The ball stands at a focus

A ball is resting on a floor, and a lamp is on somewhere above it. The shadow is an ellipse.

There is exactly one point of that arrangement a person could mark without doing any geometry at all: the place where the ball meets the floor. Reach under it with a pencil and there it is.

That point is a focus of the ellipse.

Not approximately. The distance between the contact point and the nearer focus, computed for a range of lamp positions, is a couple of parts in ten thousand million million of a metre, which is the arithmetic saying zero.

Where it comes from, which is a theorem about ice cream #

The result is Dandelin’s, from 1822, and it is normally met in a completely different room.

A cone is cut by a plane and the section is a conic. Germinal Dandelin’s construction inscribes a sphere in the cone, on one side of the cutting plane, large enough to touch that plane — and shows that the point where it touches is a focus of the section. It is the proof that the two definitions of a conic agree: the one that says slice a cone and the one that says the sum of the distances to two fixed points is constant. Every textbook draws it with a cone and a plane through it, and the spheres go by the name Dandelin spheres.

Now look at the ball on the floor again and read the same picture.

The lamp is a point. The rays from it that graze the ball form a cone — the tangent cone, with the lamp at its apex. The floor is a plane cutting that cone. And the ball is a sphere inscribed in the cone, touching the plane.

Every element of Dandelin’s construction is present, and the sphere is not an auxiliary object dropped in to make the proof work. It is the ball. So its point of contact with the floor is a focus of the section, and the section is the shadow’s edge.

That the geometry is available at all is the point of this site’s habit of computing both routes. The shadow’s outline can be arrived at by casting the silhouette circle point by point, or by writing the tangent cone as a quadratic form and intersecting it with the plane. This site does both, and the first is what draws and the second is what the foci come out of.

Why it is worth more than a curiosity #

Because it is the whole of what the shadow says.

Take the ellipse alone — the shape on the floor, with no ball and no lamp visible. What can be recovered? The instinct is that the shadow of a ball is highly informative: it is a conic, it has an eccentricity, it has a size and an orientation, and all of that came from somewhere.

It came from six numbers: where the lamp is, in three; where the ball’s contact point is, in two; and how big the ball is, in one. Six numbers went in and a conic in a plane is five. So the map from arrangements to shadows loses one dimension, and there is a one-parameter family of arrangements behind every shadow ellipse.

The theorem says which parameter. The contact point is pinned: it is a focus, and the two foci of an ellipse are computable from the ellipse. Everything else is loose. And the loose part turns out to be one of the tidiest curves in the subject.

The lamps that would have done it #

Fix the shadow ellipse. Which lamps could have cast it, with a ball of the appropriate size resting at the focus?

The answer is a hyperbola, in the vertical plane through the ellipse’s major axis:

$$\frac{u^2}{c^2} - \frac{y^2}{b^2} = 1$$

where u runs along the major axis from the ellipse’s centre, y is height above the floor, b is the ellipse’s semi-minor axis and c is its focal distance. Its vertices are the ellipse’s two foci. Its own foci are the ellipse’s two vertices. This is the classical focal pair of a conic, and it has arrived here as the answer to a question about lamps.

One branch sits over each focus, which is the second half of the theorem: the two Dandelin spheres of a cone touch the plane at the two foci, one from above and one from below, and here the branch over a focus is the family of lamps for which the ball rests there.

Running out along a branch, the lamp gets further away and the ball must grow to keep the shadow the same. A lamp twenty-five centimetres up needs a ball of eleven and a half centimetres. A lamp at sixty metres needs one of forty and a half. Every one of them casts precisely the same ellipse — the largest departure across that whole ladder is a few parts in a thousand million million of a metre, which is again the arithmetic saying they are identical.

So a photograph of a ball’s shadow, on its own, does not say how big the ball is or where the light is. It says where the ball is standing, exactly, and it draws a curve through space and says the lamp is somewhere on this.

The sun is the asymptote #

A hyperbola has asymptotes, and running the lamp out along a branch approaches one. A lamp infinitely far away in a fixed direction is the sun, and the direction the asymptote points is therefore the direction of a sun that casts the same ellipse.

Its altitude is arctan(b/c) — the ratio of the ellipse’s semi-minor axis to its focal distance — and that is a number a reader can take off the shadow with a ruler.

It checks. A sun set at a stated altitude, its shadow ellipse computed, the asymptote’s angle read back out: the two agree to within a hundred-millionth of a degree across every direction tried. The shadow’s own shape names the sun’s altitude and says nothing whatever about how far away the sun is, which is right, because the sun’s distance is exactly the quantity a parallel projection has thrown away.

This is worth pausing on, because the site has spent several essays on the difference between a lamp and the sun and this is the sharpest form of it. A light far enough away measures the distance at which a lamp becomes indistinguishable from a sun, by watching the shadow lines’ meeting point run off to the horizon. Here the sun is not a limit that is approached and never reached; it is a point of the same curve, the point at infinity of one branch, and the quantity the shadow supplies at that point — an altitude — is exactly the quantity the family still has when the position has gone.

What breaks the tie #

One number closes it: the ball’s radius.

Knowing the radius picks a single point of the branch, because the radius is a strictly increasing function of the lamp’s height along it. So ball of a known size, plus its shadow, gives the lamp — which is a single-view metrology result of the kind the metrology field is made of, and it needs no post, no vertical, no calibrated camera and no measurement in the picture except the ellipse.

