I have a sector of a circle split into 16 equal segments. I am trying to calculate the "hypotenuses" of the triangles formed by 2 intersecting straight lines (for example, triangle $LEQ$), which sort encloses the circular sector in rectangular boundaries.
- Radius of the circular segment is known.
- Angle of the sector (and hence the segments) is known.
- Lengths $UE1$ and $JE1$ are known.
- $EJ$ is parallel to the X-axis.
- Assume α is the angle for each segment.
- Calculate $KJ$, $KJ = tan(α) \cdot EJ$
- From here, line $EK = KJ / sin(α)$
- To find the "opposite" of the next segment I do $EJ\cdot(tan(2α) - tan(α))$
- Repeat step number 2 with the new value.
The red segments' "opposites" are parallel to the Y-axis. That all works fine until I reach segment 9, where the point $E1$ is. The "opposites" for the blue segments now have to be parallel to the X-axis, and continuing with my approach I only get them parallel with the Y-axis.
Using first calculation as a reference, how can I find the blue "opposites" such as $UT, TS, OE1, NO$, etc..?
EDIT: Great answer! How would one relate length of the radial lines (such as $EN$) in a function of x: f(x) where x = segment number?