition with no cone in it: the ellipse is the set of points whose distances to two fixed points add to a constant. Dandelin’s spheres are what reconcile the two accounts.

This essay is about what that focal definition forces, which turns out to be a fact about mirrors.

Every ray, in every direction. The rays in the figure are not chosen; they are aimed at points spread evenly round the curve, and the reflected direction is computed from the curve’s own normal by subtracting twice the component along it — the same arithmetic a ray tracer does. Where the reflected ray goes is then measured, not drawn.

The tangent makes equal angles #

The reflection law says the angle of incidence equals the angle of reflection, both measured from the surface. So a ray from one focus leaves along the other focal radius exactly when the tangent makes the same angle with both radii.

That equality is the whole content of the reflection property, and there is an argument for it that uses no calculus at all.

Consider the shortest path from one focus to the other that touches a given straight line. The classical solution is to reflect the second focus across the line and draw a straight segment: the shortest path bounces at the point where that segment crosses, and at that point the two legs make equal angles with the line.

Now take the line to be the tangent to the ellipse at a point P. Every point of that tangent except P lies outside the ellipse, so the sum of distances from it to the two foci is greater than the constant sum — and at P it equals the constant. So P is the point of the tangent line minimising the sum, which by the reflection argument is the point where the two legs make equal angles.

The reflection property is therefore a consequence of the focal definition plus the observation that a tangent touches without crossing, and no derivative appears anywhere.

The same argument run backwards produces the ellipse rather than assuming it, and this is the version a gardener knows. Pin the two ends of a string of fixed length and draw with the string taut: the pencil traces the points whose two distances add to the string’s length, which is the curve. The taut string is why the construction works and also why the reflection follows — at every position the string is bent at the pencil, and the two halves make equal angles with the direction the pencil is free to move, since any other angle would mean the string could be tightened. A fixed length of boundary turns up again as the thing that forces a shape, which is the same manoeuvre one dimension down.

Where one focus has gone to infinity #

A parabola has one focus and a line. Push the second focus of an ellipse further and further away, keeping the near focus fixed, and the rays coming from it arrive more and more nearly parallel; in the limit the curve is a parabola and the far focus is nowhere.

That is the property every dish is built on. A satellite dish, a headlamp, a solar concentrator and a reflecting telescope all use the same fact: parallel rays along the axis converge on the focus, and rays from the focus leave parallel.

It is worth being precise about the direction. The property holds for rays parallel to the axis and not for rays arriving at an angle, which is why a telescope has to be pointed and why an off-axis star is imaged imperfectly — the aberration called coma is exactly the failure of this property away from the axis.

The figure also checks the other definition of a parabola on the way: each point of it is as far from the focus as from a fixed line, to twelve decimal places. That equality is what makes the reflection work, by the same shortest-path argument as before with the line playing the part of the second focus.

The same equal angle, elsewhere on the site #

The equal-angle condition is a local statement about a curve and a pair of directions, and it turns up whenever a path has to be optimal.

The inscribed angle is the neighbouring result and the comparison sharpens both. There, an angle stays the same as its vertex moves along an arc; here, two angles stay equal as a point moves along an ellipse. Both are invariance statements, both are usually proved by a construction rather than a computation, and both have the property that once the invariance is noticed the theorem is almost finished.

The difference is what the invariance buys. The inscribed angle’s constancy is used to prove things about circles; the equal-angle property of an ellipse is used to build things. That an equally elementary fact should be structural in one case and industrial in the other is not a pattern, but it is worth noticing how far a statement about two angles can be made to travel.

Four curves, one number #

Once a focus and a line are in hand, all four conics arrive from a single dial.

The ratio is the eccentricity. A circle has eccentricity 0, an ellipse between 0 and 1, a parabola exactly 1, and a hyperbola above 1 — the same four curves the cone produces, from a definition mentioning neither cones nor sums of distances.

Three definitions of one family, then: slice a cone, fix the sum of two distances, or fix the ratio of a distance to a line. The Dandelin spheres connect the first to the second, and a short calculation connects the second to the third. That a family of curves has three unrelated-looking descriptions is the reason it has been studied for two thousand years without exhausting anybody.

What the property is used for #

The reflection property is the reason these curves are built into things.

Optics. Parabolic mirrors in telescopes and headlamps, elliptical mirrors in laser cavities and in lamps designed to transfer light from one point to another, and hyperbolic secondary mirrors in the standard telescope design — where the hyperbola’s own reflection property sends rays aimed at one focus off toward the other.

Sound. A whispering gallery is an elliptical room: a whisper at one focus is audible at the other and nowhere else, because every path leaving the first focus arrives at the second having travelled the same total distance and therefore arriving in step. The equal-length property is doing as much work there as the equal-angle one — the sound would arrive whatever the timing, and it is the constant sum that makes the arrivals reinforce rather than cancel.

Medicine. A lithotripter is an ellipsoidal reflector with a shock source at one focus and the patient’s stone at the other, which is the same fact used to concentrate energy rather than light.

Orbits. A planet moves on an ellipse with the star at one focus, which is a different theorem entirely — it comes from an inverse-square force rather than from reflection — and the coincidence of vocabulary has confused generations of readers. Nothing bounces.

