The shadow rules that hold here

The drawing manuals give a construction for a shadow and it is the same one everywhere. Drop a vertical from the point to its foot. From the point’s mark, run a line at the light’s angle. From the foot’s mark, run a line along the light’s plan direction. Where they cross is the shadow.

Every step of that is done with a set square on the paper, and nothing in it knows where the point is in the room.

The forty-five degree shadow took the perspective version of that rule and found it hopeless: four posts drawn by the recipe imply four different suns, between sixteen and twenty-nine degrees up, casting shadows between 1.77 and 3.46 times their heights, and no single drawing angle brings them together — the best of seven leaves eleven degrees of disagreement.

Here is the other half of that finding.

In a parallel drawing it is exact #

The construction and the projected shadow agree to within a few parts in ten thousand million million of a unit, at every post, in isometric, cabinet, dimetric and the military projection alike.

The reason is one line and it is the reason the whole parallel field exists. A parallel drawing is a linear map, and a linear map takes the intersection of two lines to the intersection of their images. The paper construction crosses two drawn lines; the scene crosses the same two lines in space; and the image of the crossing is the crossing of the images.

That is not true of a perspective projection — it is, for the intersection itself, since a projectivity also preserves incidence — but the angles the construction uses are not preserved, and the construction is stated in angles. Which brings the second half.

And one set square does the whole drawing #

Every ray in the figure is drawn at the same angle. Not nearly the same: the spread across the four posts is zero to arithmetic noise.

A parallel projection sends every copy of a given direction to the same drawn direction, wherever it is in the scene, because the map is linear and linear maps do not know where anything is. So the light’s direction has one image, and a draughtsman lays the set square once and uses it everywhere.

In a perspective picture the light’s rays are a family of parallel lines with a vanishing point, and their drawn directions fan out from it. A post near the vanishing point has a steeply drawn ray and one far from it has a shallow one, and any fixed paper angle is right for at most one post. That fan is exactly what the four-suns measurement was measuring.

So the manual’s instruction is not a bad approximation to a perspective rule. It is an exact rule for a different kind of drawing, transplanted.

What the construction is actually doing #

It is worth writing out the construction in scene terms once, because the paper version hides how little it needs.

The shadow of a point P on the ground is where the ray through P along the light’s direction meets the plane y = 0. That ray lies in the vertical plane through P containing the light’s plan direction, and so does the vertical from P to its foot F. So three things — P, F and the shadow — are coplanar, in a vertical plane, and the shadow is where two lines in that plane cross: the ray from P and the ground line from F.

The paper construction is that vertical plane’s own geometry, drawn. It works because the drawing maps that plane to the page as a whole, taking its two lines to two lines and their crossing to their crossing, and it needs nothing about the plane’s position. Every post in the scene has its own vertical plane; all of them map to the page with the same pair of drawn directions, because all of them contain the same two world directions — the vertical, and the light’s plan.

Which is why the construction is one pair of set-square settings rather than four. The four vertical planes are different planes and their two relevant directions are identical, and a parallel drawing draws a direction the same way wherever it is.

Which is where the rule came from #

The chronology is not this site’s business — the historical record of drawing practice belongs to whoever writes that subject, and nothing here reconstructs it. What the geometry can say is narrower and is enough:

A rule that sets a quantity on the paper is asserting that the quantity does not depend on position in the scene. In a parallel drawing almost everything has that property, and in a perspective picture almost nothing does.

That is the transferable form the taught-and-unmeasured essays arrived at from the perspective side, stated from the parallel side. The reason the manuals are full of paper rules is that the paper rules are correct in the drawing system the manuals grew up in, and the reason they fail in a perspective picture is that position-independence is precisely what perspective destroys.

Every taught rule this site has measured has that shape. The reflection measured down from the waterline, seven tenths along the diagonal for a circle, the third vanishing point placed by eye: each is a fixed number substituted for a function of position, and each has a parallel-projection ancestor in which the function is constant.

What a paper angle names #

If the construction is exact, then a paper angle is not merely a drawing convention — it names something in the room. Which is a claim that can be checked by running it backwards.

Take the drawn ray direction, take the light’s plan azimuth, and solve for the direction in space whose image is that drawn direction. There is exactly one, because a drawn direction is the image of a line’s worth of scene directions and the azimuth picks one of them off it. Run a real sun through the construction, read the paper angle it produces, and put the paper angle back through the inversion: the altitude comes back at 40.0000° against a sun set at forty degrees, in every system.

So the rule is honest. A paper angle in a parallel drawing does name one sun, the same one for every point, and the drawing is internally consistent in a way the perspective version is not.

What it does not name is the number it looks like it names.

The forty-five degrees is not forty-five degrees #

A ray drawn at forty-five degrees on the page is not a sun at forty-five degrees of altitude. It is a sun at whatever altitude that drawn direction is the image of, and the answer depends on the system and on the light’s azimuth.

Put a plan azimuth of thirty degrees through the inversion and a forty-five-degree paper ray means a sun at 45.00° in isometric — and at 48.05° in cabinet, and at 59.90° in cavalier. Change the azimuth to zero and the same paper ray means 20.10° in isometric and 54.74° in cavalier. Change it to ninety and isometric gives 53.79°.

Every one of those is a real sun and the drawing is consistent with it. But a draughtsman who lays off forty-five degrees because “the sun is at forty-five” has produced a drawing of a different sun, and a second draughtsman working on the elevation with a protractor will produce a third.

That is a smaller failure than the perspective one, and it is a different kind. In perspective the rule is inconsistent within one drawing. Here it is consistent within the drawing and mislabelled.

