Solving trigonometric system of equations

[Alibeg]

Hi. I'd like to know whether is it possible to do the following, and if so, how...(and also, whether is it possible to solve similar problems)

I have parametric equation of a curve and I need to find its intersection with a ray that starts at the origin of the coordinate system and makes known angle with the positive x-axis.

Curve:
x = ( R-r )( cos(\varphi) - cos(\theta) ) + D cos(\varphi (R-r)/r - \theta (R-r)/r )
y = ( R-r )( sin(\varphi) - sin(\theta) ) + D sin(\varphi (R-r)/r - \theta (R-r)/r )

R, r and D are known constants.

Ray:
y = tan(\beta) x

\beta, \theta and \varphi are angles, \varphi is my parameter, \theta and \beta are some variable angles.

I need to find the coordinates of the intersection of curve and the ray as a function of \theta and \beta.

In short I want to know coordinates of intersection but without the parameter \varphi in them.

 

[Alibeg]

This is the picture of my problem for r = 5, R= 6 and D = 7.
Note that \theta is not visible on this picture, but it had a fixed value while I was taking the picture.
Making \theta change would cause curve to change position and orientation.

Thanks :D

 

[jackmell]

How about numerically? It's not too hard to design an algorithm that zeros into the roots without having to manually select starting values.

 

[Alibeg]

Hmmm. I need distance of that intersection from the origin of the coordinate system. But I need it as a function of a \theta so I can find its minimal value for different \thetas . (i would have to use derivatives and other tricks later)

 

[jackmell]

Well what, that ain't no hill neither. But first you need to make clear if you are wiling to settle for a numeric approximation. If not, then I can't do it.

 

[Alibeg]

I am sorry, but numeric solution doesn't help me, since I cannot analyze the data further that way, and I also don't think it's an easy task.

Maybe I am using a wrong approach.Here more description of my problem:
I have to change \theta from 0 to some value. While changing it my (green) curve moves (the origin of the coordinate system is always inside of it).
If I trace my curve, I get a unusual white shaped area. I want to find the equation for the boundary line of that area.