Atom-Spiral

Author

Eric W. Weisstein
April 13, 2021

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/Atom-Spiral.nb.

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

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

Sources

Plots

Positive and negative θ

GraphicsGrid[Partition[Table[PolarPlot[(q/(q-a)),{q,-20,20},Exclusions->a,Ticks->None,PlotLabel->Row[{Style["a",Italic]," == ",a}],LabelStyle->Directive[Black,FontFamily->"Times"],PlotStyle->Red],{a,{0,1/16,1/4,1,4,8}}],UpTo[3]],Dividers->All]

Positive θ

Plot[(q/(q-1)),{q,-2,2}]

GraphicsGrid[Partition[Table[PolarPlot[(q/(q-a)),{q,a,20},Ticks->None,PlotLabel->Row[{Style["a",Italic]," == ",a}],LabelStyle->Directive[Black,FontFamily->"Times"],PlotStyle->Red],{a,{0,1/16,1/4,1,4,8}}],UpTo[3]],Dividers->All]

Properties

ArcLength

ArcLength[(q/(q-a)){Cos[q],Sin[q]},{q,0,q0}]

Sqrt[FullSimplify[#.#&@D[(q/(q-a)){Cos[q],Sin[q]},q],a>0]]

(1/(((a-q))^2))Assuming[0<a<b,Integrate[(Sqrt[-2 a (q^3)+(q^4)+(a^2) (1+(q^2))]),{q,0,b}]]

Curvature

FullSimplify[Curvature[(t/(t-a)){Cos[t],Sin[t]},t],0<a<t]

NormalVector

FullSimplify[NormalVector[(t/(t-a)){Cos[t],Sin[t]},t],0<a<t]

ParametricEquations

ParametricPlot[(q/(q-1)){Cos[q],Sin[q]},{q,1,20}]

PolarCurvature

FullSimplify[Curvature[(q/(q-a)),q],q>a]

PolarEquation

PolarPlot[(q/(q-1)),{q,1,20}]

PolarSlope

FullSimplify[Slope[(q/(q-a)),q],q>a]

PolarTangentialAngle

FullSimplify[TangentialAngle[(q/(q-a)),q],q>a]

Slope

FullSimplify[Slope[(t/(t-a)){Cos[t],Sin[t]},t],0<a<t]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[(t/(t-a)){Cos[t],Sin[t]},t],0<a<t]

TangentialAngle

TangentialAngle[(t/(t-a)){Cos[t],Sin[t]},t]

TangentVector

FullSimplify[TangentVector[(t/(t-a)){Cos[t],Sin[t]},t],0<a<t]

VectorLength

FullSimplify[VectorLength[(t/(t-a)){Cos[t],Sin[t]},t],0<a<t]