Neoid

Author

Eric W. Weisstein
June 6, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/Neoid.nb.

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Original notebook by Margherita Barile
June 21, 2002

Definition

neoid[a_,b_][t_]:=a t+b

Equation

With[{a=1/2,b=1},ContourPlot[x^2+y^2==a ArcTan[x,y]+b,{x,-3,3},{y,-3,3}]]

Plot

PolarPlot[t/2+1, {t,-4Pi,4Pi},PlotStyle->Red,Ticks->None,PlotLabel->r[θ]==θ/2+1,ImageSize->200]

PolarPlot[#1 t+#2, {t,-4Pi,4Pi},PlotStyle->Red,Ticks->None,ImageSize->200]&@@@{{1,1},{1,2},{2,1}}

Properties

curve=(a t+b) {Cos[t],Sin[t]};

Arc Length Function

v5

ArcLength[neoid,θ]//Timing

...ran for >8 hours with no result...

v6

FullSimplify[(Sqrt[(((a θ+b))^2)+((D[a θ+b,θ])^2)]),θ∈Reals&&a>0&&b>0]

ArcLength[neoid[a,b][t],t,Assumptions->t>0]//Timing

...crashes a local kernel when run in a FE...

ArcLength[neoid[a,b][t],t,Assumptions->a>0&&b>0&&t∈Reals]//Timing

FullSimplify[(1/(2 a))(-b (Sqrt[(a^2)+(b^2)])+b (Sqrt[(a^2)+(b^2)+2 a b t+(a^2) (t^2)])+a t (Sqrt[(a^2)+(b^2)+2 a b t+(a^2) (t^2)])-(a^2) Log[b+(Sqrt[(a^2)+(b^2)])]+(a^2) Log[b+a t+(Sqrt[(b^2)+2 a b t+(a^2) (1+(t^2))])]),a>0&&b>0&&t∈Reals]

s=%/.{a->1,b->1}

Plot[s,{t,-10,10}]

Curvature

Parametric

Curvature[neoid[a,b][t],t]

Polar

FullSimplify[Curvature[a q+b,q],(a|b|q)∈Reals]

Slope

Slope[(a t+b) {Cos[t],Sin[t]},t]//FullSimplify

Slope[a q+b,q]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0**b>0&&t∈Reals]

Tangential Angle

v6

TangentialAngle[neoid[a,b][t],t]

Vector Properties

parm=(a t+b) {Cos[t],Sin[t]};
assum=(a|b|t)∈Reals;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};

VectorLength

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

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]/.xform]]

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]/.xform]]