The forty-five degree shadow

The rule is the first thing anybody is told about drawing shadows. Put the sun at forty-five degrees, run the shadow off at forty-five degrees, and make it as long as the object is tall.

In a plan it is exactly right. A sun halfway up the sky casts a shadow equal to the height of what casts it, and a shadow running at forty-five degrees across the plan is a sun in the corresponding direction. Nothing is wrong with the rule where it was written.

Applied to a perspective drawing — a fixed angle on the page, a length equal to the drawn height — it puts every object in the picture under a different sun.

Reading a drawn shadow back #

The measurement here is not a matter of opinion, because a drawn shadow can be interrogated.

The tip of a shadow is a point on the ground, and a point on the ground in a picture has exactly one place it can be — the camera’s ray through that mark, met with the floor. So each drawn tip names a world position. The top of the post is a known world point. The line from the top to the tip is the light ray that would have had to produce that shadow, and its direction has an altitude and an azimuth.

Four shadows, four light directions. If the drawing depicts a scene lit by the sun, the four must agree, because the sun is at infinity and its rays are parallel.

The four suns the rule produces on the picture drawn here differ by 13.3° in altitude and 31.5° in azimuth. The four real shadows, cast by one sun and projected, read back as one sun to within four parts in a hundred trillion of a degree, which is the control that makes the first number mean something.

Why it cannot work #

The reason is structural rather than a matter of tuning, and it is worth stating because it rules out every repair of the same kind.

The rule sets two things on the paper: the direction the shadow runs and its length. Both of the quantities it is standing in for depend on where the object is.

The direction the shadow runs in the picture is the direction toward the shadow’s own vanishing point — the point on the horizon belonging to the sun’s azimuth, which is what where shadows vanish establishes. Lines running to a single point on the horizon are not parallel on the page: they fan. So a fixed drawing angle means each post’s shadow runs to a different point on the horizon, which means a different azimuth for each.

The length in the picture is foreshortened by the depth. A near post’s shadow is drawn long and a far post’s short, for the same world length. So a rule that makes the drawn shadow equal to the drawn height gives the far post a shadow far longer in the room than the near one.

A single number cannot be a function of position. Setting either quantity on the page guarantees the other one varies.

No angle rescues it #

The obvious response is that forty-five is the wrong angle for this camera, and some other angle would do better.

None does.

The curve has a minimum, because at some angle the fan of implied azimuths happens to straddle the true one most evenly, and it never reaches zero, because the shape of the failure does not depend on the angle chosen.

There is also a limit at the steep end that is worth recording, because it is a refusal rather than an error. Drawn steeply enough upward — away from the viewer — the rule places the shadow tip above the horizon, and a point above the horizon has no ground position at all. The rule is then asking for a shadow infinitely far away, which the machinery here declines to supply rather than returning a large number.

Where the rule is right, and it is not a region #

The sweep finds a best angle and a spread that never reaches zero, and it is worth asking the complementary question: for a chosen angle, where in the picture is the rule right? The answer is that it is right at isolated points, and knowing that explains why no amount of tuning helps.

The rule fixes two things and each is one equation.

The direction is right along a single line. All true shadows run to one point $V$ on the horizon — the azimuth’s own vanishing point — so the drawn direction from a foot $F$ is the direction of $FV$. Requiring that to equal the rule’s fixed angle $\theta$ picks out the feet lying on the line through $V$ at angle $\theta$. One line, through the horizon point, and along the whole of it the rule’s direction is exactly correct.

The length is right along a curve. The true shadow has world length $h/\tan\alpha$, and its drawn length is that foreshortened by the depth of the tip, while the drawn height is the post’s own height foreshortened by the depth of its top. Requiring the two to be equal is one equation in the foot’s two ground coordinates, so its solutions are a curve — not a line in general, because the two foreshortenings have different denominators.

