What Shape Does a Catenary Shadow Form on a Cylinder?

[tasp77]

I saw a shadow of a catenary (or close enough) on a cylinder and wondered about the shape.

Specifically, it was a powerline strung between 2 utility poles, and the sun angle would have been around 45 degrees or so, and the cylindrical grain bin was between and north (I am in northern hemisphere) of the poles. The powerline I assume is a catenary, although it would be close to a parabola, right?, and the grain bin would be a vertically oriented cylinder to pretty high precision (neglecting corrugations).

The shadow of the powerline appeared to define a plane on the grain bin, but that is just to me and my bifocals. And the shadow also looked from a distance to be a segment of a circle, but I find that difficult to fathom, as how would a circular cylinder (so to speak) cancel out (so to speak) the catenariness of the shadow?

Anyhow, is this an interesting (and previously identified) shape?

 

[Eynstone]

It's definitely an interesting shape and very likely to be a conic - catenaries have hyperbolic cosines in their equation & the projection is likely to be a parabola/hyperbola.

 

[tasp77]

Hey thanks!

I don't think the powerline guys or the farmer oriented their items to amuse me, the alignment is just accidental, and it is clearly visible from the adjoining highway I travel on frequently, on sunny days.

 

[tasp77]

BTW, is a catenary formed when the wire weighs the same per unit if distance, or is it when the weight per unit of distance the wire crosses the same?

Which is which, and what do we call the other one?

Examples would be for the first, just a wire hanging between 2 poles, for the second, it would be closer to a suspension bridge with the deck being of uniform weight all along its length.

 

[blue_raver22]

I find this observation intriguing and worth further investigation. The shape of the shadow on the cylinder is indeed interesting and can be explained by the principles of geometry and optics.

Firstly, the powerline strung between the two poles is likely to be a catenary, which is a curve formed by a hanging chain or cable under its own weight. However, due to the tension in the powerline, it may also approximate a parabola. This is because the tension in the cable creates a force that is perpendicular to the tangent of the curve, resulting in a shape that is similar to a parabola.

The cylindrical grain bin, on the other hand, can be viewed as a cylinder with a circular cross-section. When the sun's rays hit the cylinder at an angle of around 45 degrees, the shadow cast on the surface of the cylinder will appear as a segment of a circle. This is because the circular cross-section of the cylinder acts as a lens, focusing the light onto a smaller area and creating a circular shadow.

Regarding the cancellation of the catenary shape in the shadow, it is important to note that the shadow is not a perfect representation of the shape of the powerline. This is due to the cylindrical shape of the grain bin and the angle at which the sun's rays hit the surface. Therefore, the shadow may appear to be more circular than the actual shape of the powerline.

In conclusion, while this shape may have been previously identified, it is still interesting to observe and analyze from a scientific perspective. It showcases the principles of geometry and optics, and further investigation could potentially reveal more insights into this phenomenon.