Concho-Spiral

Author

Eric W. Weisstein
April 8, 2021

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpaceCurves/Concho-Spiral.nb.

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

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

Plot

With[{μ=1.08,a=1,c=1},ParametricPlot3D[{μ^u a Cos[u],μ^u a Sin[u],μ^u c},{u,0,30},PlotStyle->Red,Boxed->True,Axes->False,ViewPoint->{2,0,.5}]]

Function[{μ,a,c},ParametricPlot3D[{μ^u a Cos[u],μ^u a Sin[u],μ^u c},{u,0,30},PlotStyle->Red,Boxed->True,Axes->False]]@@@{{1,2,3},{1.05,1,1},{1.05,.5,1},{1.05,.5,2}}

Properties

curve={(μ^t) a Cos[t],(μ^t) a Sin[t],(μ^t) c};

With[{x=(μ^t) a Cos[t],y=(μ^t) a Sin[t],z=(μ^t) c},(((x^2)+(y^2))/(a^2))==((z^2)/(c^2))]//Simplify

FullSimplify[Eliminate[{x==(μ^t) a Cos[t],y==(μ^t) a Sin[t],z==(μ^t) c},t],c>0&&a>0&&μ>0&&(x|y|z)∈Reals]

ArcLengthFunction

ArcLength[curve,{t,0,t0},Assumptions->a>0&&c>0&&t>0&&μ>0]/.t0->t

Simplify[ArcLength[curve,u,Assumptions->a>0&&c>0&&u>0&&μ>0],μ>0&&u>0]

ArcLength[curve,u,Assumptions->a>0&&c>0&&u>0&&0<μ<1]

ArcLength[curve,u,Assumptions->a>0&&c>0&&u>0&&μ>1]

BinormalVector

FullSimplify[BinormalVector[curve,t],μ>0&&a>0&&c>0&&t∈Reals]

Curvature

FullSimplify[Curvature[curve,t],μ>0&&a>0&&c>0&&t∈Reals]

NormalVector

FullSimplify[NormalVector[curve,t],μ>0&&a>0&&c>0&&t∈Reals]

TangentVector

FullSimplify[TangentVector[curve,t],μ>0&&a>0&&c>0&&t∈Reals]

Torsion

FullSimplify[Torsion[curve,t],μ>0&&a>0&&c>0]

TotalCurvature

FullSimplify[TotalCurvature[curve,t],μ>0&&a>0&&c>0&&t∈Reals]

VectorLength

FullSimplify[VectorLength[curve,t],μ>0&&a>0&&c>0&&t∈Reals]