
We know that:
The center of the profile arc and the toolpath locus is at (0,R).
The center of the locus of all points equidistant from the peak of the ridge is at (0,e)
The equation of a circle of radius r with center at (h,k) is:
(x  h)² + (y  k)² = r²

Then the equation of the ridge locus is:
x² + (y  e)² = r² [1]
and the equation of the toolpath locus is:
x² + (y  R)² = t² = (R  r)² [2]
Subtracting [1] from [2] yields:
(y  R)²  (y  e)² = (R  r)²  r² [3]
Which can be expanded and then simplified to:
y = (2Rr  e²)/(2R  2e) [4]
We can substitute this value of y from [4] into [1], expand, and determine that the values for x are:
x = ± sqrt(r²  (y  e)²)
And so, as expected, we have intersections on both sides of the origin.

If we define a to be one half of the angle A, we can use the right intersection point to determine:
tan(a) = x/(R  y)
which we can write as:
a = arctan(x / (R  y))
and to get the entire angle:
A = 2a = 2arctan(x / (R  y))
Note for new CNC woodworkers:
I normally use the largest round nose or core box router bit (typically 1" diameter) that I have available and set e = 0.001" for my production work. A 1/1000" ridge is, for all practical purposes, undetectable.
