A wire with a corner in its shadow

Every shadow this site has cast so far has come off something convex: a point, a post, a flat card, a ball. Their shadows are legible because the outline of the shadow is the image of something obvious — a tip, an edge, a silhouette circle — and because nothing about the shape does anything surprising on the way down.

A wire is the smallest departure from that, and it already has something the convex cases do not.

The wire has no corner. There is no kink in it, no join, no place where the tangent direction jumps; it is a helix, and its tangent turns as continuously as anything in this subject. The shadow has a corner — a genuine one, a cusp, at which the shadow’s own direction reverses.

What the shadow of a curve is #

Nothing new. A shadow is a projection from the lamp, so a curve’s shadow is the point shadow applied along the curve: for each point of the wire, run the ray from the lamp through it and take where it meets the floor. That is the same projectToGround every post in the light field uses, called once per sample.

Writing it out is what shows where the corner comes from. The shadow of the point at parameter s is

$$q(s) = L + t(s)\,\bigl(p(s) - L\bigr)$$

with t chosen so the result lands on the floor. Differentiate and the shadow’s velocity has two terms — one from the point moving along the wire, one from t changing as it does — and they cancel exactly when the wire’s tangent runs along the ray.

At such a point the shadow stops dead. Not slows: stops, and then comes back the way it went. That is what a cusp is.

The test is therefore one number: the sine of the angle between the tangent at s and the ray from the lamp to p(s). It is never negative, so a cusp is a minimum touching zero rather than a sign change. That mattered more than it should have, and it is recorded further down.

The lamps that do it are a wall #

The condition is that the lamp lies on a tangent line of the curve. Not near one — on one.

A tangent line is one-dimensional and there is one for every point of the curve, so the tangent lines sweep out a surface: the curve’s tangent developable. The lamps that give a cusp are exactly the points of that surface, and a surface in a room is something a person walks through rather than something a person hits by luck.

That is the difference between this and a coincidence. A cusp is not generic — a lamp put down at random is not on the surface — but it is codimension one, and every path across the room that crosses the surface produces a shadow that develops a corner, holds it for an instant and loses it again.

Where the wall is, for a helix #

The tangent developable is easy to say and hard to picture, so it is worth locating for the wire in the figures, which is a helix of radius forty-two centimetres standing on the floor.

Take the tangent at any point of it. Its horizontal part is tangent to the circle the helix runs round — that is what a helix’s tangent does — so the tangent line’s shadow on the floor is a line tangent to a circle of forty-two centimetres. A tangent line to a circle never comes closer to the centre than the radius.

So no tangent line of this helix ever passes within forty-two centimetres of its axis, and a lamp standing inside the cylinder the helix winds around cannot be on any of them. Hang a bulb straight down the middle of a spiral staircase and its shadow has no corner in it at any height. Move the bulb out past the handrail and the corners become available.

The wall, for this wire, is a surface wrapped around outside the cylinder and rising with it, and it is not a wall anybody would have guessed the shape of. It is however a wall a person could find by experiment in about a minute, which is the usual situation with codimension-one things.

Two cusps, or none #

Crossing a wall changes something by a whole number, and here the whole number is the count of cusps — which goes up and down in twos.

The reason is the same one that makes a tangent line tangent: at a lamp on the surface, the sine of the angle between tangent and ray touches zero and comes back. Move the lamp to one side and the minimum lifts off zero, and there is no cusp anywhere near. Move it to the other and the minimum dips below — except that the sine cannot go below zero, so what actually happens is that the single tangency splits into two nearby ones, and the shadow acquires a pair of cusps with a short doubled-back piece between them.

That doubled-back piece is a swallowtail, and it is the standard picture of what happens to a projected curve as a viewpoint crosses a tangent surface. It is worth naming because it explains the thing a reader notices first about a near-cusp shadow, which is that the shadow seems to overlap itself for a little while. It does. Two arcs of the wire are casting onto the same stretch of floor, and the fold between them is the pair of cusps.

And it relaxes rather than switching off #

The corner is not a threshold phenomenon, which is the part worth having.

Move the lamp off the surface and the shadow’s sharpest turn falls away smoothly. Three millimetres off, the shadow still turns through most of a straight angle. Two centimetres off it is turning through about a hundred and seven degrees. Twelve centimetres off it is about twenty-two degrees, and at a metre and a fifth it is a single degree and unremarkable.

So “the shadow of that wire has a kink in it” is a statement with a tolerance attached, and the tolerance is the reader’s eyesight rather than anything in the geometry. This is the ordinary situation for a degeneracy: the degenerate case is measure zero and the nearly degenerate case is a neighbourhood, and everything interesting happens in the neighbourhood.

A flat curve cannot do it at all #

Bend the wire into a circle and lay it on a table, or hold it upright, or tip it at any angle — and no lamp anywhere off its plane produces a cusp.

The reason is one line: the tangent lines of a plane curve all lie in that plane. So the tangent developable of a plane curve is the plane itself, and a lamp not in the plane is not on any tangent line. The closest a tangent comes to pointing at a lamp above a tilted ring, across the whole ring, is about twelve and a half degrees of angle — a long way from zero and not getting closer.

