Find the Dots

Math problems are Mohr fun when they involve circles.

Given 0 < tx < Rx and 0 < ty < Ry, two concentric but not necessarily similar ellipses are defined as x2/Rx2 + y2/Ry2 = 1 and x2/(Rx - tx)2 + y2/(Ry - ty)2 = 1.

Two circles, C₁ and C_N are defined as (x - Rx + tx/2)2 + y2 = (tx/2)2 and (y - Ry + ty/2)2 + x2 = (ty/2)2.

In between these circles are (N-2) circles denoted C₂ through C_N-1 with respective radii r₂ through r_N-1 whose edges are tangent to both ellipses and whose maximum overlap with each adjacent circle is given by m/2 × (rn + rn+1) where 0 ≤ m < 1, r_n is the radius of one circle and r_n+1 is the radius of the adjacent circle.

Find the number of circles N, the minimum possible constant value of the overlap ratio m, and the center coordinates and radii of all the circles.

Alternate problem: Instead of starting with C₁ and C_N defined as circles centered on the axes, C₁ and C_N are tangent to both ellipses and overlap the x and y axes, respectively, by mr1 and mrN, respectively.

Why do we care? By drawing every other circle, we can create a dotted CSS borders for curved corners. :)