Radius of circle inside a funnel

Question

Given a random point in a funnel area in 2d I'm trying to find the radius of the circle that has center in the bisector of that funnel.

This is for a geometry program so I can extract a lot of information. I think I'm just missing a small detail but not sure what.

This is the situation. I'm given the center of a funnel and there is a circle that can goes along the bisector of that funnel, expanding or contracting until the edges of the funnel depending on the distance. I'm given a point P as the image shows. I'm trying to find the radius of circle that this point belongs to. Obviously there are 2 circles that contain that point however I want the one showed in the image, to the right of the point P. I have the distance D to that point from the center of the funnel and the angle alpha it makes with the bisector. All the letters A, B, C, E are easily obtainable however I can't seem to find the a way to use them to calculate the radius of the circle. I also want to calculate the distance from the circle center to the center of the funnel but that is easy if I have the radius. Any ideas or hints?

Answer 1

CONSTRUCTION based on a homothety:

Denote $\mathcal{K}$ the circle you want to construct, and H its center. Denote V the vertex of the funnel. Construct an arbitrary circle $\mathcal{L}$ tangent to the arms of the tunnel, with center S at the bissector of the funnel and such that |SV|<|HV|.
The half-line VP cuts this small circle at two points. That one which is closer to P (denote it R) is an image of P through a homothety. The small circle $\mathcal{L}$ is image of $\mathcal{K}$ through this same homothety. Thus, the segments RS and PH are parallel. We have R, S, P, thus we can easily construct H.

Answer 2

Denote with $\gamma$ the acute angle that radius R (shown in your picture) creates with the funnel bisector. Denote the half angle of the funnel with $\beta$.

$$R\cos\gamma=D\cos\alpha-\frac{R}{\sin\beta}$$

$$R\sin\gamma=D\sin\alpha$$

Square these two equations and add them. Unknown angle $\gamma$ will vanish and you will get a simple quadratic equation with R being the only unknown.