The last of those deserves its own sentence, because the two facts are so often run together. That an orbit is an ellipse and that an ellipse reflects to its other focus are unrelated statements about the same curve: the first is a consequence of a force law and would be false for any other exponent, while the second is a consequence of the curve’s definition and would hold if no force existed at all. What they share is the word focus, and the word was attached to the geometric point centuries before anybody knew a planet went near one.

Why the four are one family #

The three definitions are worth putting side by side once, since each makes a different property obvious and hides the others.

The cone makes the family obvious and the foci invisible. Four curves from one construction with one parameter, and no hint that any of them has a special point.

The focal sum makes the reflection obvious and the family invisible. The ellipse’s definition is about two points; the parabola’s is about a point and a line; the hyperbola’s is about a difference of distances. Three definitions that look unrelated.

The eccentricity makes the family obvious again and the reflection nearly so. One focus, one line, one number, and the four curves are the four ranges of that number — but the second focus, which is what the reflection property is about, is nowhere in the description.

That no single description makes everything obvious is the ordinary condition of a well-studied object, and it is the reason the cone-to-focus bridge was worth building in 1822 when both halves had been known for two millennia. A translation between two descriptions is not a restatement; it lets a fact that is easy on one side be exported to the other.

Where it fails, and what the drawing hides #

Three conditions are quietly in force.

The mirror must be the whole curve. A ray leaving a focus reaches the other focus after one bounce, and after the second bounce it returns to the first, and so on. A partial mirror — an arc rather than the whole ellipse — reflects only the rays that hit the arc, and the rest escape.

The equality is exact and the mirror is not. A real reflector is a manufactured surface with a tolerance, and the concentration achieved is limited by how closely the surface matches the ideal. That is why large telescope mirrors are specified to a fraction of a wavelength.

Reflection is a ray property. Light is a wave, and a perfect elliptical mirror does not focus a point source to a point but to a spot of a size set by the wavelength and the aperture. The geometry is exactly right and the physics puts a floor under the answer that the geometry cannot see.

The figures also hide a limitation of their own. The rays drawn are line segments computed from a reflection law; nothing in the picture is a wave, an intensity, or a beam of finite width. What is checked is a geometric incidence and nothing more.

There is one further gap between the picture and the claim, and it is the usual one on this site. Thirteen rays are drawn and the property holds for all of them; the theorem is about every direction, of which there are infinitely many. What closes that gap is the equal-angle argument, which refers to no particular ray — and the figure’s thirteen are evidence that the argument has been implemented correctly rather than evidence for the argument. A reader who wants more rays can have them; a reader who wants all of them needs the proof.

The choice of thirteen is worth a word too. An even number spread evenly round the curve would put a pair of rays exactly along the major axis, where the reflected ray retraces the incoming one and the picture shows a degenerate case rather than a general one. Odd counts avoid the coincidence, which is the sort of detail that separates a figure that happens to work from one that was checked.

What it costs #

Each ray in the figures costs a handful of arithmetic operations: a point on the curve, a normal from the gradient, a subtraction of twice a projection, and a check that the focus lies on the resulting line. That check is the assertion the figure would fail on — the distance from the focus to the reflected ray is required to be below a billionth of a drawing unit.

Computing the same thing for a general curve is harder in one specific way. The normal comes from a derivative, and for a curve given only as a list of points a numerical derivative has to be estimated, with an error that then propagates into the reflected direction. Ray tracing against curves defined by equations is exact for the same reason it is fast: the normal is a formula.

The total path length claim costs nothing at all. For an ellipse it is the defining property, so every path in the first figure is exactly the same length by construction — the figure computes each one and requires them to agree, which is a check of the drawing rather than a discovery about ellipses.

The ladder from here #

Rungs above: the reflection property proved by calculus, with the tangent’s slope from implicit differentiation. The hyperbola’s version, where a ray aimed at one focus reflects away from the other. Confocal conics, and the fact that an ellipse and a hyperbola with the same foci meet at right angles — which is what makes them a coordinate system. The optical properties in the wave picture, with the caustic that forms when the curve is not exactly right. Conics as projective objects, where all four are the same curve seen from different places and the classification collapses. Kepler’s laws, where the ellipse arrives from a force law rather than from a definition. And the reflection property of the Reuleaux curves and other constant-width shapes, which behave quite differently.

The shape of the idea #

The argument at the centre of this essay is a minimisation in disguise, and that is worth extracting because it recurs.

The reflection law says the angles are equal. Fermat’s observation is that the equal-angle path is the shortest path from one point to another via a line, so light reflecting is light taking the quickest route. And the tangent to an ellipse touches at exactly the point where the sum of the two distances stops decreasing, so the point of tangency is the minimiser and the equality of angles follows without any differentiation.

Three statements, one about mirrors, one about lengths and one about tangency, and they are the same statement. The same pattern makes the circle enclose the most area — a shape is optimal exactly when a local rearrangement stops improving it — and the technique of recognising a geometric property as an optimum is one of the most reliable ways of explaining why a curve behaves the way it does rather than merely establishing that it does.