How to make curved line into torus?

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SUMMARY

This discussion focuses on modifying the equations of a trefoil knot to create a toroidal shape. The original trefoil knot equations are defined as x = (2 + cos 3t)cos 2t, y = (2 + cos 3t)sin 2t, and z = sin 3t. To transform this into a torus, the equations x(u, v) = (R + r cos v) cos u, y(u, v) = (R + r cos v) sin u, and z(u,v) = r sin v are utilized, with parameters set to R=2 and r=1, and the substitutions (u,v) = (2t,3t) are proposed for the transformation.

PREREQUISITES
  • Understanding of parametric equations in 3D geometry
  • Familiarity with the mathematical concepts of knots and toroidal shapes
  • Knowledge of trigonometric functions and their applications in modeling
  • Basic skills in mathematical transformations and substitutions
NEXT STEPS
  • Explore the mathematical properties of toroidal shapes in 3D space
  • Learn about parametric surface equations and their applications
  • Investigate the visualization of knots and toroidal structures using software like Mathematica or MATLAB
  • Study the implications of varying R and r on the shape of the torus
USEFUL FOR

Mathematicians, 3D modelers, and anyone interested in geometric transformations and knot theory will benefit from this discussion.

CosmicVoyager
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Greetings,

I am trying to figure out how to modify the equations for a trefoil knot to make it toroidal.

A trefoil knot is:
x = (2 + cos 3t)cos 2t
y = (2 + cos 3t)sin 2t
z = sin 3t

A torus is:
x(u, v) = (R + r cos v) cos u
y(u, v) = (R + r cos v) sin u
z(u,v) = r sin v

where

u, v are in the interval [0, 2π),
R is the distance from the center of the tube to the center of the torus,
r is the radius of the tube.

Thanks
 
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A comparison of the equations suggests that we set R=2, r=1 & (u,v) =(2t,3t).
 

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