Intrinsic 3D Curves

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The French mathematicians Joseph Alfred Serret and Jean Frédéric Frenet found a way to represent a parametrized curve by intrinsic equations. At each point of the curve (parametrized by arclength ), three mutually perpendicular unit vectors are defined (called a TNB frame). The tangent shows the direction of motion of the point, the normal points toward the direction in which the curve bends, and the binormal is a vector perpendicular to both. Another two quantities are introduced: curvature to measure how quickly the curve is changing its direction, and torsion to measure how quickly the curve is leaving the TN plane. In this Demonstration, the functions (kappa, for curvature) and (tau, for torsion) can be adjusted using the parameters and and can be chosen to build a ribbon-like surface (in fact, a ruled surface) of a selected width and length, in discrete steps applied times. This is possible because the other edge has the same TNB frame, but displaced.

Contributed by: Enrique Zeleny (December 2014)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The Serret–Frenet equations that relate the three vectors with the curvature and torsion of a curve are

,

,

.

References

[1] R. Rucker. "How Flies Fly: Kappatau Space Curves." (Dec 5, 2014) www.cs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm.

[2] Wikipedia. "Frenet–Serret Formulas." (Dec 5, 2014) en.wikipedia.org/wiki/Frenet–Serret_formulas.



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