Ellipse Parallel Curves

Author

Eric W. Weisstein
June 27, 2013

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

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

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

Equation

par=FullSimplify[ParallelCurve[{a Cos[t],b Sin[t]},t,k],a>0&&b>0]

GraphicsRow[Function[{a,b},ParallelCurvesPlot[{a Cos[t],b Sin[t]},{-1.5,1,.2},{t,0,2Pi}]]@@@{{1,.5},{1,.9}}]

Show[With[{a=1,b=.5},Table[ParametricPlot[{Cos[t] (a+((b k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]))),Sin[t] (b+((a k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)])))},{t,0,2π},PlotStyle->Hue[(k+1.2)],Ticks->None],{k,-1.2,-.2,.1}]
],PlotRange->All]

Equation

eqn=Together[{((((x/(Cos[t]))- a))^2)-(((((b k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]))))^2),((((y/(Sin[t]))-b))^2)-(((((a k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]))))^2)}/.{Cos[t]->cost,Sin[t]->sint,Sec[t]->1/cost,Csc[t]->1/sint}]//Numerator

(gb=Factor/@GroebnerBasis[Append[eqn,sint^2+cost^2-1],{x,y},{cost,sint},MonomialOrder->EliminationOrder])//Timing

gb1= ((a^8) (b^4)-2 (a^6) (b^6)+(a^4) (b^8)-2 (a^8) (b^2) (k^2)+2 (a^6) (b^4) (k^2)+2 (a^4) (b^6) (k^2)-2 (a^2) (b^8) (k^2)+(a^8) (k^4)+2 (a^6) (b^2) (k^4)-6 (a^4) (b^4) (k^4)+2 (a^2) (b^6) (k^4)+(b^8) (k^4)-2 (a^6) (k^6)+2 (a^4) (b^2) (k^6)+2 (a^2) (b^4) (k^6)-2 (b^6) (k^6)+(a^4) (k^8)-2 (a^2) (b^2) (k^8)+(b^4) (k^8)-4 (a^6) (b^4) (x^2)+6 (a^4) (b^6) (x^2)-2 (a^2) (b^8) (x^2)+6 (a^6) (b^2) (k^2) (x^2)-8 (a^4) (b^4) (k^2) (x^2)+4 (a^2) (b^6) (k^2) (x^2)-2 (b^8) (k^2) (x^2)-2 (a^6) (k^4) (x^2)+4 (a^4) (b^2) (k^4) (x^2)-8 (a^2) (b^4) (k^4) (x^2)+6 (b^6) (k^4) (x^2)-2 (a^4) (k^6) (x^2)+6 (a^2) (b^2) (k^6) (x^2)-4 (b^4) (k^6) (x^2)+6 (a^4) (b^4) (x^4)-6 (a^2) (b^6) (x^4)+(b^8) (x^4)-6 (a^4) (b^2) (k^2) (x^4)+10 (a^2) (b^4) (k^2) (x^4)-6 (b^6) (k^2) (x^4)+(a^4) (k^4) (x^4)-6 (a^2) (b^2) (k^4) (x^4)+6 (b^4) (k^4) (x^4)-4 (a^2) (b^4) (x^6)+2 (b^6) (x^6)+2 (a^2) (b^2) (k^2) (x^6)-4 (b^4) (k^2) (x^6)+(b^4) (x^8)-2 (a^8) (b^2) (y^2)+6 (a^6) (b^4) (y^2)-4 (a^4) (b^6) (y^2)-2 (a^8) (k^2) (y^2)+4 (a^6) (b^2) (k^2) (y^2)-8 (a^4) (b^4) (k^2) (y^2)+6 (a^2) (b^6) (k^2) (y^2)+6 (a^6) (k^4) (y^2)-8 (a^4) (b^2) (k^4) (y^2)+4 (a^2) (b^4) (k^4) (y^2)-2 (b^6) (k^4) (y^2)-4 (a^4) (k^6) (y^2)+6 (a^2) (b^2) (k^6) (y^2)-2 (b^4) (k^6) (y^2)+6 (a^6) (b^2) (x^2) (y^2)-10 (a^4) (b^4) (x^2) (y^2)+6 (a^2) (b^6) (x^2) (y^2)+4 (a^6) (k^2) (x^2) (y^2)-6 (a^4) (b^2) (k^2) (x^2) (y^2)-6 (a^2) (b^4) (k^2) (x^2) (y^2)+4 (b^6) (k^2) (x^2) (y^2)+6 (a^4) (k^4) (x^2) (y^2)-10 (a^2) (b^2) (k^4) (x^2) (y^2)+6 (b^4) (k^4) (x^2) (y^2)-6 (a^4) (b^2) (x^4) (y^2)+2 (a^2) (b^4) (x^4) (y^2)-2 (b^6) (x^4) (y^2)-2 (a^4) (k^2) (x^4) (y^2)+2 (a^2) (b^2) (k^2) (x^4) (y^2)-6 (b^4) (k^2) (x^4) (y^2)+2 (a^2) (b^2) (x^6) (y^2)+2 (b^4) (x^6) (y^2)+(a^8) (y^4)-6 (a^6) (b^2) (y^4)+6 (a^4) (b^4) (y^4)-6 (a^6) (k^2) (y^4)+10 (a^4) (b^2) (k^2) (y^4)-6 (a^2) (b^4) (k^2) (y^4)+6 (a^4) (k^4) (y^4)-6 (a^2) (b^2) (k^4) (y^4)+(b^4) (k^4) (y^4)-2 (a^6) (x^2) (y^4)+2 (a^4) (b^2) (x^2) (y^4)-6 (a^2) (b^4) (x^2) (y^4)-6 (a^4) (k^2) (x^2) (y^4)+2 (a^2) (b^2) (k^2) (x^2) (y^4)-2 (b^4) (k^2) (x^2) (y^4)+(a^4) (x^4) (y^4)+4 (a^2) (b^2) (x^4) (y^4)+(b^4) (x^4) (y^4)+2 (a^6) (y^6)-4 (a^4) (b^2) (y^6)-4 (a^4) (k^2) (y^6)+2 (a^2) (b^2) (k^2) (y^6)+2 (a^4) (x^2) (y^6)+2 (a^2) (b^2) (x^2) (y^6)+(a^4) (y^8));
gb2=((a^12) (b^4)+4 (a^10) (b^6)+6 (a^8) (b^8)+4 (a^6) (b^10)+(a^4) (b^12)-2 (a^12) (b^2) (k^2)-10 (a^10) (b^4) (k^2)-20 (a^8) (b^6) (k^2)-20 (a^6) (b^8) (k^2)-10 (a^4) (b^10) (k^2)-2 (a^2) (b^12) (k^2)+(a^12) (k^4)+8 (a^10) (b^2) (k^4)+23 (a^8) (b^4) (k^4)+32 (a^6) (b^6) (k^4)+23 (a^4) (b^8) (k^4)+8 (a^2) (b^10) (k^4)+(b^12) (k^4)-2 (a^10) (k^6)-10 (a^8) (b^2) (k^6)-20 (a^6) (b^4) (k^6)-20 (a^4) (b^6) (k^6)-10 (a^2) (b^8) (k^6)-2 (b^10) (k^6)+(a^8) (k^8)+4 (a^6) (b^2) (k^8)+6 (a^4) (b^4) (k^8)+4 (a^2) (b^6) (k^8)+(b^8) (k^8)-4 (a^10) (b^4) (x^2)-10 (a^8) (b^6) (x^2)-10 (a^6) (b^8) (x^2)-6 (a^4) (b^10) (x^2)-2 (a^2) (b^12) (x^2)+6 (a^10) (b^2) (k^2) (x^2)+20 (a^8) (b^4) (k^2) (x^2)+18 (a^6) (b^6) (k^2) (x^2)-2 (a^4) (b^8) (k^2) (x^2)-8 (a^2) (b^10) (k^2) (x^2)-2 (b^12) (k^2) (x^2)-2 (a^10) (k^4) (x^2)-8 (a^8) (b^2) (k^4) (x^2)-2 (a^6) (b^4) (k^4) (x^2)+18 (a^4) (b^6) (k^4) (x^2)+20 (a^2) (b^8) (k^4) (x^2)+6 (b^10) (k^4) (x^2)-2 (a^8) (k^6) (x^2)-6 (a^6) (b^2) (k^6) (x^2)-10 (a^4) (b^4) (k^6) (x^2)-10 (a^2) (b^6) (k^6) (x^2)-4 (b^8) (k^6) (x^2)+6 (a^8) (b^4) (x^4)+6 (a^6) (b^6) (x^4)+3 (a^4) (b^8) (x^4)+(b^12) (x^4)-6 (a^8) (b^2) (k^2) (x^4)-10 (a^6) (b^4) (k^2) (x^4)-10 (a^2) (b^8) (k^2) (x^4)-6 (b^10) (k^2) (x^4)+(a^8) (k^4) (x^4)+3 (a^4) (b^4) (k^4) (x^4)+6 (a^2) (b^6) (k^4) (x^4)+6 (b^8) (k^4) (x^4)-4 (a^6) (b^4) (x^6)+2 (a^4) (b^6) (x^6)+2 (b^10) (x^6)+2 (a^6) (b^2) (k^2) (x^6)+2 (a^2) (b^6) (k^2) (x^6)-4 (b^8) (k^2) (x^6)+(a^4) (b^4) (x^8)-2 (a^2) (b^6) (x^8)+(b^8) (x^8)-2 (a^12) (b^2) (y^2)-6 (a^10) (b^4) (y^2)-10 (a^8) (b^6) (y^2)-10 (a^6) (b^8) (y^2)-4 (a^4) (b^10) (y^2)-2 (a^12) (k^2) (y^2)-8 (a^10) (b^2) (k^2) (y^2)-2 (a^8) (b^4) (k^2) (y^2)+18 (a^6) (b^6) (k^2) (y^2)+20 (a^4) (b^8) (k^2) (y^2)+6 (a^2) (b^10) (k^2) (y^2)+6 (a^10) (k^4) (y^2)+20 (a^8) (b^2) (k^4) (y^2)+18 (a^6) (b^4) (k^4) (y^2)-2 (a^4) (b^6) (k^4) (y^2)-8 (a^2) (b^8) (k^4) (y^2)-2 (b^10) (k^4) (y^2)-4 (a^8) (k^6) (y^2)-10 (a^6) (b^2) (k^6) (y^2)-10 (a^4) (b^4) (k^6) (y^2)-6 (a^2) (b^6) (k^6) (y^2)-2 (b^8) (k^6) (y^2)+6 (a^10) (b^2) (x^2) (y^2)+10 (a^8) (b^4) (x^2) (y^2)+10 (a^4) (b^8) (x^2) (y^2)+6 (a^2) (b^10) (x^2) (y^2)+4 (a^10) (k^2) (x^2) (y^2)+10 (a^8) (b^2) (k^2) (x^2) (y^2)+82 (a^6) (b^4) (k^2) (x^2) (y^2)+82 (a^4) (b^6) (k^2) (x^2) (y^2)+10 (a^2) (b^8) (k^2) (x^2) (y^2)+4 (b^10) (k^2) (x^2) (y^2)+6 (a^8) (k^4) (x^2) (y^2)+10 (a^6) (b^2) (k^4) (x^2) (y^2)+10 (a^2) (b^6) (k^4) (x^2) (y^2)+6 (b^8) (k^4) (x^2) (y^2)-6 (a^8) (b^2) (x^4) (y^2)-2 (a^6) (b^4) (x^4) (y^2)+12 (a^4) (b^6) (x^4) (y^2)-2 (a^2) (b^8) (x^4) (y^2)-2 (b^10) (x^4) (y^2)-2 (a^8) (k^2) (x^4) (y^2)-2 (a^6) (b^2) (k^2) (x^4) (y^2)+12 (a^4) (b^4) (k^2) (x^4) (y^2)-2 (a^2) (b^6) (k^2) (x^4) (y^2)-6 (b^8) (k^2) (x^4) (y^2)+2 (a^6) (b^2) (x^6) (y^2)-2 (a^4) (b^4) (x^6) (y^2)-2 (a^2) (b^6) (x^6) (y^2)+2 (b^8) (x^6) (y^2)+(a^12) (y^4)+3 (a^8) (b^4) (y^4)+6 (a^6) (b^6) (y^4)+6 (a^4) (b^8) (y^4)-6 (a^10) (k^2) (y^4)-10 (a^8) (b^2) (k^2) (y^4)-10 (a^4) (b^6) (k^2) (y^4)-6 (a^2) (b^8) (k^2) (y^4)+6 (a^8) (k^4) (y^4)+6 (a^6) (b^2) (k^4) (y^4)+3 (a^4) (b^4) (k^4) (y^4)+(b^8) (k^4) (y^4)-2 (a^10) (x^2) (y^4)-2 (a^8) (b^2) (x^2) (y^4)+12 (a^6) (b^4) (x^2) (y^4)-2 (a^4) (b^6) (x^2) (y^4)-6 (a^2) (b^8) (x^2) (y^4)-6 (a^8) (k^2) (x^2) (y^4)-2 (a^6) (b^2) (k^2) (x^2) (y^4)+12 (a^4) (b^4) (k^2) (x^2) (y^4)-2 (a^2) (b^6) (k^2) (x^2) (y^4)-2 (b^8) (k^2) (x^2) (y^4)+(a^8) (x^4) (y^4)+2 (a^6) (b^2) (x^4) (y^4)-6 (a^4) (b^4) (x^4) (y^4)+2 (a^2) (b^6) (x^4) (y^4)+(b^8) (x^4) (y^4)+2 (a^10) (y^6)+2 (a^6) (b^4) (y^6)-4 (a^4) (b^6) (y^6)-4 (a^8) (k^2) (y^6)+2 (a^6) (b^2) (k^2) (y^6)+2 (a^2) (b^6) (k^2) (y^6)+2 (a^8) (x^2) (y^6)-2 (a^6) (b^2) (x^2) (y^6)-2 (a^4) (b^4) (x^2) (y^6)+2 (a^2) (b^6) (x^2) (y^6)+(a^8) (y^8)-2 (a^6) (b^2) (y^8)+(a^4) (b^4) (y^8));

