I have the general situation illustrated below. I know angles $\Theta_{1}, \Theta_{2}$ and line lengths $A$, $B$ and $C$, (where $A = a_{1} + a_{2}$ etc.). I'm trying to calculate the largest semi-circle I can fit on line $B$ (i.e., calculate r), that will (always?) be tangent to line $A$ and $C$ when the angles are acute.
So I think: $$ r = \frac{B.sin\Theta_{1} .sin\Theta_{2} } { sin\Theta_{1} + sin\Theta_{2} } $$
For the cases where $A > a_{1}$ and $C > c_{1}$ (meaning A is longer than the theoretical tangent point at the angle $\Theta_{1}$ etc.). But what if $A$ was very short? Does $r$ simply become the distance to the end point of $A$? And how do I 'know' when this is the case mathematically?
EDIT: To add some clarification...
- This is part of a larger effort to find the largest semi-circle that will fit inside an arbitrary closed polygon.
- The lines illustrated are thus three consecutive segments of that polygon
- They may be any angle and length, and don't form a triangle unless my polygon is a triangle.
- Clearly, if the angle at either or both ends is larger than $\pi$, the semi-circle can go right to that vertex and isn't impinged upon by the next or previous segment.
The part I'm mostly struggling with is incorporating the line length in determining $r$ and the centre point. Obviously, it has an effect at any given angle:
My ultimate idea is to draw lines between every polygon vertex and calculate the largest semi-circle that will fit inside the polygon along that line. I hope that will give me a good guess at the largest generally that will fit.
I know the blue values, how do I calculate $r$ (and thus $b_{2}$ and $b_{3}$)? ($h$ and $b_{1}$ being trivial). It seems like it should be easy...but I just can't seem to get my head around it.