Given points P1, P2, and P3, I need to calculate the center point of 2 tangent circles, C1 and C2, with radius R. Line P1P2 is tangent to circle C1 at P2, line P2P3 is tangent to C2, and C1 and C2 are tangent to each other.
I am plotting points along a route as a vehicle travels from P1 to P2 and P3. It will turn with a fixed radius R, along C1, towards P3 after it have passed P2, and will continue onto C2 and then onto line P2P3.
How do I calculate the center point of the two circles? Bonus points for also calculating the path arc angles.
Let the angle of inclination of $P_2P_3$ be $\alpha$, a known quantity.
The center of $C_2$ lies on the line whose slope is $\tan \alpha$ and its x-intercept is given by $\dfrac {-R}{\sin \alpha}$.
The center of $C_2$ also lies on the circle (centered at (R, 0) and radius = 2R).
Solving the above two will give us the location of the center of $C_2$.
By applying sine law to $\triangle XHK, \beta$ (and therefore $\gamma$ and $\delta$) can be found.
The arc lengths of the red and blue arcs can then also be found.