Block[{a=1,b=.5,k=-.5},ContourPlot[gb1==0 ,{x,-2,2},{y,-2,2},Evaluated->True]]

Manipulate[ContourPlot[((a^8) (b^4)-2 (a^6) (b^6)+(a^4) (b^8)-2 (a^8) (b^2) (k^2)+2 (a^6) (b^4) (k^2)+2 (a^4) (b^6) (k^2)-2 (a^2) (b^8) (k^2)+(a^8) (k^4)+2 (a^6) (b^2) (k^4)-6 (a^4) (b^4) (k^4)+2 (a^2) (b^6) (k^4)+(b^8) (k^4)-2 (a^6) (k^6)+2 (a^4) (b^2) (k^6)+2 (a^2) (b^4) (k^6)-2 (b^6) (k^6)+(a^4) (k^8)-2 (a^2) (b^2) (k^8)+(b^4) (k^8)-4 (a^6) (b^4) (x^2)+6 (a^4) (b^6) (x^2)-2 (a^2) (b^8) (x^2)+6 (a^6) (b^2) (k^2) (x^2)-8 (a^4) (b^4) (k^2) (x^2)+4 (a^2) (b^6) (k^2) (x^2)-2 (b^8) (k^2) (x^2)-2 (a^6) (k^4) (x^2)+4 (a^4) (b^2) (k^4) (x^2)-8 (a^2) (b^4) (k^4) (x^2)+6 (b^6) (k^4) (x^2)-2 (a^4) (k^6) (x^2)+6 (a^2) (b^2) (k^6) (x^2)-4 (b^4) (k^6) (x^2)+6 (a^4) (b^4) (x^4)-6 (a^2) (b^6) (x^4)+(b^8) (x^4)-6 (a^4) (b^2) (k^2) (x^4)+10 (a^2) (b^4) (k^2) (x^4)-6 (b^6) (k^2) (x^4)+(a^4) (k^4) (x^4)-6 (a^2) (b^2) (k^4) (x^4)+6 (b^4) (k^4) (x^4)-4 (a^2) (b^4) (x^6)+2 (b^6) (x^6)+2 (a^2) (b^2) (k^2) (x^6)-4 (b^4) (k^2) (x^6)+(b^4) (x^8)-2 (a^8) (b^2) (y^2)+6 (a^6) (b^4) (y^2)-4 (a^4) (b^6) (y^2)-2 (a^8) (k^2) (y^2)+4 (a^6) (b^2) (k^2) (y^2)-8 (a^4) (b^4) (k^2) (y^2)+6 (a^2) (b^6) (k^2) (y^2)+6 (a^6) (k^4) (y^2)-8 (a^4) (b^2) (k^4) (y^2)+4 (a^2) (b^4) (k^4) (y^2)-2 (b^6) (k^4) (y^2)-4 (a^4) (k^6) (y^2)+6 (a^2) (b^2) (k^6) (y^2)-2 (b^4) (k^6) (y^2)+6 (a^6) (b^2) (x^2) (y^2)-10 (a^4) (b^4) (x^2) (y^2)+6 (a^2) (b^6) (x^2) (y^2)+4 (a^6) (k^2) (x^2) (y^2)-6 (a^4) (b^2) (k^2) (x^2) (y^2)-6 (a^2) (b^4) (k^2) (x^2) (y^2)+4 (b^6) (k^2) (x^2) (y^2)+6 (a^4) (k^4) (x^2) (y^2)-10 (a^2) (b^2) (k^4) (x^2) (y^2)+6 (b^4) (k^4) (x^2) (y^2)-6 (a^4) (b^2) (x^4) (y^2)+2 (a^2) (b^4) (x^4) (y^2)-2 (b^6) (x^4) (y^2)-2 (a^4) (k^2) (x^4) (y^2)+2 (a^2) (b^2) (k^2) (x^4) (y^2)-6 (b^4) (k^2) (x^4) (y^2)+2 (a^2) (b^2) (x^6) (y^2)+2 (b^4) (x^6) (y^2)+(a^8) (y^4)-6 (a^6) (b^2) (y^4)+6 (a^4) (b^4) (y^4)-6 (a^6) (k^2) (y^4)+10 (a^4) (b^2) (k^2) (y^4)-6 (a^2) (b^4) (k^2) (y^4)+6 (a^4) (k^4) (y^4)-6 (a^2) (b^2) (k^4) (y^4)+(b^4) (k^4) (y^4)-2 (a^6) (x^2) (y^4)+2 (a^4) (b^2) (x^2) (y^4)-6 (a^2) (b^4) (x^2) (y^4)-6 (a^4) (k^2) (x^2) (y^4)+2 (a^2) (b^2) (k^2) (x^2) (y^4)-2 (b^4) (k^2) (x^2) (y^4)+(a^4) (x^4) (y^4)+4 (a^2) (b^2) (x^4) (y^4)+(b^4) (x^4) (y^4)+2 (a^6) (y^6)-4 (a^4) (b^2) (y^6)-4 (a^4) (k^2) (y^6)+2 (a^2) (b^2) (k^2) (y^6)+2 (a^4) (x^2) (y^6)+2 (a^2) (b^2) (x^2) (y^6)+(a^4) (y^8))==0 ,{x,-2,2},{y,-2,2},PlotRange->Full,AspectRatio->Automatic],{{a,1},0,1},{{b,.5},0,1},{{k,0},-10,10}]

