I have two quadratic curves sharing one control point $A$. The control points for the first curve are $A, B, C$ and for the second curve: $A, D, E$.
The two quadratic curves are defined as:
- $P(t) = (t - 1)^2A - 2t(t - 1)B + t^2C$
- $P(u) = (u - 1)^2A - 2u(u - 1)D + u^2E$
I need to find the $t$ value(s) on the curve $P(t)$ for the input value $r$ which is the radius of the circle that touches the tangent lines of both curves (note that there can be more than one circle that matches the criteria). In other words, I need to find the $t$ value where the offset curves (with offset $r$) of the two curves intersect.
I've got far enough to get the equations for the offset curves and a relation between $u$ and $t$ for the intersection points but I'm having trouble getting some sort of relation between $r$ and $t$, which will help me calculate or approximate the intersection points and the maximum radius of the circle that fits between the curves.
Just a suggestion:
You can construct the offset curve of one of the Beziers (with offset $2r$) and intersect with the other. To implement this, flatten the two curves and intersect the segments pairwise (do not flatten then offset).
A brute-force solution will take effort $O(nm)$ for $n$ and $m$ segments in the respective polylines, but you can reduce this by plane-sweep techniques (Bentley-Ottmann).