If the torus is centered around the z-axis then it can be written under the usual quartic form

(x² + y² + z² + R² - r²)² - 4R²(x² + y²) = 0

The straight line is defined using a point P(xo, yo, zo) and a vector V(a,b,c).  Thus, it can  can be written in parametric form

x(t) = xo + at

y(t) = yo + bt

z(t) = zo + ct

where t is a real variable.

Plugging the above parametric form in the quartic form, you will get a quartic equation involving the single variable t and the constants of the problems. Then, taking into account that t is real, you will get 0 to 4 solutions.

Hope this helps.

If you have the point and the vector of your line you can easily get the parameter form of it.

So I would guess, you just equal that to the parameter form for your torus and solve the resulting equations component like (just make sure that you are in the right coordinate system). - Wouldn't that do the trick?

If the torus is centered around the z-axis then it can be written under the usual quartic form

(x² + y² + z² + R² - r²)² - 4R²(x² + y²) = 0

The straight line is defined using a point P(xo, yo, zo) and a vector V(a,b,c).  Thus, it can  can be written in parametric form

x(t) = xo + at

y(t) = yo + bt

z(t) = zo + ct

where t is a real variable.

Plugging the above parametric form in the quartic form, you will get a quartic equation involving the single variable t and the constants of the problems. Then, taking into account that t is real, you will get 0 to 4 solutions.

Hope this helps.

I think you can use the formulas of Mr. Lehtihet  and mathematical software. I tried to solve this task using commercial software – Mathematica. It gives 4 solutions but they are very complex. I recommend you Wolfram Alpha http://www.wolframalpha.com/ and simplified calculation for fixed x0, y0, …, etc. I attached the exemplary solution for a, b, … =1. You can try other values of course.

Thanks a lot for your contributions.

Would it help to first consider intersection of a torus with a plane? The plane can be chosen to be parallel to one of the axes.