The shadow of a ball is a conic

A lamp is a centre of projection. That identification is the whole of a shadow is a second projection and it is what lets one piece of code draw both the picture and the shadows in it. Its consequences for a ball are more specific than the general statement suggests, and they are not the ones a reader expects.

The rays from the lamp that graze the ball form a cone. The shadow’s edge is where that cone meets the ground. A cone cut by a plane is a conic section — an ellipse, a parabola or a hyperbola — so the boundary of a ball’s shadow is one of those three, and which one is a fact about where the lamp is.

The condition, and how simple it turns out to be #

Write $\alpha$ for the cone’s half-angle, so $\sin\alpha = R/D$ with $R$ the ball’s radius and $D$ the distance from lamp to ball centre. Write $\delta$ for the angle by which the cone’s axis descends — the angle below horizontal of the line from the lamp to the ball’s centre, so $\sin\delta = (L_y - c_y)/D$.

The shallowest ray in the cone descends by $\delta - \alpha$. It fails to reach the ground when that is zero or negative, and both sines have the same $D$ underneath, so the $D$ cancels:

$$\delta \le \alpha \iff L_y - c_y \le R$$

The shadow stops closing exactly when the lamp is level with the top of the ball. No distance appears in the condition. A lamp a hundred metres away and a lamp at arm’s length give shadows of wildly different sizes, positions and eccentricities, and both cross from closed to open at the same height — the height of the top of the ball.

That is worth a moment because it is not the intuition. The intuition is that a distant lamp behaves like the sun and so gives a bounded shadow whatever its height, and it does not: a distant lamp low down gives a shadow that runs to the horizon just as a near one does.

The claim is checked the way this site checks things: the crossing is found by bisection on the fitted section’s own discriminant, without being told where to look, and it agrees with the closed form. And the absence of the distance is checked separately by putting three lamps at three distances, all at the critical height, and requiring the discriminant to be zero for all three.

The sun cannot do it #

Take the lamp to infinity and the cone becomes a cylinder. A cylinder cut by a plane is always an ellipse — it has no unbounded sections at all unless the plane is parallel to its axis, which for sunlight would mean the sun on the horizon.

So here is a difference between a lamp and the sun that is qualitative rather than quantitative, and it is the sharpest one this site has found. Almost everything else about a distant lamp converges smoothly on the sun’s behaviour as the distance grows. This does not: for any finite lamp there is a height below which the shadow is unbounded, and for the sun there is none. The transition is at $R/D$, which goes to zero and never arrives.

Reading the sun’s altitude off the shadow #

The ellipse the sun casts has a property worth having on its own. The cylinder is circular with radius $R$, so its section by the ground has minor semi-axis exactly $R$ and major semi-axis $R/\sin(\text{altitude})$. The ratio of the two axes is therefore

$$\frac{\text{minor}}{\text{major}} = \sin(\text{altitude})$$

and the ball’s size divides out. A shadow twice as long as it is wide was cast at 30°. A shadow of any ball on any flat ground carries the sun’s altitude, needing no reference length and no ruler — only the ratio of two distances in the same picture.

That is a round trip rather than a formula: the outline is sampled by casting the silhouette circle, a conic is fitted to the sampled points, its semi-axes come out of the fit, and the altitude is recovered from their ratio. Across five altitudes it comes back to nine decimal places, and the minor axis comes back as the ball’s own radius, which is the control — a fit that got the shape right and the size wrong would still return the correct altitude and would mean something quite different was happening.

The silhouette is not a great circle #

One detail in the machinery deserves to be said out loud because it is the same fact the terminator rung is entirely about.

The set of points where the tangent rays touch the ball is a circle, and it is not a great circle. It lies at $R^2/D$ from the centre, toward the light, with radius $R\sqrt{1 - R^2/D^2}$. For the sun that offset is zero and the radius is $R$: the silhouette is a great circle, as everyone assumes. For a lamp two radii away the offset is half the radius and the silhouette circle is only 0.866 of the ball’s radius across.

Casting that circle is what gives the shadow. Casting a great circle instead gives a shadow that is close and wrong, and the error grows exactly as the lamp comes in — which is the regime where the shadow is most interesting.

What the conic type is good for #

There is a practical reading of all this, and it is about pictures rather than about balls.

A photograph or a painting containing a ball with a closed elliptical shadow says the light was above the top of the ball. If the shadow runs out of frame in a way that is widening rather than closing, the light was below it. These are checkable against everything else in the picture: the shadows of parallel posts must converge to a point, and that point must be on the horizon for the sun and below it for a lamp, and the ball’s shadow must be a section of a cone with its apex at the same light.

