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The forty-five degree shadow

Draw the shadow at forty-five degrees and make it as long as the object is tall. In the plan that is exactly a sun halfway up the sky. Applied on the paper it puts four posts under four different suns — altitudes from sixteen to twenty-nine degrees, azimuths thirty-two degrees apart, and shadows between one and three-quarters and three and a half times the height. No drawing angle brings them together.
17 min read 13 figures Taught, and never measuredOne machine used twiceProjected, not constructed

Worth reading first: Where shadows vanish · A shadow is a second projection · The lamp, out of the picture.

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.

The rule's shadows and the sun'sFour posts, each with a shadow drawn at 45° on the paper and as long as the post is drawn tall, and beside each the shadow a single sun 45° up actually casts. The rule's shadows are the same on the page and the sun's are not, which is the whole of the difference.correct from 19 cm, at 160 mm wide46° across
Fig. 1 Four posts of one height at four places on the floor. Each has a shadow drawn by the rule — the same angle on the paper, the same length as its own drawn height — and beside it the shadow a single sun actually casts. The rule’s shadows are alike on the page and the sun’s are not, which is the whole difference.
The rule's shadows and the sun'sFour posts, each with a shadow drawn at 30° on the paper and as long as the post is drawn tall, and beside each the shadow a single sun 45° up actually casts. The rule's shadows are the same on the page and the sun's are not, which is the whole of the difference.correct from 19 cm, at 160 mm wide46° across
Fig. 2 The same four posts with the rule drawn at thirty degrees instead. The shadows are shorter on the page and the disagreement between the four implied suns is larger, not smaller.

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.

Four shadows, four sunsEach drawn shadow read back as a light: altitudes from 16.1° to 29.4°, and shadows 1.77 to 3.46 times the post's height, from a rule that promised one sun and shadows as long as the posts are tall. The four true shadows agree to 4e-14°.post at 5.0 m24.4° up2.20× tallpost at 8.5 m19.2° up2.87× tallpost at 4.2 m29.4° up1.77× tallpost at 11.0 m16.1° up3.46× tallthe sun each drawn shadow impliesthe rule promised one sunit drew 4
Fig. 3 Each drawn shadow read back as the light behind it. The four altitudes run from about sixteen degrees to about twenty-nine, and each shadow is between one and three-quarters and three and a half times its post’s height — from a rule that promised one sun and shadows as long as the objects are tall.
Four shadows, four sunsEach drawn shadow read back as a light: altitudes from 1.8° to 3.4°, and shadows 17.03 to 31.71 times the post's height, from a rule that promised one sun and shadows as long as the posts are tall. The four true shadows agree to 4e-14°.post at 5.0 m3.3° up17.37× tallpost at 8.5 m2.3° up24.99× tallpost at 4.2 m3.4° up17.03× tallpost at 11.0 m1.8° up31.71× tallthe sun each drawn shadow impliesthe rule promised one sunit drew 4
Fig. 4 The other reading of the same instruction, read back the same way. Laid off away from the viewer the rule implies suns within a few degrees of the horizon and shadows tens of times the objects’ heights.

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.

Six posts in sunlight from 55°The shadows are parallel in the world, so in the picture they meet at one point on the horizon — found from the drawn shadows to 1e-12 px.horizonshadows meet at x = -627, off the frameon the horizon, as it must be
Fig. 5 The fact the rule contradicts. Shadows cast by the sun are parallel in the room, so in the picture they converge on one point of the horizon. A drawing whose shadows are parallel on the paper is a drawing whose shadows converge nowhere.

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.

No angle rescues itAt every drawing angle the four posts imply four different suns, and the closest they come is 11.0° of altitude apart. A rule that sets the shadow on the paper is one number; where the shadow goes depends on where the post is.051015204060the angle the rule is drawn at, degrees below the horizontalspread of the sun altitudes the four shadows imply (degrees)one number for a quantity that varieswith the post's own position
Fig. 6 The spread of implied sun altitudes against the angle the rule is drawn at. There is a shallow minimum and it does not come near zero: at the best angle tried the four posts still disagree by eleven degrees about where the sun is.

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.

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.

The rule's shadows and the sun'sFour posts, each with a shadow drawn at 45° on the paper and as long as the post is drawn tall, and beside each the shadow a single sun 45° up actually casts. The rule's shadows are the same on the page and the sun's are not, which is the whole of the difference.correct from 19 cm, at 160 mm wide46° across
Fig. 7 The other reading of the same instruction — the same angle, laid off away from the viewer instead of toward. The drawn shadows are short on the page and enormous in the room, because a line running up toward the horizon covers ground very fast.

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.

