If you think about a different angle things can look a lot simpler.
The B in my diagram is in the above formulae.
Even if you know P, A and r I think you'll need a numerical method to find .
Dear community, this is my first post on mathhelpforum and I would be really very grateful for some assistance with this problem. I am going quite crazy!
It concerns a simple 2D shape which is made of 2 intersecting circles of equal radius and distance of centres separation . I have attached a diagram of this shape as a PNG.
When angle radians, the shape is a single circle, because the two circles intersect completely. When radians, the shape is two independent circles which are just touching each other.
I have come up with expressions for the total area , and total circumference of this object:
So, given radius , and the distance separating the centres of the circles, using the above expressions, you can easily calculate the total area , and the circumference .
I however need to solve the inverse problem. That is, I know the total circumference , and the total area , and I instead would like to calculate value pairs of radius and separation distance which fulfill these constraints.
Any ideas on an analytical or numerical solution very welcome!
Thanks tutor for taking the time to respond to my post.
However, I think taking the path you propose ultimately leads to the same problem I have.
I will illustrate the problem with a graph.
The graph shows two circles which are separated by a fixed distance of d=1. It shows what happens to the total area A (blue line) and circumference C (green line) of the circle pair as the radius r is increased from 1.
You can see that by fixing d, their are only certain distinct (C, A) value pairs which lead back to a value of the radius r.
It is thus not possible to get a full reverse mapping from an arbitrary (C, A) pair to a value for r for a fixed d, because only 1 distinct (C,A) pair will lead back to a value of the radius r.
You could relax the constraint that you are fixing d, but then you have to numerically solve a 3d function, and this does not appear straightforward to me.
In the case you propose, I think it is the same: you are trying to solve in terms of r and theta, whereas I am trying to solve in terms of r and d. Both cases require numeric solution in 3d.
It is amazing that a task seemingly simple can turn out to be so complex.