Calculate intersect point between arc and line

Question

I want to calculate the intersect point between arc and line. I have all the data for line and arc.

For line : start and and end point.
For arc : start/end poin

Answer 1

A point on the arc has coordinates

R.cos(t) + Xc
R.sin(t) + Yc

Using the implicit form of the line equation (either given or obtained from two given points),

A.X + B.Y + C = 0

then

A.R.cos(t) + B.R.sin(t) + A.Xc + B.Yc + C = 0

To solve this trigonometric equation, first divide both members by R.√A²+B², giving

c.cos(t) + s.sin(t) = d

which can be rewritten, with tan(p) = s/c and d = cos(q):

cos(t-p) = cos(q)

then

t = p +/- q = arctan(B/A) +/- arccos(-(A.Xc + B.Yc + C)/R.√A²+B²)

Eventually, you will need to check if these values of t fall in the range (start, end), modulo 2π.