Conical Spiral

Author

Eric W. Weisstein
July 1, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Inititalization

Plot

spiral=With[{
h=1,
r=.8,
a=1.5
},
ParametricPlot3D[
{(h-z)/h  r Cos[a z],
(h-z)/h r Sin[a z],z},
{z,0,70},Boxed->True,Axes->False,
PlotStyle->Red,ViewPoint->{2,0,.5}]
]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["conicsp.m",spiral]

Equations

Parametric

curve={(h-z)/h  r Cos[a z],(h-z)/h r Sin[a z],z};

Block[{h=1,r=.8,a=1.5},ParametricPlot3D[curve,{z,0,20}]]

Natural

Solve[s==(a r t Sqrt[1+r^2 (1+a^2 t^2)]+(1+r^2) ArcSinh[(a r t)/Sqrt[1+r^2]])/(2 a r),t]

Implicit

Block[{h=1,r=.8,a=1.5},Show[{
ParametricPlot3D[curve,{z,0,10},Evaluated->True,PlotStyle->Directive[Red,Thickness[.02]]],

ContourPlot3D[x==(h-z)/h  r Cos[a z],{x,-10,10},{y,-10,10},{z,0,10},ContourStyle->Opacity[0.8],Mesh->None],

ContourPlot3D[y==(h-z)/h r Sin[a z],{x,-10,10},{y,-10,10},{z,0,10},ContourStyle->Opacity[0.8],Mesh->None]
}]]

Properties

parm={t r Cos[a t],t r Sin[a t],t};
assum=r>0&&a>0&&t>0;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n,Sqrt[x_^(n_/;Mod[n,2]==0)]:>Abs[x]^(n/2)};

ParametricPlot3D[curve/.{r->1,a->1},{t,0,20}]

Arc Length Function

FullSimplify[ArcLength[curve,t,Assumptions->a>0&&r>0&&t>0],a>0&&r>0&&t>0]

Binormal Vector

b=FullSimplify[BinormalVector[curve,t],a>0&&r>0&&t>0]

FullSimplify[b.b]

Curvature

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

Normal Vector

n=FullSimplify[NormalVector[curve,t],a>0&&r>0&&t>0]

FullSimplify[n.n]

Tangent Vector

tan=FullSimplify[TangentVector[curve,t],a>0&&r>0&&t>0]

FullSimplify[tan.tan]

Torsion

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

TotalCurvature

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

VectorLength

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

Properties: h

curve={(h-z)/h  r Cos[a z],
(h-z)/h r Sin[a z],z};

Arc Length

ArcLength[curve,z,Assumptions->h>0&&r>0&&a>0&&z>0]

Curvature

FullSimplify[Curvature[curve,z],h>0&&r>0&&a>0&&z>0]

Torsion

FullSimplify[Torsion[curve,z],h>0&&r>0&&a>0&&z>0]