The edge of a shadow is drawn on the object

A ball sits on a floor with a lamp above it, and there is a dark patch on the floor with a definite edge. Ask what that edge is the shadow of and the natural answer — the edge of the ball — turns out to name two different curves, neither of which is an edge and only one of which is doing the casting.

The curve that casts #

A ball has no edge. Every point of it is like every other, and what makes some of it lit and some of it dark is not a property of the surface but a relation between the surface and the lamp: a point is lit when its outward normal leans toward the lamp and unlit when it leans away.

The boundary between those two regions is where the normal is exactly perpendicular to the ray — where the surface turns away — and that boundary is a curve on the object. It has a name in the vision literature, the contour generator, and it is the thing whose shadow is the edge of the cast shadow.

For a ball the curve is a circle, and it is not the obvious circle. A lamp at distance d from the centre of a ball of radius R touches the ball along the circle at an angular radius of arccos(R/d) from the lamp’s own direction. That is the circle where the tangent cone from the lamp meets the sphere, and the tangent cone is the same object the shadow of a ball is a conic intersects with the floor.

Two consequences fall straight out and both are the wrong way round from the usual picture.

It is smaller than a great circle. At two radii from the centre the circle sits at sixty degrees from the lamp’s direction; at ten radii, a shade over eighty-four; at a hundred, eighty-nine and a half. It opens to a full ninety degrees only for a lamp at infinity, which is to say the sun. Every drawing that shows a ball lit exactly half is drawing sunlight, whether or not it means to.

It moves. Carry the lamp round and the circle goes with it, sliding over a surface on which nothing is marked. This is the whole of what “the edge of a shadow is drawn on the object” means: there is a curve, it is on the object, and it is not a property of the object.

The site already has one half of that statement, at a lower rung. A lamp lights less than half a ball computes the lit fraction — how much of the surface a near lamp reaches — and the number there is the same geometry read as an area. What that essay does not say is what the boundary of the lit region is for, which is that it is the caster.

There is a third consequence which is not about the ball at all. The quantity arccos(R/d) is the complement of the tangent cone’s half-angle, and the tangent cone is what the whole of the sphereshadow ladder is built on — so this circle is not a new object introduced to talk about terminators. It is the contact circle of a cone the site has been using since the shadow of a ball was first drawn, looked at from the object’s side instead of the floor’s.

The curve the camera sees #

Now do the same thing with an eye instead of a lamp.

A camera looking at the ball has its own contour generator: the circle where the surface turns away from the eye, at arccos(R/D) from the eye’s direction for an eye at distance D. Its image is the outline of the ball in the photograph — the disc’s rim, the thing a person would trace with a pencil.

Two circles, then, on one ball, and both of them determined by something outside the ball. In the figure above they are about eighty-nine degrees apart, which is very nearly the largest they can be while both remain visible; move the lamp round toward the camera and they close up; move it behind and they open out.

The angle between them is exactly the angle the lamp and the eye subtend at the ball’s centre, and that is a statement with a closed form to be checked against — which it is, in the site’s own gate, because a walked curve agreeing with a formula is the difference between having found the contour generator and having found a level set of the parametrisation used to look for it.

Walked, not assumed #

The circle at arccos(R/d) is a closed form, and a closed form is exactly the thing this site refuses to draw from. So the curve in the figures is not that circle. It is found by walking the ball’s own surface on a parameter grid, evaluating the dot product of the normal against the direction to the lamp, and keeping every place it changes sign.

Then the two are compared, and the comparison is the check. Across lamps from just above the surface to a hundred radii out, the walked curve’s mean angular radius agrees with arccos(R/d) to within a millionth of a degree, and — this is the half that matters — the spread of the walked points about that mean is smaller still.

The spread is the real test. A walk that had accidentally found a level set of the parametrisation rather than the tangency would still produce a closed curve on the ball, would still be smooth, and would still move when the lamp moved. What it would not do is have every one of its points at the same angle from the lamp’s direction, at every lamp. The spread is what says the walk found a circle rather than something circle-shaped.

Reading the lamp off the two circles #

If both curves are visible in one photograph, the lamp’s direction comes out of them.

The ball’s outline gives the eye’s own circle, which is a known consequence of where the camera is. The terminator gives the lamp’s. Each of the two circles lies in a plane, and the axis of each is perpendicular to its plane through the ball’s centre — so the lamp is along the terminator’s axis, and how far along it is set by the circle’s angular radius through arccos(R/d).

That is a complete recovery of a lamp from one ball in one picture, and it is a different route from the one the light field already has. The lamp out of the picture recovers a lamp from drawn shadow lines: the joins of post-tops to shadow-tips meet at the lamp’s image, and the joins of feet to tips meet at the image of its foot. That construction needs posts and a ground plane and knows nothing about the objects except that they stand upright.

This one needs a single ball and no ground at all. What it needs instead is a shaded photograph — the terminator has to be findable, which means the surface has to be smooth enough to have one and lit clearly enough for it to be located. The two routes fail in different places, which is why having both is worth something: the shadow-line construction is defeated by a scene with no vertical in it, and this one by a scene with no smooth solid.

They coincide exactly once #

Put the eye at the lamp. Then the two axes are the same axis, the two circles are the same circle, and the shadow’s edge is the object’s outline.