There is a smaller consequence of the same fact which is worth having, because it disposes of an argument a reader may reach for. It might be thought that the paper angle is at least monotone in the altitude — that a steeper drawn ray always means a higher sun, whatever the system, so a draughtsman comparing two drawings is at least ordering them correctly.

It is monotone, for a fixed azimuth. It is not comparable across azimuths, and the spread is large: the same drawn forty-five degrees runs from about twenty degrees of altitude to about fifty-five across the azimuths in one system. So two shadows drawn at the same paper angle in the same drawing, for lights coming from different directions, are two different suns — and a drawing containing both is not the projection of any single-lit scene at all, which is the parallel world’s version of the four-suns failure.

The rule a ruler can carry out #

There is a second rule in the manuals, and it is arithmetic rather than construction: set the ray so that its rise equals its run, and the sun is at forty-five degrees.

That is a statement about two measured lengths, and it is right in exactly one of these systems.

A military projection draws the ground plan at true scale and the verticals at true length, so both legs of the light’s right triangle are on the paper undiminished and arctan(rise ÷ run) is the altitude exactly — zero error, at every sun tried.

No other system does. Isometric foreshortens the vertical by 0.8165 and the plan run by a direction-dependent factor, and the same measurement is out by up to 7.2° over a spread of suns. Cavalier reaches 13.4°, dimetric 19.8°, and cabinet 21.5°.

One thing is worth being precise about, because it is easy to state the military result too strongly. A protractor laid on the drawing does not read the altitude, in any system including that one, because the drawn plan and the drawn vertical are not at right angles on the page. The rule is about two lengths and only a ruler can carry it out. That distinction is the same one a ruler on an isometric drawing turns on: what a drawing gives to a ruler depends entirely on the direction the ruler is laid along, and the military projection’s ground plane is the one plane in this field where the answer is the same in every direction.

Shadows on things that are not the ground #

The construction above lands the shadow on the ground because the ground is where the second line was drawn. Nothing in it is special to the ground, and the parallel case makes the generalisation cheap in a way the perspective case does not.

To cast onto a wall, replace the ground line from the foot with the wall’s own trace. To cast onto a sloping roof, replace it with the roof’s. In each case the shadow is where the drawn ray from the point crosses the drawn line of the receiving plane’s section through the same vertical plane — and each of those is a drawn line at a fixed direction, so it is another set-square setting and not another construction.

The consequence is that a parallel drawing’s shadows across a stepped or folded receiver are laid out with a small fixed collection of angles: one for the light ray, one for the light’s plan, and one per receiving surface. A draughtsman with four set-square settings can shadow an entire building.

In a perspective picture the same job needs the light’s vanishing point, the vanishing point of the light’s plan, and the vanishing line of each receiving plane, and every one of those has to be constructed. A shadow across an edge measures what happens when the receiver folds: the shadow is straight on each face and kinks at the crease, and the kink is the image of the crease rather than anything about the caster. Both worlds have that fact; only one of them can act on it with a set square.

What is exact, in a list #

Stated together, because the boundary is sharper than any of the pieces.

Exact in every parallel system. The intersection construction — foot, ray, plan, cross — reproduces the projected shadow. One paper angle serves the whole drawing. A paper angle names exactly one sun. Shadows of parallel lines are drawn parallel. A shadow’s length in the drawing is in fixed ratio to its true length along any single direction.

Exact in the military projection only. Rise over run gives the altitude. Plan distances and plan angles come off with a ruler and a protractor.

Exact nowhere. A protractor reading the drawn ray against the drawn plan. Any claim that the paper angle is the altitude.

And exact in no parallel system, but exact in perspective. Nothing — which is the asymmetry worth ending on. Perspective’s own constructions, the ones this site’s construction field is made of, are exact in perspective and meaningless in a parallel drawing, because they are built on vanishing points and a parallel drawing has none. The two families do not share a construction, and every rule that appears in both books is a parallel rule that has been carried across.

Reading a drawing that somebody else made #

The inversion has a use beyond checking the site’s own arithmetic, and it is the one a reader is most likely to want.

Given a parallel drawing with shadows already on it, the light can be recovered. Measure the drawn ray’s direction with a protractor and the drawn plan’s direction with the same protractor; those two drawn directions, together with the system’s own three drawn axes, determine the light’s direction in space up to the one-parameter freedom the kernel always leaves — and the plan direction removes it. The answer is a sun with an azimuth and an altitude, and it is the sun the drawing is consistent with whether or not it is the one the draughtsman intended.

Two consistency checks come free with it. If the drawing’s several shadows do not all use the same drawn ray angle, the drawing is not the parallel projection of any single-lit scene, and the departure is measurable in degrees. And if the recovered altitude is negative, the drawing depicts a light below the horizon — which is not a subtlety, it is a picture in which the shadows fall on the wrong side of their objects, and it happens whenever a paper angle is laid off in the wrong sense.

Neither check exists in the perspective world in that form, because there the drawn angles are supposed to differ and a spread proves nothing. There, the equivalent test is that the shadow lines meet at a single point below the horizon, which is what the lamp out of the picture is built on.

The check, and what it had to reject #

The claim that the construction is exact needs a case where it is not, or it is a claim about arithmetic working.

Two are available and both are used. The construction is run in four systems and the agreement is arithmetic noise in all four — which would also be true if the construction had accidentally been implemented as project the true shadow, so the second check is the one that matters: the drawn light ray and the drawn plan direction are computed from the light’s own image rather than from the shadow, and the intersection is solved for. A version that took the answer from the scene would not have a determinant to check, and this one refuses when the drawn ray and drawn plan come out parallel on the page — which happens for a light whose direction images onto its own plan’s image, and is the degenerate case a draughtsman meets as a construction that will not close.