A line and a curve meet at isolated points. So for any drawing angle there are a few positions on the floor where the rule draws exactly the right shadow, and nowhere else. That is why sweeping the angle moves the minimum spread around and never removes it: rotating $\theta$ slides the line about $V$ and slides the intersections along the curve, and four posts cannot all sit on a set of isolated points.

Two consequences follow, and the second is the practical one.

The error grows with distance from those points, smoothly, so a drawing in which every object is clustered near one of them looks nearly right and one whose objects are spread over the floor does not. That is the reason the rule survives in illustration and fails in a measured picture: an illustration usually has its subject in one place.

And the two errors are independent. A shadow can have the right direction and the wrong length — anywhere on the line but off the curve — or the right length and the wrong direction. So a reader testing a drawing has two separate tests rather than one, and they fail in different ways: a wrong direction is caught by extending the shadows and finding they do not concur, which is the recovery this collection runs on real pictures, and a wrong length is caught by reading each shadow back to its own altitude, which is what the second projection makes possible.

The pair also explains the sweep’s shallow minimum. The best angle is the one whose line passes nearest to the most feet, so it depends entirely on where the objects happen to stand — a fact about the composition rather than about the geometry. A different arrangement of the same four posts would put the minimum somewhere else and would not lower it, which is the same conclusion the taught depth rule reaches about its own free parameter: a constant standing in for a function is exact on a set of measure zero, and which set depends on the drawing.

The two readings of the same instruction #

There is a second thing the wording does not settle, and it makes a large difference.

“Forty-five degrees” on the paper can mean down-and-across — the shadow coming toward the viewer — or up-and-across, running away into the picture. Both are drawn in books.

Drawn toward the viewer, the implied suns are between sixteen and twenty-nine degrees up and the shadows are one and three-quarters to three and a half times the height. Drawn away from the viewer, the implied suns are between 1.8° and 3.4° above the horizon, and the shadows are seventeen to thirty-two times the objects’ heights.

A sun two degrees up is a sun at the moment of setting. So one reading of the rule draws mid-afternoon and the other draws the last minute of daylight, and the instruction as printed does not choose.

That is the same structure as the two readings of the eight-point circle rule in the circle in the square wants a number — one instruction, two ways to carry it out, both consistent with the words, and a large difference in what results. It is beginning to look like the characteristic defect of a rule stated in terms of the paper: the paper has more degrees of freedom than the rule mentions.

What the rule should have said #

The correct construction is not much longer than the wrong one and it is entirely a matter of joins and meets.

Pick the sun’s azimuth: mark a point on the horizon. Every shadow on the ground runs to that point, so from each object’s foot, draw the line to it.

Pick the sun’s altitude: mark a point above or below the horizon, on the vertical through the azimuth point. That is the vanishing point of the light rays themselves. From each object’s top, draw the line to it.

Where the two meet is the shadow’s tip. Two lines per object, and two points chosen once for the whole picture.

That construction is exact, it produces a single consistent sun by construction, and it is the same construction as the one in a shadow is a second projection, where the sun is treated as a second camera and the shadow as its picture. Nothing about it is harder than the rule it replaces; it is one extra dot on the paper.

The lamp version, which is worse #

Everything above is for the sun. For a lamp inside the scene the rule is not merely inconsistent but qualitatively wrong.

A lamp’s shadows are not parallel in the room at all: they radiate from the point directly under the lamp. So in the picture they run to different points on the horizon and their directions vary as strongly as the geometry does. Any fixed drawing angle is wrong for all but one object, and the pattern of the failure is one nobody would produce by accident.

The recovery machinery is the same either way, and it distinguishes the two cases with one number: the shadow lines, extended, meet on the horizon for the sun and below it for a lamp — as far below as the lamp is close. That is a fact a reader can check on a photograph with a ruler and no arithmetic.

What a picture with four suns in it means #

It is worth pausing on what the measurement above actually says, because “four suns” sounds like a figure of speech and is not one.

