Seiffert's Spherical Spiral

Author

Eric W. Weisstein
December 23, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpaceCurves/SeiffertsSphericalSpiral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SeiffertsSphericalSpiral.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

Inititalization

Plot

V5

With[{k=.5},ParametricPlot3D[{
JacobiSN[s,k^2] Cos[k s],JacobiSN[s,k^2] Sin[k s],
JacobiCN[s,k^2],
Red
},{s,0,100},Ticks->None,PlotPoints->500,ImageSize->300]]

V6

With[{k=.5},ParametricPlot3D[{
JacobiSN[s,k^2] Cos[k s],JacobiSN[s,k^2] Sin[k s],
JacobiCN[s,k^2]
},{s,0,100},Ticks->None,PlotStyle->Red,ImageSize->300]]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["sieffert.m",plot]

Equations

Parametric

curve=r{JacobiSN[t,k^2] Cos[k t],JacobiSN[t,k^2] Sin[k t],JacobiCN[t,k^2]};

Block[{k=.5,r=1},ParametricPlot3D[curve,{t,0,100}]]

Implicit

{x,y,z}=r{JacobiSN[t,k^2] Cos[k t],JacobiSN[t,k^2] Sin[k t],JacobiCN[t,k^2]}

FullSimplify[x^2+y^2+z^2]

FullSimplify[y/x==z]

Properties

curve=r{JacobiSN[t,k^2] Cos[k t],JacobiSN[t,k^2] Sin[k t],JacobiCN[t,k^2]};

FullSimplify[Norm[curve],k>0&&t>0&&r>0]

FullSimplify[%/.Abs[x_]^2:>x^2]

Arc Length Function

ArcLength[curve,t,Assumptions->k>0&&t>0]

Arc Length

Binormal Vector

(b=FullSimplify[BinormalVector[curve,t],k>0&&r>0&&t∈Reals])//Timing

FullSimplify[b/.Abs[x_]^2:>x^2,r>0&&k>0&&t∈Reals]//Timing

FullSimplify[b.b,r>0&&k>0&&t∈Reals]

Curvature

FullSimplify[Curvature[curve,t],r>0]

Normal Vector

(n=FullSimplify[NormalVector[curve,t],k>0&&r>0&&t∈Reals])//Timing

FullSimplify[n/.Abs[x_]^2:>x^2,r>0&&k>0&&t∈Reals]//Timing

FullSimplify[n.n,r>0&&k>0&&t∈Reals]

Tangent Vector

(tan=FullSimplify[TangentVector[curve,t],k>0&&r>0&&t∈Reals])//Timing

FullSimplify[tan/.Abs[x_]^2:>x^2,r>0&&k>0&&t∈Reals]//Timing

FullSimplify[tan.tan,r>0&&k>0&&t∈Reals]

Torsion

FullSimplify[Torsion[curve,t],k>0&&r>0]

TotalCurvature

FullSimplify[TotalCurvature[curve,t],k>0&&r>0]

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]