ukmuiik

Question

Out for a walk yesterday, I noticed something curious about the shadow cast by a certain wall.

This wall had a height that grew approximately linearly. The base of the wall followed a smooth curve (hugging a bend in the road).

The sun was fairly high in the sky behind the wall, causing the wall to cast a shadow. The extent of this shadow remained at approximately a constant perpendicular distance from the wall.

In some sense, the curve in the wall "cancelled out" the increasing height of the wall.

My question: what curve causes this phenomenon?

I have managed to sketch out the setup in Geogebra, at the following link:

https://ggbm.at/jtrnv9ds

... and hereÃ¢Â€Â™s a pencil sketch of the setup:

pencil-sketch

... but I haven't got very far analytically. Can anybody help me? Thanks a lot!

may be the sun was so high that the thickness of the shadow was close to minimum and it looked like the shadow had the same height. — Jul 24 at 13:20
Nope, sorry! Obviously the shadow didn't exactly have a constant width irl, I'm just modelling it as such. — Jul 24 at 17:41
Is there a constant minimum width between wall foundation and edge of shadow? Can you upload a picture of wall and its shadow? — Jul 24 at 18:37
Do we assume in the model, that the curve of the base of the wall, as well as the wall's shadow, lie in one horizontal plane, and that the height of the wall increases by equal amounts for equal distances along the base of the wall, whence the top of the wall does not lie in a single slanting plane but rises somewhat helically? I'm trying to get clear on the conditions here, at least generically. — Jul 29 at 22:40
@EdwardPorcella - yes, all of your assumptions are correct! The height of the wall is a function of the curve's arc length. — Jul 30 at 11:36

Edward Porcella 1,0731411 · Answer 1 · 2018-08-02 19:10:47Z

If a wall is plane and of fixed height, the edge of its shadow is straight and parallel to the wall's base.

If the wall is plane but of uniformly increasing height, the edge of the shadow is straight but diverges from the wall's base as the wall's height increases.

What curve in the base of a wall of uniformly increasing height produces a shadow whose edge is a parallel curve, i.e. a curve whose minimum distance from the wall's base is the same at every point? Not the arc of a circle it seems.

shadow of circular wall
In the figure, $BJHKE$ is the wall viewed directly from above, with $BJ$, $JH$, $HK$, $KE$ equal arcs along the wall's base. From its height at $B$, the wall is higher by equal amounts at $J$, $H$, $K$, $E$. Denote the corresponding points at the top of the wall $B'$, $J'$, $H'$, $K'$, $E'$. From its great distance, light from the sun arrives at the wall in parallel rays, casting shadows of $B'$, $J'$, $H'$, $K'$, $E'$ at $C$, $N$, $O$, $P$, $F$ lying on curve $CD$. Since the shadow falls farther out from the base of the wall in proportion to the wall's height at the point of incidence, distances $JN$, $HO$, $KP$, $EF$ increase by equal amounts from $BC$. But in concentric circles, segments of chords intercepted between parallels drawn through points marking equal arcs on one of the circles do not differ by equal amounts.

Though not a formal proof, the following may be useful:

$$JN>BC$$and$$HO>JN$$but$$JN-BC<HO-JN$$ because $J$ rises higher in its run from $B$ than $N$ does in its run from $C$, but $H$ rises even higher from $J$ than $O$ does from $N$, and so on. The slopes where the parallels cross the circle at $J$, $H$, $K$, ... are greater, and increase faster, than the slopes where those parallels cross the arc of a concentric circle which is larger, and hence of smaller curvature, at $N$, $O$, $P$, .... In short, there is increasingly more rise for the run along the arc of the smaller circle than along the arc of the larger, so that $JN$, $HO$, $KP$, ... do not increase by equal amounts for equal arc increases $BJ$, $JH$, $HK$, ....

photo circular wall shadow
In the above photo of a handsaw blade clamped to a circular trashcan lid, the sunlight is perpendicular at the narrow end of the blade as in the previous diagram. The edge of the shadow is evidently not the arc of a circle concentric with the can lid; the length of a perpendicular drawn from a point on the shadow's edge to the blade increases for a while with the increasing height and then decreases to nothing.

I conclude that under the given conditions the base of the wall cannot trace a circular arc. The proof sketched above relied on the fact that an arc parallel to a circular arc is itself circular. This is not true for curves generally, however. E.g. a curve parallel to a parabola is not parabolic.