A picture is a projection of a scene. The question “what scene is this a picture of” always has an answer for the objects — a drawn post is a picture of some post — and for the shadows it need not. A shadow is not an object; it is a relation between an object, a light and a floor. A drawn mark on the floor is a picture of a real mark, but whether that mark is the shadow of the post beside it is a claim, and the claim can fail.

So a drawing whose shadows were put in by the rule is a picture of a floor with four dark patches on it, none of which is the shadow of anything under any single illumination. It is not a picture of a badly lit room. There is no room.

That distinction matters because it decides what can be recovered from such a picture. Everything this collection does with shadows — the lamp out of the picture, the depth from a shadow, un-casting a shadow to find the object — starts by assuming the marks are shadows and fits a light. Handed the rule’s marks, the fit returns something, because a fit always does, and the something is the light that best explains four incompatible pieces of evidence.

The residual is what says so. Two shadows always determine a light exactly, because two lines meet; the third and fourth are what can disagree, and their disagreement is the only signal that the marks were never shadows. A drawing with three or more shadows in it is therefore a drawing that can be tested, and a drawing with two is not.

The one case where the rule is exact #

For completeness, the rule does have a case, and it is narrower than “in the plan”.

Take a sun at forty-five degrees of altitude whose azimuth runs exactly across the picture — parallel to the picture plane. Then the shadows in the room are all parallel to the picture plane, so they are parallel on the page too, and their vanishing point is at infinity in the picture rather than on the horizon. A fixed drawing direction is now correct for every object.

The lengths still are not. A shadow parallel to the picture plane at depth d is drawn at a length proportional to 1/d, and so is the post’s height, so the ratio of drawn shadow to drawn height is the same for every post — and equal to the world ratio, which for a forty-five degree sun is one.

So in that one arrangement both halves of the rule are exact: the direction is fixed and the drawn length equals the drawn height. It is the arrangement in which every diagram illustrating shadow construction is drawn, and it requires the sun to be in one particular place relative to the camera.

Move the sun’s azimuth off that line and the direction rule breaks. Keep the azimuth and move the sun’s altitude and the length rule breaks. The rule is a point in a two-parameter family, presented as the family.

Why it looks all right #

The rule survives, and the reason it survives is worth stating because it is the same reason as everywhere else in this collection.

A drawing with shadows put in by the rule looks lit. Everything has a shadow, the shadows all go the same way, they are all in proportion to their objects, and the picture reads immediately. The errors are consistent — every shadow is wrong in the same style — and a consistent error is exactly the kind a viewer’s eye normalises away.

What gives it away, to somebody looking for it, is that the shadows do not converge. In a photograph of a sunlit street the shadows of the lamp posts visibly fan out toward a point; in a drawing made by the rule they are parallel on the page like a hatching pattern. Once seen it is not unseeable, and it is the quickest test there is for whether a drawing’s light was constructed or applied.

The second tell is the length under a low sun. The rule makes every shadow proportional to its object’s drawn height, so a distant figure gets a short shadow. A real low sun gives a distant figure a shadow that stretches most of the way across the picture, because the shadow’s world length is fixed by the sun and the drawn length is whatever the projection makes of it.

The general statement #

Three of the rules this collection has measured now fail the same way and the way is worth naming.

A rule that sets a quantity on the paper is asserting that the quantity does not depend on position in the scene, and in a perspective picture almost nothing has that property. A drawn ellipse’s lean is the same complaint about a different quantity, and its error is a product of two angles rather than a fixed number for the same reason.

The shadow angle depends on where the object stands. The shadow length depends on the depth. Seven tenths along a diagonal depends on the foreshortening. The vertical drop from a waterline depends on the tilt of the picture plane. Every one of them is a fixed number substituted for a function, and every one of them is exactly right at one place — which is where the diagram illustrating it was drawn.

The repair in each case is the same in shape: find the vanishing point the quantity belongs to, and draw to it. A vanishing point is a fixed thing that produces a varying result, which is precisely what a fixed rule cannot do.