None of these is a subtle test and all of them are routinely failed by constructed images, because a shadow drawn by hand is drawn to look right rather than computed. The reason to put numbers on them is not to catch anybody out; it is that a claim with a number attached can be wrong, and “the shadow looks about right” cannot.

Which conic, in terms a reader can check #

The type is decided by the discriminant of the section, and the discriminant is a number in a computation. There is a version of the same statement that can be checked in a picture with no computation at all.

Ellipse. The shadow closes. Its far end is a curve that comes back, and the whole shadow is a bounded region of the floor. This is what a lamp well above the object gives, and it is what the sun always gives.

Parabola. The shadow’s two sides run away parallel — never meeting again, never diverging. This is the knife-edge case and it is not observable in practice, because a real lamp is exactly at the top of a real ball for exactly one lamp height.

Hyperbola. The two sides diverge and the shadow widens as it goes. A shadow that gets wider the further it runs is the signature, and it is not subtle at any distance.

So the three cases are distinguishable by eye and the boundary between them is a statement about the lamp’s height alone. That is unusually convenient: most consistency tests on a picture need a construction, and this one needs a look.

What the sun’s ellipse is doing, geometrically #

The sun case is worth deriving rather than asserting, because the derivation says why the ball’s size drops out and the assertion does not.

Sunlight around a ball of radius $R$ is a circular cylinder of radius $R$ — every ray grazing the sphere is parallel to every other, so the tangent surface is a cylinder rather than a cone, and its cross-section perpendicular to the sun’s direction is a circle of exactly the sphere’s radius.

Cut a cylinder of radius $R$ by a plane at an angle. The section is an ellipse whose minor semi-axis is $R$ — the cut is at full width across the cylinder’s own width — and whose major semi-axis is $R$ divided by the sine of the angle between the plane and the cylinder’s axis. For the ground and the sun, that angle is the sun’s altitude.

So the minor axis is the ball’s diameter and the major is that divided by the sine, and the ratio of the two has no $R$ in it. That is why a shadow of any ball carries the altitude: the size cancels between the two axes because both are proportional to it.

The same cancellation is why the measurement is robust to something a reader might worry about. The ball need not be a sphere of known radius, need not be a known distance away, and need not have its own scale in the picture. Only the two axes of its shadow are needed, and only their ratio.

The five numbers, and which of them are readable #

A conic has five degrees of freedom, and the point of using a ball rather than a post is that all five carry something.

Two go on position — where the shadow’s centre is on the floor, which locates the ball’s own position relative to the light and the ground.

Two go on shape — the axis ratio and the orientation. Under the sun the ratio is the sine of the altitude and the orientation is the azimuth, so those two numbers are the sun’s direction, read off the shadow with no reference length anywhere.

And one goes on size, which is the ball’s radius as scaled by the geometry — the one number that needs something else known before it means anything, since a picture of a large ball far away and a small one near by are the same picture.

Reading that list against a post’s shadow, which is a segment and carries two numbers, says why a ball is the right object to put in front of a light. A post gives the light’s direction and nothing else; a ball gives the direction, the position, the type and — with one length — the size.

The consistency test, written out #

For a picture containing a ball, its shadow, and at least one vertical object with its own shadow, four conditions have to hold together, and each is checkable with a straightedge.

The ball’s shadow must be a conic — five points of its outline determine the conic and the rest of the outline must lie on it.

Its type must agree with the light’s height as read from the vertical object’s shadow: closed if the light is above the top of the ball, open if below.

The shadows of the vertical objects must converge to a single point, on the horizon under the sun and below it under a lamp.

And under the sun, the ball’s shadow’s axis ratio must be the sine of the altitude that the vertical’s shadow length implies.

Four conditions and one light. A picture that satisfies all four was lit by something; a picture that fails one was not, and which one it fails says what kind of mistake was made.

The one place the analogy strains #

A shadow is a projection from the lamp onto the ground, and this essay has been treating the ball’s tangent cone as the object. That is a small sleight: the cone is not part of the scene, it is the boundary of the set of rays the ball blocks. What is projected is the silhouette circle, and the cone is the family of rays through it.

The reason it works out is that the projection of a circle from a point is exactly the section of the cone through that circle, so the two descriptions are the same object approached from either end. It is worth noticing only because the same substitution fails as soon as the occluder is not convex: a ring’s shadow is not the section of anything, and a shadow with a hole in it is not a conic. Convexity is doing quiet work, as it does in the hull the shadow of a box is, where a shadow drawn as a bow-tie went unnoticed for the whole life of a figure because nothing was asking whether the region was a region.

The reason a ball is the right object to ask about is exactly that. A post’s shadow is a segment and a segment carries one number. A ball’s shadow is a conic, a conic has five, and the five are all readable — which makes a ball the most informative thing anybody can put on the ground in front of a light.