# Find intersection of two lines given subtended angle

I'm tinkering with a bit of graphics software.

I want to be able to nudge the value of a single angle in a polygon, then redraw the new polygon.

The intersection point of the two lines creating the angle will change, and their lengths will change, but I can't work out how to calculate the new coordinates.

All other intersections in the polygon will be unchanged as to position. (Obviously, the angles of the adjacent corners shall have to be modified as well as the corner that I explicitly "nudge.")

In summary:

Find the coordinates of the point of intersection of two lines, given a single coordinate pair for each line and the measure of the angle of intersection.

Geometrically I think there will be more than one possible coordinate pair for any angle other than $180$ degrees. If I can choose the coordinate pair nearest to the previous coordinate pair, I think that would give the result I seek. But, I'd be happy just to know what general approach to take.

This problem is not the same as finding intersection for lines where both slopes are known.

• That feel when your summary paragraph is just as long LOL – TorsionSquid Dec 14 '16 at 2:58
• The following should be added, straight lines in plane.... Regular polygon... You should pehaps say given points $P,Q$ co-ordinates, the means of dividing $2 \pi$ into an $n$ arbitrary number of parts. You can have $n= 2,4, 5,,6,8,12, 16,17, ..$ only, Right? – Narasimham Dec 14 '16 at 3:21

## 1 Answer

The point of intersection cannot be uniquely determined. This is because of the "angle in the same segment" theorem. See below. • A good point, but since he originally stated that no other intersection should change, I believe he intends minimal change to the angles of the adjacent corners and probably minimal change to the coordinates of the corner in question. So you might expand this. :) – Wildcard Dec 14 '16 at 4:08
• @Wildcard I think I have replied to the OP’s (newest) bold faced summary. Maybe a diagram will help to clarify the intention (especially about the polygon part). – Mick Dec 14 '16 at 4:30
• Thanks for helping to clarify my question. The re-wording exactly reflects my intended specification; that the single angle is "nudged" which changes the angles of two adjacent intersections, adjacent segment lengths, but no coordinate values other than the coordinates of the "nudged" angle. To explain a bit more, the user drags might drag a join to approximate a right-angle. But the angle might be shown to be 89.7. Hence the value of small increments. – Vince Stewart Dec 14 '16 at 6:44
• @Mick, @Wildcard; Thanks for the answer. I can now proceed to resolve the issue by adding an extra constraint. – Vince Stewart Dec 14 '16 at 7:07
• see follow up at:math.stackexchange.com/q/2058959/399067 – Vince Stewart Dec 15 '16 at 4:44