Sorry I made a mistake in my post by using the word "line" which implies a straight line. The proper word is "curve". Doh!

Thanks for the links!

I don't think am looking for volume since that would involve integration right?? I am just looking for a parametric or vector curve representation of the intersection **result**.

I thought perhaps you may be studying 3d graphics, because your questions reminded me of the topic "constructive solid geometry" which is a way of ray-tracing shapes. Representing a volume involves the use of inequalities, for example the volume of a sphere would be x^2+y^2+z^2 ≤ r^2. But this isn't your goal.

Using equations similar to yours...

\(\displaystyle \boldsymbol{s_{v1}} = r_1\cos(t_1)\boldsymbol{v_1} + r_1\sin(t_1)\boldsymbol{v_2} + s_1\boldsymbol{v_3}\)

\(\displaystyle \boldsymbol{s_{v2}} = r_2\cos(t_2)\boldsymbol{v_4} + r_2\sin(t_2)\boldsymbol{v_5} + s_2\boldsymbol{v_6}\)

The first thing to do is equate them...

\(\displaystyle \boldsymbol{s_{v1}} =\boldsymbol{s_{v2}} \)

\(\displaystyle r_1\cos(t_1)\boldsymbol{v_1} + r_1\sin(t_1)\boldsymbol{v_2} + s_1\boldsymbol{v_3} = r_2\cos(t_2)\boldsymbol{v_4} + r_2\sin(t_2)\boldsymbol{v_5} + s_2\boldsymbol{v_6}\)

Next split out the x,y,z components and you'll end up with three equations. The unknowns are \( t_1, t_2, s_1, s_2 \). Using the three equations it

*should* be possible to eliminate two of the unknowns and be left with an equation of the form:<something in terms of 2 unknowns> = 0. This "locks" the values of the two unknowns together onto a curve, and you can use these values to determine the eliminated unknowns by back substitution.

That's my suggestion for a "high level strategy", but the devil might be in the detail. I expect that a quadratic will show up somewhere, because it's possible to have two separated curves (imagine one cylinder has a smaller radius and it pierces the bigger cylinder in two separate locations). I guess you should aim to use the identity sin^2+cos^2=1. If you agree with this strategy then please give it a try and post your work.

I'm still a bit curious, are you doing this for interest? Do you have a computer controlled welder/ cutter?