Hi Michael,

After much study, I believe that I grasp your post and formulas :-), so I made a few notes.

(Note, swap Brian's point labels A and B, to conform to Michael's code. I believe that Michael's code, and arccontinue work in 3D. )

Notes on the code:
In the cplane Frame, for purposes of the computation of r or R, in order to find the center point of the arccontinue scripting, the value of R represents the y coordinate B2D.y, and so can be positive or negative. If R is negative, point A2D lies below the x-axis, and so A2D.y is negative. Or perhaps I should say, if point A2D lies below the x-axis, then R is negative. (Of course radii of circles are usually considered as being of positive distance.)
I had two quibbles with the last two sentences of the code.
The formula for the circle is essentially the pythagorean code.
Using R with its associated plus or minus value:
Forming a triangle for point A2D, gives the pythagorean formula:
((-1) * (R - B2D.y))^2 + (B2D.x)^2 = R^2. which is exactly the same
form as Michael derivation. So Michael's Radius =((B2d.x *...) applies. So keep the sign of R, to get the value of centerPt.y, and centerPt.x = 0, in cPlane Frame.

Convert centerPt back to world coordinates, and run arccenter with centerPt, pointA, and pointB, to get the script equivalent to pointpicker arccontinue command.

The last quibble is with the final sentence, which was confusing to myself. IMHO, drop the "original" text, and it should read something like:
The circle's origin is on the cPlane frame's y axis at point y = calculated R value, (maintaining the + or - value calculated.)

*************

It belatedly comes to mind that the script could calculate and create the line segment, and other points needed, and then ask the user to select the upper end of the line segment, with code using pointpicker.
Asking the user for the bottom end point, with code for a second pointpicker, would enable another arccontinue from point C to point D.

Can the pointpicker dialog be done in the "middle" of a script???,
after some geometry has been added to the geometry database?

(But I prefer the script to calculate the entire curve profile, without pointpicker interaction.)

- Brian