How do I detect intersections on roads that are made out of points?

So I'm working on a city building game again, and I have a few problems concerning roads. Road intersections. There is a method ive been looking up for making roads. I am making the straight roads out of two points, that then generates the geometry at the start and the end, and the cars lerp between the start and end points. For curved roads, the roads are part of an arc of the circle. Using basic geometry, I have found the length of the arc that is the road, and then divided the road by a number of segments equal to its length. This gives a pretty accurate, clean edged curve. Since the straight roads are made up of two nodes, and the curved roads are made up of multiple nodes, how do I create intersections? At the minute, I know how to draw roads, so long as none of them touch each other. But how would I go about snapping the roads to existing roads, or making intersections? I could make it snap to an existing roads node if it is within a certain radius of it, but then it wouldn't snap to straight roads (only a node at the start and end) and would snap to every metre along curved roads rather than anywhere on the road. Sorry that's not very well explained, but I'm not sure how I could go about this. Thanks for any help you can give me.