What families of curves are in these pictures?

Question

Check out these photographs of churches:

http://www.slate.com/blogs/behold/2013/10/13/richard_silver_vertical_churches_captures_churches_from_altar_to_entrance.html

The photographer is effectively projecting the ceilings and walls of these churches onto a cylinder which axis is perpendicular to the main centerline of the churches.  Given that these churches have the following geometrical elements:

1) Horizontal lines running parallel to the main centerline
2) Horizontal lines running perpendicular to the main centerline
3) Vertical lines running orthogonal to 1) and 2)

What families of curves are 1), 2), and 3) are making? 

Note:  Many of the "vertical columns" veer off on separate curves at the top, disregard those separate curves.  Only consider the orthogonal straight line elements 1), 2), 3)

Fred · Accepted Answer

Do you know that's how it was done, geometrically, or did you surmise it in some way? Because there are other ways to get these sorts of shots, and it will make a difference in the result.

It's all about projections from 3-D space onto 2-D (a plane) -- a topic that includes map projections.

When an ordinary photo is taken, it can be idealized as a projection of the (3-D) scene through a point, onto a plane. If we idealize this scene as consisting of a 3-D gridwork of lines parallel to each of the 3 coordinate axes, and supposing that the camera is being aimed in the z-direction (i.e., that the focal plane, F, is parallel to the xy-plane), then the x-parallels and the y-parallels would be straight lines (still parallel to their respective axes), and the z-parallels would be radial straight lines, emanating from the central point of F. Clearly, that can't be the case here, since the images of the y- and z-parallels aren't straight.

I will label your numbered directions ("classes" of lines):
1) y
2) x
3) z

So let's go with your cylinder idea. Then instead of an xy-plane being used to project the image, take instead, a cylinder, whose axis is || to x, and passes through the center-point, P, of the lens. The class-2 lines will still be straight lines, in the x-direction. Classes 1 and 3 will be transformed in the same way (as each other). You can idealize one of these as a line || to a perpendicular through the cylinder axis, which is then projected through P, onto the cylinder. This will produce either an azimuthal circle (which will "unroll" into a straight line) in the case where the original line had x=0, or an ellipse (which will "unroll" into a sinusoid).

And this appears to be a good description of the result, and validate your statement at the top, about the cylinder.

So I'd say that your answers are:
1) sinusoids, plus one straight line when x=0
2) straight lines; generators of the cylinder
3) sinusoids, plus one straight line when x=0

What families of curves are in these pictures?

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