I have this [on hold] question (#857264) re-phrased. Hope that the content is more meaningful now. The following is the picture modified from the original.
The question is a rectangular piece of metal sheet of width W. It is then rolled into a tube with base radius R such that 2πR = W. After the above operation, P goes to P’, Q goes to Q’, T to T’ etc. The question is asking for a relation between the images (after the rolling) w.r.t. its originals.
I think a 2D solution will be sufficient. We therefore let the cross-section of the tube be placed on a rectangular co-ordinate system as shown. It is represented by the yellow circle ($C : x^2 + (y – R)^2 = R^2$)
The relation (NOT a function because it will not pass the vertical line test) has to be divided (and described) into 4 parts. The non-shaded part is the symmetric image of the colored part (and hence omitted.)
I. The pink-shaded part with x in [0, 0.5πR] and y in [0, R].
P’(X, Y) is ‘mapped’ by P(x, 0) and the relation between them are
x = Rθ; X = R sin θ; and Y = R - R cos θ for θ in [0, π/2].
θ has to be eliminated, and both X and Y have to be expressed in terms of x to reflect the relation (x, 0) -> (X, Y).
II. The green-shaded part should be treated similarly.
The question is how to express such a relation?
EDIT: The previous version has bugs in it. This is an update. Sorry for the inconvenience.