Two things are worth saying about that, and the second is the awkward one.

First, the recovery is a choice of focus. An ellipse has two, and each carries a branch, so the same shadow with the same ball radius admits a lamp on the left and a lamp on the right — mirror arrangements that produce the identical mark on the floor. Nothing on the floor separates them. What separates them is the ball: it is standing on one focus and not the other, and a person in the room can see which.

Second, the conditioning is worst where the ellipse is nearly a circle. A lamp almost straight overhead gives an ellipse with the two foci nearly coincident, and then “which focus” is a question about a difference of a few millimetres and the branch is nearly vertical, so the radius pins the height loosely. This is not a defect of the method. It is the geometry saying that a shadow directly under a ball carries little information about where the light is, which is also what a reader would say from the doorway.

The round trip, done both ways #

The site’s habit is that a claim of this kind is not finished until the machinery has been made to give back what it was given. Here the round trip is short and the two halves are genuinely independent.

Forward: a lamp, a ball and a floor produce a tangent cone, and the cone’s intersection with the floor is written as a quadratic in two variables. Nothing in that route mentions a focus.

Backward: the quadratic is turned into a centre, two semi-axes and an angle, the focal distance follows from the semi-axes, and the two foci are placed. Nothing in that route mentions the ball.

The two meet at the contact point, and they meet there for every lamp tried — high, low, close in, far out to the side, and at eccentricities from a shadow that is very nearly a circle to one that is nearly a parabola. That is the check. A construction that produced the focus from the ball’s position would be restating its own input; these two do not share a line of arithmetic.

The same discipline gives the hyperbola its own test, and it is not the algebra. Take a point on the branch, work out the ball that goes with it — in the vertical plane the ball is the circle inscribed in the triangle whose apex is the lamp and whose base runs between the ellipse’s two vertices — and then cast that ball’s shadow from that lamp, forward, through the same tangent-cone machinery that drew the first one. The ellipse that comes back is the one the branch was derived from, to a few parts in a thousand million million of a metre, at every rung of the ladder.

The other cases #

An ellipse has foci. A parabola has one. A hyperbola has two, one on each branch.

The shadow of a ball is not always an ellipse, and the shadow of a ball is a conic settles when it is which: the shadow closes if and only if the lamp is above the top of the ball, and that condition contains no distance at all. So take a lamp and lower it. The ellipse grows, its eccentricity climbs, its far focus runs away — and at the moment the lamp is level with the top of the ball the shadow is a parabola, the far focus has gone to infinity, and the near one is still the contact point.

Lower the lamp further and the shadow is a hyperbola. The branch that reaches the floor is the shadow; the other branch is the region the ball’s cone would light if the cone were extended backwards through the lamp. The contact point remains a focus of the whole conic, and it is the focus of the branch the ball is standing in.

None of that needed re-deriving. Dandelin’s theorem is about a cone cut by a plane, and it does not care whether the section closes.

And for the sun the same statement is a different fact #

For a sun the tangent “cone” is a cylinder — parallel rays grazing the ball — and Dandelin’s construction still applies, with one difference worth having.

A cone’s two inscribed spheres have different radii, which is why the two foci of a plane section are geometrically distinguishable once a sphere size is known. A cylinder’s inscribed spheres all have the same radius. So for the sun, the two Dandelin spheres are congruent, both of them exactly the size of the ball, and the shadow ellipse is symmetric between its two foci in a way that a lamp’s never is.

The consequence is small and clean: under the sun, the ball’s contact point is a focus, and a ball of the same size touching the plane at the other focus — from underneath, which is not a place a ball can be — would give the same ellipse. Under a lamp, the ball at the other focus is a different size. The lamp’s asymmetry is what makes the near focus recoverable at all.

What this does not say #

It says nothing about brightness. The edge of a shadow is a geometric boundary here, and a real one is a penumbra whose width is the lamp’s own image through the grazing point. A person marking the contact point on a photograph is marking the middle of a soft band, and the accuracy of that mark is set by the band’s width rather than by anything above.

It says nothing about what makes a shadow look like a shadow, and this site has no standing on that.

And it does not make the shadow a measuring instrument on its own. A shadow ellipse plus a known ball gives a lamp; a shadow ellipse alone gives a contact point and a curve. The distinction matters because the shadow looks informative — it is a whole conic, five numbers, drawn on the floor for anyone to measure — and five numbers is one short of what was needed.

The transferable form #

Every recovery on this site has ended with a statement of what it left free, and most of them left a free parameter that was awkward to describe: a one-parameter relief, a family of eyes, a scale nobody can supply. This one leaves a free parameter that is a named classical curve, and the reason is worth carrying:

When a family of arrangements produces one picture, the family is usually a curve or a group with a name, because the picture was made by a construction that already had one.

The lamps consistent with a shadow ellipse form the ellipse’s focal hyperbola because the thing that had to stay fixed was a right circular cone through a fixed conic, and the apexes of right circular cones through a conic are a classical locus that predates every question in this essay by a century and a half. The free parameter was not discovered by pushing numbers around until something stopped moving. It was recognised.

That is the same move the marks name the place, not the height makes about anamorphs, where the family of eyes explaining a set of floor marks turns out to be exactly the plane’s own projective group. Both times, the way to find out what a picture left free was to ask what group or curve the construction was already standing in.