A box and its shadow, both projections from a pointThe rays from the lamp to the corners are the same construction as the rays from the eye to the corners — one operation, two centres.correct from 26 cm, at 160 mm wide34° across
Fig. 8 The general machinery. A shadow is a projection from the light, so a lamp is a second eye and its picture is the shadow. Once that is the frame, “where does this shadow go” is the same question as “where does this point go”, and the answer is two lines.

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.

The same construction, in sunlightThe lines through top and shadow tip are parallel in space, so their images meet at a vanishing point — and the lines through foot and tip meet ON the horizon, 5e-12 px off it. A light whose foot is on the horizon is a light at infinity.the shadow lines meet on the horizon — a light at infinityhorizoncorrect from 25 cm, at 160 mm wide3 posts · foot 5e-12 px off the horizon
Fig. 9 Where a light was, from the drawn shadows alone. Two facts do all the work and neither needs a calibrated camera: the ray lines meet at the image of the light, and the ground lines meet at the image of the point beneath it.
The lamp is the second eyeOne camera, one lamp, one point. The camera's ray through the point's image fixes it on a line; the image of its shadow fixes where the lamp's ray through it meets the floor; and two lines that are not parallel meet. The point comes back at 1e-15 m of closest approach and 9e-16 m from where it was put — depth out of a single photograph, with no second camera and nothing assumed about the object. The rays cross at 28.5°, and that angle is what the measurement is worth.horizonthe pointits shadowcorrect from 22 cm, at 160 mm widerays cross at 28.5° · recovered to 9e-16 m
Fig. 10 And what separates a lamp from the sun in a picture. A lamp’s shadow lines meet below the horizon and the sun’s meet on it, and the signed distance from the horizon is the quantity that says which.

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.

Shadow length against the sun's elevationA 1.8 m post casts a 2 m shadow at 45° and a 10.2 m shadow at 10°. The curve is a cotangent and it has no upper bound.0102030204060elevation of the sun (degrees)length of the shadow of a 1.8 m post (m)45° — shadow equals heightcot of the elevationunbounded as the sun sets
Fig. 11 How shadow length depends on the sun and not on the object’s position. The world length is height divided by the tangent of the altitude, one number for the whole scene, and it is the picture’s job to foreshorten it — which is exactly the job the rule takes away from it.
The shadow of a ball, under a lamp 2.40 m upThe lamp's tangent cone cuts the ground in an ellipse. The top of the ball is at 0.84 m and the lamp at 2.40 m: above that height the shadow closes, below it the shadow reaches the horizon, and the ball's distance from the lamp appears nowhere in the condition.the lamp is above the top of the ball — an ellipsecorrect from 22 cm, at 160 mm wideaxis ratio 0.4012 · discriminant -6.23e-1
Fig. 12 What a correctly constructed shadow is, for scale. The shadow of a ball is a conic whose type is decided by the sun’s altitude, and every one of its points is where a ray from one light meets the floor.

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.

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.

The outline, the shadow, and the outline againA point light is a centre of projection, so the map from the occluder's plane to the floor is a plane projectivity and has an inverse. Running each shadow point back through the lamp returns the outline to 3e-16 m — the recovered curve is drawn over the original and cannot be seen apart from it.the outline, its shadow, and the outline recovered from itcorrect from 20 cm, at 160 mm widerecovered to 3e-16 m
Fig. 13 And a reminder that the machinery runs both ways. Given a shadow and the light, the object comes back; given the object and the shadow, the light comes back. Neither is available from a picture whose shadows were put in by a rule, because there is no light to find.

Where this goes next

horizonshadows meet at x = -58, off the frameon the horizon, as it must be Where shadows vanish The shadows of parallel posts under the sun meet at a point, and that point must lie on the horizon. Under a lamp they meet at the lamp's foot instead. Both are checkable in any photograph, and a picture whose shadows fail the check was not lit by anything. the shadow lines meet below the horizon — a lamp in the roomhorizoncorrect from 25 cm, at 160 mm wide3 posts · foot 351 px below the horizon The lamp, out of the picture Two posts and their shadows put the light's position in a photograph, exactly, with nothing given but the drawn lines and the camera's own horizon. Two posts. One gives a residual of zero and an answer that is a whole one-parameter family — the sharpest counter-example this site has to the idea that a small residual means a right answer, met again in a new field. correct from 26 cm, at 160 mm wide34° across A shadow is a second projection The construction that puts a shadow on the ground is the construction that puts the scene on the picture plane, with the lamp where the eye was. Shadow drawing is taught as a separate set of recipes and it is one operation with the centre moved, which is why the same code draws both.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

AltitudeAzimuthForeshorteningGround planeHorizonLight recoveryProjection from a pointShadowSunVanishing point