# Wolfram Function Repository

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Plot intrinsic curves in 3D

Contributed by:
Enrique Zeleny

ResourceFunction["Intrinsic3DCurve"][ τ,a,b,w,ds,n]plot an intrinsic curve with curvature with parameters τa and b and width w in discrete steps ds applied n times. |

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 arc length), 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.

The starting point is {0,0,0}.

Options are the same as Graphics3D.

A simple curve:

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Increasing width, we get a ribbon-like surface (in fact, a ruled surface—the other edge has a displaced TNB frame):

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With a constant function:

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A higher constant function:

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A function varying rapidly needs small discrete steps to plot smoothly:

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Avoid undefined values like log(0):

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Wolfram Language 11.3 (March 2018) or above

- 1.0.0 – 07 March 2019

This work is licensed under a Creative Commons Attribution 4.0 International License