Note too that, if we assume as I have in the figure for simplicity, that the sunlight is perpendicular to the wall at its lowest point, then the base of the wall can bend at most through $90^o$. Shadows of any extension beyond that will eclipse portions of the shadow preceding (or fall to the other side of the wall?), as the wall rises ever higher. So we are looking for a finite portion of a non-circular curve, if it is a closed curve. Then again, from its principal vertex to any point however far out on a hyperbola, the change in direction is less than half the angle of the asymptotes, i.e. less than some acute angle; and from its principal vertex to any point however far out on a parabola, the change in direction is less than, although arbitrarily close to, $90^o$. So if either of these curves is the solution, the shadow edge could be parallel to the wall base for an unlimited arc of the curve.

Finally, it appears that to compensate for the non-uniform width of the shadow evident in the photo above, the curvature of the wall's base must diminish in some way as the wall's height increases, as in OP's sketch and in the figure below.
non-circular wall base

$EHKG$ is the top of the curved wall, rising linearly from $E$ to $G$, i.e. the height of the wall increases by equal amounts for equal distances along the wall's base $CILSF$. Area $DMFC$ is in shadow. $DOJPNM$ is the curved edge of the shadow. $OC=PI$ are normals from points on the shadow's edge to the base of the wall.

$D$, $J$, $N$, $M$ are the shadows of points $E$, $H$, $K$, $G$. And since the sun's rays $ED$, $HJ$, $KN$, $GM$ are virtually parallel, then right triangles $ECD$, $HIJ$, $KLN$, $GFM$ are similar, and $CD$, $IJ$, $LN$, $FM$ increase linearly with the height of the wall at points $E$, $H$, $K$, $G$, respectively. Therefore, $JI$, $NL$, increase linearly with distances $CI$, $CL$ along the base of the wall. More precisely, supposing $CE=1$, and that $m$ is the constant steepness of the rising curve $EHKG$, then$$HI=1+mcdot CI$$and$$KL=1+mcdot CL$$Therefore$$fracJINL=fracHIKL=frac1+mcdot CI1+mcdot CL$$We are looking for a curve whose distance from a parallel curve, measured along parallel intercepts, increases linearly with its arc length.

photo non-circular wall shadow
The saw blade proved well suited to simulate the wall, because of its temper as well as its taper. Since under a given force it bends more at the narrow end than at the broad end, it was possible to verify, as the second photo indicates, that the shadow's width can indeed be more nearly constant when the curvature of the wall diminishes as its height increases.

But even for a given curve, variation or constancy of shadow width seems to depend on the wall's compass-orientation to the sun. To achieve the near-constant shadow width of the second photo I had sunlight striking more directly at the narrow end of the blade than at the broad end, but nowhere perpendicular. But that near-constant width was not seen in other orientations. It is easier to move a saw than a wall, if you have to do it before the sun moves.

The above is of course an incomplete answer to the posted question, and may contain no more than the OP already knows, but perhaps it may be useful to someone.

This is a stunning attempt, Edward; thank you and well done! I especially love your use of the saw blade. That second photo almost perfectly reproduces the phenomenon. I'm going to have a longer think about your points when I have time at the weekend. — Aug 2 at 22:57

score 1 · Answer 2 · 2018-08-01 05:00:10Z

This is a comment but not yet an answer.Trying to find traces of upper and lower ends of a constant ascent spiral wall shadow. Please comment for further 3D geometry understanding, I do not get how constant width comes about.

Edges_Helix

EDIT1:

Please ignore the above. In the following sketch

enter image description here

Constant normal curves/surfaces are parallel Bertrand curves/surfaces. The mutual common normal (red to black) is of constant length.

Translational or surfaces dragged/extruded parallel to themselves can be defined.

In your case it is a combination of these two types. Ground intersection of canal coincides with Bertrand curve.

There is no unique upper edge to the wall. Any line can be drawn on the canal surface. Wall height can vary arbitrarily with respect to arc so long it is on the canal surface.

Differential equations with two independent constants can be written to describe the situation.

Thanks for having a go @Narasimham. IÃ¢Â€Â™ve attached a pencil sketch to the question which I hope will help. — Jul 31 at 18:32
In your hand-sketch, is sun rays' inclination $theta = 45^circ?$ — Jul 31 at 21:12
..and what you described can be observed from morning to evening? — Aug 1 at 4:53
Not necessarily $theta=45Â°$. But $theta$ is arbitrary, since $mathrmtantheta$ just becomes a constant multiplier for the shadow length parallel to the sun's rays. This $mathrmtantheta$ could be absorbed into the constants $alpha$ and $k$ (the gradient and initial height of the wall). Hence WLOG we could assume $theta=45Â°$. — Aug 1 at 18:48
No, it's not observable throughout the day, only at a specific time. $theta$ (the sun's zenith angle) is arbitrary in the setup, but the sun's bearing (angle from North) is not. — Aug 1 at 18:51

Edward Porcella 1,0731411 · Answer 3 · 2018-08-02 19:10:47Z