I'm trying to avoid a clash during a parametric design for 3D printing that involves the intersection of multiple shapes. The design can be simplified to consider only surfaces and then simplified further to consider a single plane, reducing the clash avoidance issue to a 2D problem. That's the problem I'm writing about here.
Two concentric circles with radii R and R–s intersect with an ellipse in a particular way. The vertices of the ellipse are on the circle with radius R at points A and B, creating chord AB with length equal to the ellipse major axis 2a where a<R. A third circle of radius r with s<r<a is centered at point A. This circle intersects the circle with radius R–s at two points C1 and C2. Call the point closer to the ellipse C1. The semi-minor axis of the ellipse, b, is chosen such that C1 remains outside the ellipse at a distance of t from the ellipse. What is the function b(a, r, R, s, t)?
Other parameters in the design define particular values for r, s, and t. The locations of the center of the concentric circles and of the points A and B are also defined, thereby defining R and a and the location of points C1 and C2. I also know—as previously stated—that 0<s<r<a<R, and I also know that 0<t<r. The value of t may be less than, equal to, or greater than s.
The free parameter is b, the semi-minor axis of the ellipse. The clash is avoided when b is sufficiently close to zero to create an ellipse that remains a distance t or greater from C1. However, a value of b that is too small results in a 3D shape that is aesthetically unpleasant for other reasons. I can tell by visual inspection that a particular value of b is required to have an ellipse that is a distance t from C1 when a, r, R, s, and t are known, but having the function to describe the relationship would be very useful. I rarely deal with problems at this level of intricacy, and I'm getting a bit stuck. Any help would be useful!
The given data fix the positions of points $A$, $B$ and $C_1$, which I'll rename $C$ for the sake of brevity. Set up then a coordinate system where the coordinates of those points are: $$ A=(-a,0),\quad B=(a,0),\quad C=(x_C,y_C). $$ The equation of the ellipse can then be written as: $$ {x^2\over a^2}+{y^2\over b^2}=1, \tag1 $$ where $b$ is to be found. If $P=(x,y)$ is the point on the ellipse nearest to $C$, then we have another equation: $$ a^2(x-x_C)y=b^2(y-y_C)x. \tag 2 $$ Finally, we also require $CP=t$, leading to a third equation: $$ (x-x_C)^2+(y-y_C)^2=t^2. \tag3 $$ By solving the above system of three equations one can find the unknowns $x$, $y$ and $b$. But it's a system of sixth degree (at least), so I don't expect an explicit solution to exist.