This is not a curiosity; it is the reason a photograph taken with a flash mounted at the lens shows no shadow. The shadow is still there — the floor behind the ball is still dark — but it is exactly hidden behind the ball from that viewpoint, because the boundary of the dark region and the boundary of the ball’s image are the same curve on the film. The whole umbra lies inside the ball’s silhouette.

So the arrangement in which the two curves agree is precisely the arrangement in which the disagreement between them cannot be seen. That is the shape of a great many things on this site: the case in which a rule is exact is the case in which the rule was never needed. The forty-five degree shadow is exact in the plan, where nobody is standing.

What it costs to get this wrong #

An illustrator drawing a ball lit from one side has to decide where the terminator goes, and the two available answers differ by an amount that is easy to compute and not always small.

Draw it as a great circle and the lit region is exactly half the sphere. Draw it correctly for a lamp two radii away and the lit cap is a cone of sixty degrees rather than ninety — visibly less than half, and by an amount that changes the shape of the highlight and the placement of the darkest part of the drawn edge.

Whether a viewer notices is not a question this site has any standing on. What it can say is that the two are different pictures of two different arrangements, and that the great circle is the picture of a sun. A drawing that puts a lamp in the room and a sun’s terminator on the ball is internally inconsistent, in the same way and to the same degree as a drawing whose four posts imply four suns.

The general case, and where the argument stops #

A ball is the case where the contour generator is a circle and everything has a closed form. For a general smooth solid it is the set where (p − L)·n(p) = 0 on the surface, and there is nothing tidy about it:

• it need not be connected — a non-convex solid can turn away from a lamp in several separate bands;
• it need not be planar, and for anything that is not a quadric it usually is not;
• and points of it can be hidden from the lamp by other parts of the same solid, in which case they are on the boundary of nothing and cast no edge at all.

That last one is the honest limit of the tidy statement. The full sentence is not the shadow’s edge is the image of the contour generator but the shadow’s edge is the image of the parts of the contour generator the lamp can actually see, and deciding which parts those are is an occlusion question rather than a differential one. For the convex casters in this essay the two sentences agree, because a convex body never hides its own contour generator from the light that defined it.

Why the shadow grows when the lamp comes in #

The figure above answers a question the two circles make natural. A lamp brought closer casts a bigger shadow, and the reason is now visible rather than a fact about similar triangles: bringing the lamp in shrinks the contour generator toward the lamp’s side of the ball, which widens the tangent cone, and a wider cone cuts the floor in a bigger conic.

Both effects are the same arccos. The angular radius arccos(R/d) is what the cone’s half-angle is the complement of, so the curve on the ball and the spread of the shadow are two readings of one number. That is the sort of thing the site looks for and does not always find: a quantity that turns up in two different-looking claims because there is only one quantity.

The band, rather than the curve #

Everything above is drawn with a point lamp, and a point lamp is a fiction with a known error term.

A source of finite size does not produce a terminator; it produces a band, because a point of the surface can see part of the source and not the rest. The band’s angular width at the ball is the angular width of the source seen from that point, and the site has computed exactly this quantity once before: the penumbra is the lamp’s image shows that the soft edge of a cast shadow is the source imaged through the occluding edge, which is the same statement read on the floor rather than on the ball.

So the two curves in the figures should be read as the middles of two bands. For the sun the band is about half a degree wide, which on a ball is a strip a couple of hundredths of the radius across and negligible for anything here. For a bare bulb at two radii it is wide enough to see, and for a window it is wide enough that the word terminator stops being useful.

What does not change is the geometry of the middle. The band is centred on the point lamp’s answer, because the source’s own centre is a point lamp, and everything above is a statement about it.

The sun, and the one time the drawing is right #

The lit half of a ball under the sun is exactly half, and the terminator is exactly a great circle, and this is the one case in which the picture everybody draws is the correct one.

It is worth saying why it is exact rather than very close. The sun subtends about half a degree, so a naive expectation would be that the terminator is a great circle to within some fraction of a degree. But the relevant distance is not the sun’s angular size, it is the sun’s distance in ball radii — a hundred and fifty million kilometres against a ball a few centimetres across — and arccos(R/d) for that ratio differs from ninety degrees in the fourteenth decimal place. The great circle is right to well past any precision a drawing has.

Which puts the boundary in an unexpected place. The terminator’s departure from a great circle is a near-field effect, invisible for anything astronomical and obvious for a table lamp, and the crossover is at a few tens of radii rather than anywhere far away. A lamp three metres from a football is already within it.

The habit this is a case of #

Three essays in this row are about the same mistake, made about three different features of a shadow, and it is worth stating in the form that transfers.

A feature you can see in a shadow is a feature of an arrangement, not of the thing casting it. Take away the lamp, or move it, and the feature is somewhere else or gone.

The cusp in a wire with a corner in its shadow is at a point of the wire that changes when the lamp moves. The hole in a ring’s shadow, in a hole is not preserved, is present or absent depending on how obliquely the lamp sees the ring. And the edge here is cast by a curve that is on the ball and belongs to the lamp.

Each of the three is easy to state and each is routinely got wrong in the same way — by treating the shadow as a picture of the object rather than as a picture of the object and the lamp and the floor, jointly. A shadow is a projection, and a projection has a centre; the centre is half of what is being seen.