That is the same statement as a much more familiar one. A plane curve’s shadow is its image under a projection between two planes — a projectivity — and a projectivity takes smooth curves to smooth curves and conics to conics. Which is exactly why the shadow of a ball is a conic, and why a circle of wire casts an ellipse, a parabola or a hyperbola and nothing else, and why the shadow of a plane curve carries so little information about how the curve is oriented in space: the whole of the shadow is one plane map applied to one flat thing.

A space curve’s shadow is not a projectivity of anything. There is no plane to be mapped from. The shadow is a genuinely new curve, and it can do things — cusp, self-intersect, change its number of inflections — that no projective image of the original could.

Two routes, and the disagreement that found a bug #

The tangency test says where a cusp should be. It does not say the shadow has one; it says a condition holds. The site’s rule is that a claim gets a test it could fail, and here the second test is the drawn shadow itself: take the emitted polyline and measure the angle it turns through at its sharpest sample.

The two agree — where the tangency vanishes, the polyline turns through very nearly a straight angle, and where it does not, the polyline turns gently. That agreement is the check that the cusp is a fact about the projection rather than about the condition that predicted it.

It also caught the first version of the tangency test, which was wrong in a way that passed everything.

The quantity is a cross product, and the first draft took a signed component of it and hunted for sign changes with a bisection. A signed component of a cross product changes sign where the cross product is horizontal, not where it vanishes. So the test reported two cusps on a tilted ring — which, by the paragraph above, has none — and the drawn shadow at those two places was turning through less than half a degree. The disagreement between the two routes is the only thing that showed it. A test hunting for zeros of a non-negative quantity has to look for minima; there is no sign to change.

And a number that was a fact about the sampling #

The second thing the two routes turned up is subtler, and it is why this collection has a gate about it.

The sharpest turn in a polyline is not the sharpest turn of a curve. It depends on how finely the curve was sampled, and near a cusp the samples crowd — the shadow is barely moving there — so the angle between consecutive samples is decided by where the samples happen to fall.

Measured at four thousand samples along the wire, the sequence of sharpest turns against the lamp’s distance from the tangent surface is not even monotone: about a hundred and sixty-one degrees at three millimetres off and a hundred and sixty-nine at eight. Measured at sixteen thousand it falls cleanly the whole way. Nothing about the shadow changed. The quantity being reported was partly a quantity about the sampler.

That is not a rounding worry. It is the difference between a measurement and a reading, and this site had quoted the reading. The figure above is drawn at the finer sampling and says so, and the collection’s convergence gate now requires any quantity taken off a grid or a polyline to be shown converging, with the ladder in the resolution given rather than assumed.

Not a crossing #

A cusp and a self-intersection are different things and both are common in these shadows, so it is worth separating them once.

A self-intersection happens when two different points of the wire cast onto the same point of the floor. It is a fact about two places at once, it needs no tangency, and it is entirely generic — the shadow of almost any bent wire crosses itself somewhere, because a curve in space has more room to avoid itself than its shadow does. A crossing is smooth: both branches run through it at a definite angle and neither of them stops.

A cusp happens at one point of the wire. Both sides of the crossing are the same piece of the wire, the shadow’s velocity vanishes there, and the two arcs leave the cusp in the same direction rather than at an angle.

The mechanical way to tell them apart on a drawing is to follow the shadow with a finger. At a crossing the finger goes straight through and comes out the other side; at a cusp it has to turn round. In the arithmetic they are just as different: the crossing is a statement about two parameters s and s′ with the same image, and the cusp is a statement about a derivative at one parameter.

Nothing in this essay is about crossings, and the helix’s shadow has several.

Why the drawing manuals do not mention it #

A drawing manual that teaches shadow construction teaches it on posts, boxes, cylinders and the occasional sphere, and none of those has a tangent developable a lamp can plausibly be on. A post’s edges are straight lines, and a straight line’s tangent lines are the line itself; being on it means the lamp is in the post. A box is made of flat faces and straight edges, and the same applies. A sphere has no distinguished curve on it until a lamp is chosen, and its silhouette circle is planar.

So the whole familiar repertoire is made of objects for which the cusp is unreachable, and the manuals are not wrong to leave it out. What they leave out with it is the reason: not shadows do not do that but the objects in the repertoire have no tangent surface a lamp can stand on.

The moment a bent wire, a spiral, a handrail or a cable enters the picture, the situation changes, and a draughtsman who has learned the repertoire has no rule that covers it. That is the shape of nearly every result in the wrong field of this site — a rule stated as general which is a rule about the case it was worked out on — arriving here from the other side, because in this instance nobody ever stated the rule at all.

Where the corner goes on the wire #

One last thing, because it is the sort of question the drawing prompts and the answer is not the obvious one.

The cusp appears at a definite point of the shadow. It does not correspond to a special point of the wire. The point of the wire whose shadow is the cusp is the point whose tangent happens to run through the lamp, and moving the lamp moves that point along the wire continuously. Mark the wire with a pen at the place its shadow has the corner, move the lamp a hand’s breadth, and the mark is no longer where the corner is.

This is the same shape of statement as the next essay’s, and it is worth naming here because the two are so easily run together. The edge of a shadow is drawn on the object is about a solid’s silhouette-from-the-lamp, which is also a curve that lives on the object and also slides when the lamp moves. Both are cases of the same habit of mind: a feature that is plainly visible in the shadow is treated as a feature of the thing casting it, and in neither case is it one.

A cusp in a shadow is a fact about a wire, a lamp and a floor, jointly. Take away any one of the three and there is nothing to point at.