What is the equation for helical path length around a torus?

  • Thread starter Thread starter BasicWill
  • Start date Start date
  • Tags Tags
    Orbits
AI Thread Summary
The discussion seeks an analytical expression for the helical path length around a torus, emphasizing the need for precision rather than approximation. It suggests starting with a parameterization of the torus, defined by a cross-section radius r and a central circle radius R greater than r. Participants are advised to differentiate this parameterization to find tangent vectors and create a vector field with undetermined coefficients. The conversation hints at considering specific "pitch" for the helical path and finding integral curves based on this setup. Overall, the focus is on deriving a mathematical formula for the helical path length on a toroidal surface.
BasicWill
Messages
2
Reaction score
0
Does anyone know the equation for the helical path length around a torus?
I need an analytical expression, not an approximation.

Thanks
 
Mathematics news on Phys.org
Hint: write a parameterization of your surface (torus of cross-section radius r centered on a circle with radius R > r). Find the tangent vectors to this surface by differentiating wrt the two parameters of your parameterization. Consider a linear combination with undetermined coefficients to obtain a vector field lying in the torus. (Did you have a particular "pitch" in mind?) Find the integral curves.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
View full post »
Thread 'Unit Circle Double Angle Derivations'

Thread 'Unit Circle Double Angle Derivations'

Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
View full post »
Thread 'Imaginary Pythagoras'

Thread 'Imaginary Pythagoras'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

Parameters of helix approximated by parts of torus
Replies
1
Views
2K
Regarding the complex solutions of a torus
Replies
5
Views
2K
I Find Two Equal Length Paths in a Tree Graph
Replies
4
Views
1K
Valid Shapes of DNA: Is A Reasonable?
Replies
12
Views
2K
Equator of a Planet with Four Spatial Dimensions is a 4D Torus
Replies
9
Views
4K
I Understanding Spinor's Helicity
Replies
3
Views
2K
± sqrt(b^2 - a^2) = ± b ± ai ?
Replies
6
Views
3K
I Const Curvature Scalar & 3-Torus: Is It Maximally Symmetric?
Replies
12
Views
3K
Heat Transfer Coefficient of a circular water duct of rectangular cross section
Replies
3
Views
2K
B How Does Group Orbit Theory Relate Torus and Cylinder Structures?
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top