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.

Answer 2

Let's define an arc and a line:

Arc:

• xa=X-coordinate
• ya=Y-coordinate
• a1=starting angle (smaller angle)
• a2=ending angle (greater angle)
• r=radius

Line:

• x1=first X-coordinate
• x2=second X-coordinate
• y1=first Y-coordinate
• y1=second Y-coordinate

From that you can calculate:

• dx=x2-x1
• dy=y2-y1
• al=arctan(dy/dx) (Angle of the line)

The arc and the line won't intersect when al < a1 or al > a2 or, in other words, the angle of the line isn't between the angles of the arc. The equations for an intersection are as follows:

• xa+rcos(al)=x1+cdx
• ya+rsin(al)=y1+cdy

where c (0 < c <= 1)is the variable we're looking for. Specifically:

• (xa+r * cos(al)-x1)/dx=c
• (ya+r * sin(al)-y1)/dy=c

The intersection point is therefore at (x1+c * dx),(y1+c * dy)

This algorithm only works when the arc and the line have one single intersection. If the line goes through the arc two times then it won't register any intersection.