CONTENTS OF LAB03 Crv3Dff
A description of the files in LAB03 Crv3Dff
Lab03Crv3Dff
1. 3DCrvEX.nb
This Mathematica notebook which introduces the curvature and torsion of (regular) curves in three-space. It gives the Mathematica code for graphing curves in three space, for computing curvature and torsion. It computes and graphs the curvature and torsion functions for the "twisted cubic" curve, a "potato chip" curve, and a torusknot curve. For curves with parameterizations depending on extra parameters (besides the "t" variable), animations of the curves obtained by varying the extra parameterizations are given. Similar animations of the curvature and torsion functions are given.
We study TorusKnots on a background torus, with parameters determining the ‘rotation’ ellipse for the torus, and (p,q)-parameters determining the number of times the torusknots loop around and over the torus. The curvature and torsion for these curves are computed.
We vary parameters in the TorusKnot and ‘background’ Torus definitions, creating animations of these objects. Corresponding animations of the curvature and torsion functions of these curves are created.
Exercises compare the above types of curves, including an extended exercise in studying the behavior of the torusknot curve and its curvature and torsion when the parameters are varied. Students must describe, geometrically and analytically, the asymptotic behavior of the curvature and torsion functions as the varying parameter tends to infinity.
2. 3DCrvFrenet.nb
This Mathematica notebook generates animations the Frenet Frame (colored tanget = red, normal = blue, binormal = green) moving along a (circular) helix with a variety of different attributes, and along a torusknot. Mathematica generates an animation of a colored Frenet Frame traveling along the helix, an animation of the Frenet frame with a "trapezoid" in the { N , B } plane colored in, and an animation in which a tube-surface about the helix is generated side-by-side with the Frenet frame.
We study the Frenet frame and its behavior for a fixed TorusKnot on a background torus. The curvature and torsion for these curves are computed.
At certain points on the torusknot, the Frenet frame is observed to be ‘spinning’ dramatically.
In an extended exercise, this phenomenon is investigated more completely. Approximate solutions a system of differential equations (like the Frenet frame equations) and their relation to actual solutions are compared via Gronwall-type inequalities. Graphics are generated which show the torusknot Frenet frame animation and an animation of the "approximate" Frenet frame obtained by solving the torusknot Frenet frame equations, except that the curvature is assumed to be zero (thus simulating the behavior of the Frenet frame at points where curvature is small and torsion is large).
An additional exercise takes up the question of precisely what happens to the Frenet frame on a regular curve which has points where the acceleration vanishes. An example of Nomizu is given of such a curve with a zero of all orders greater than one, at one point, and for which no continuous Frenet frame can exist. Students explore the analysis and Mathematica graphics of this curve.
3. 3Dkt.nb
This Mathematica notebook contains a series of cells in which Mathematica graphs, using NDSolve, unit-speed space curves which have curvature and torsion functions, kappa[t] (assumed > 0) and tau[t] which one picks ahead of time. The question of the geometric interpretation (for the curve Mathematica generates by solving the Frenet frame equations) for the functions kappa[t] and tau[t] when kappa[t] is allowed to assume nonpositive values is explored in an extended exercise.
Trying to "predict" the types of curves one might get, or even trying to understand why the curve Mathematica generates does or does not seem to agree with one's understanding of curvature and torsion, makes for fascinating exploration. By making the curvature and/or torsion functions depend on an outside parameter and varying that parameter, one gets a family of space curves in this way. Running these curves in an animation can give powerful ideas about how changing curvature and/or torsion changes the shape of a given curve (or one might say how changing curves' curvature and torsion functions change). Also included are animations of the family of curvature or torsion functions alongside the animations of the space curves which have those functions as their curvature and/or torsion.
Exercises are included which allow students to explore more examples with Mathematica and to explain the observations they make.
In an extended exercise, the question of the geometric interpretation (for the curve Mathematica generates by solving the Frenet frame equations) for the functions kappa[t] and tau[t] when kappa[t] is allowed to assume nonpositive values is explored.
4. Contents of Lab03 Crv3Dff.doc
This document.