I think we are not getting the whole problem here.
The intersection of two planes is either:
- a plane (identical to both planes)
- a line
- the null set (if they are parallel)
The line is the only really interesting case here.
A helical coil is going to need more information than that to specify it, like the rate of the rise. I am thinking of your helical coil as a spring.
The intersection of a 'spring' and a line is going to be
- zero points
- one point
- two points
- infinite points
I think you are going to need to specify more about this problem before we can really help beyond this.