Pappus Spiral

Author

Eric W. Weisstein
April 12, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Plot

https://en.wikipedia.org/wiki/List_of_spirals

https://mathcurve.com/courbes3d.gb/spiraleconic/pappus.shtml

What is the meaning of α in the definition?

With[{a=1,α=.8},ParametricPlot3D[{a Sin[α]t Cos[t],a Sin[α]t Sin[t],a Cos[α]t},{t,0,30},PlotStyle->Red,Boxed->True,Axes->False,ViewPoint->{2,0,.5}]]

GraphicsRow[Function[{a,α},ParametricPlot3D[{a Sin[α]t Cos[t],a Sin[α]t Sin[t],a Cos[α]t},{t,0,30},PlotStyle->Red,Boxed->True,Axes->False]]@@@{{1,.8},{1,.5},{2,.2}}]

Properties

<<MathWorld`Curves`

curve={a Sin[α t]Cos[t],a Sin[α t]Sin[t],a Cos[α t]};

assum=a>0&&0<α<π/2&&t>0;

ArcLengthFunction

ArcLength[curve,{t,0,t0},Assumptions->assum]/.t0->t

BinormalVector

FullSimplify[BinormalVector[curve,t],assum]

Curvature

FullSimplify[Curvature[curve,t],assum]

NormalVector

FullSimplify[NormalVector[curve,t],assum]

TangentVector

FullSimplify[TangentVector[curve,t],assum]

Torsion

FullSimplify[Torsion[curve,t],assum]

TotalCurvature

TotalCurvature[curve,t]

FullSimplify[%,assum]

VectorLength

FullSimplify[VectorLength[curve,t],assum]