Geometry is not my strongest subject and I can't find solution to my problem.
I'm writing algorithm that finds coordinates in a circle sector. There is $n$ amount of parallel lines, each with equal distance from each other, within circle sector. Below I attach image for reference:
Problem: Find coordinates of $(x_n,y_n)$ and $(p_n,q_n)$
Known variables:
$P$ - point (0, 0). Point P is the origin of the circle
$\alpha$ - triangle angle
$d$ - distance between parallel lines
$r$ - circle radius
So far I found formula for $x_n$ and $y_n$:
Values for $x_n$:
- $x_1$ = $\frac{d}{sin\alpha}$
- $x_n = n*x_1$
Values for $y_n$:
$y_n = 0$
I don't know how to find coordinates of $(p_n, q_n)$. I've been trying to solve this for a few hours now and I still can't find solution. Thank you in advance!
PS: Solidworks is only software that know how to use to show geometry. I don't know how to use interactive geometry software. Sorry for the inconvinience
The problem is easier if you rotate the figure so that the oblique radius becomes horizontal. Then you intersect the circle of equation
$$u^2+v^2=r^2$$
with the horizontals
$$v_k=-kd.$$
The intersection points are thus
$$\left(\sqrt{r^2-v_k^2},v_k\right).$$
Now you can rotate to bring the figure back in the original position. It is convenient to use complex numbers and compute
$$p_k+iq_k=(u_k+iv_k)(\cos\alpha+i\sin\alpha).$$
The $(x_k,y_k)$ are easier.