Given a Torus with major radius "R" and minor radius "r" which is centred at origin (0,0,0), and a straight line defined with a point "p(x,y,z)" and a vector "v(x,y,z)". I would like to know if there is a mathematical expression to compute the possible intersection points between them (when possible).
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.
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.
This is a good question.
In addition to the excellent answers already given, consider a tangent to a torus at a point where the boundary of a sphere and the boundary of a torus have a point in common (see the attached figure). This approach to the intersection of a line with a torus is based on
Ku-Jin Kim and Myung-Soo Kim, Torus/Sphere Intersection Based on a Configuration Space Approach:
http://3map.snu.ac.kr/mskim/ftp/tsi.pdf
See page 84 in the paper.
Another approach is to consider a hyperbolic torus with just one point missing (a tiny hole in the "donut"), the cusp in the torus. Then let a line puncture the torus by passing through the torus cusp. That line passing through the cusp will intersect the torus. For more about this, see
Mirzakhani’s Starting Point:
https://www.maths.ox.ac.uk/system/files/attachments/Caroline_series_slides.pdf
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