Figure 1


Given a line segment (x1,y1), (x2,y2), and a third line with angle g3, will the third line intersect the line segment, and if so calculate the intersect point (x3,y3) and the distance d of the intersect point (x3,y3) from the origin.
If the point (x1,y1) has angle g1 and the point (x2,y2) has the angle g2, then the line with angle g3 will intersect line segment (x1,y),(x2,y2) if g1 > g3 > g2.

The problem is useful to solve in computer graphics and 2D modelling to work out whether line segments are cut by a line.
If there are many line segments, then if g1 > g3 > g2 relation is not satified by a particular line segment, then that line segment can be dropped from further calculations.
If there are many line segments that are cut by one line, then selecting the smallest distace d also select the line segment that is closest to the origin when cut.

This type of problem is usually solved in text books  by calculating the gradients and working out the intersect point.
That method however is prone to numerical blow up when gradients become infinite.
The solution described here uses the sine law and algebra with a big helping hand from xmaxima to get final answer:

                           sin(g3) x1 y2 - sin(g3) x2 y1
           [y3 = -------------------------------------------------]
                 cos(g3) y2 - cos(g3) y1 - sin(g3) x2 + sin(g3) x1
 



                           cos(g3) x1 y2 - cos(g3) x2 y1
           [x3 = -------------------------------------------------]
                 cos(g3) y2 - cos(g3) y1 - sin(g3) x2 + sin(g3) x1




                                   x2 y1 - y2 x1
           [d  = -------------------------------------------------]
                 sin(g3) x2 - cos(g3) y2 - sin(g3) x1 + cos(g3) y1