AlgebraicDegree[gb1,{x,y}]

Block[{a=1,b=.5,k=.5},ContourPlot[gb2==0 ,{x,-2,2},{y,-2,2},Evaluated->True]]

FullSimplify[{((1-((Tan[(t/2)])^2))/(1+((Tan[(t/2)])^2))),((2Tan[(t/2)])/(1+((Tan[(t/2)])^2)))}]

eqn=Together[{((((x/(Cos[t]))- a))^2)-(((((b k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]))))^2),((((y/(Sin[t]))-b))^2)-(((((a k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]))))^2)}/.{Sec[t]->1/cost,Csc[t]->1/sint,Cos[t]->cost,Sin[t]->sint}/.{cost->(1-s^2)/(1+s^2),sint->2s/(1+s^2)}]//Numerator

(gb=Factor/@GroebnerBasis[eqn,{x,y},s,MonomialOrder->EliminationOrder])//Timing

Properties

ArcLength

Assuming[a>0&&b>0&&(k|t)∈Reals,ArcLength[par,{t,0,2π}]]

FullSimplify[(Sqrt[#.#])&[D[par,t]],a>0&&b>0&&(k|t)∈Reals]

(s=Integrate[Abs[1+((2 (Sqrt[2]) a b k)/((((a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]))^(3/2)))] (Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]),{t,0,2π},Assumptions->a>0&&b>0&&(k|t)∈Reals])//Timing

ArcLengthFunction

Assuming[a>0&&b>0&&(k|t)∈Reals,ArcLength[par,t]]

FullSimplify[(Sqrt[#.#])&[D[par,t]],a>0&&b>0&&(k|t)∈Reals]

(sf=Integrate[Abs[1+((2 (Sqrt[2]) a b k)/((((a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]))^(3/2)))] (Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]),t,Assumptions->a>0&&b>0&&(k|t)∈Reals&&0<t<2π])//Timing

Area

Area[{Cos[t] (a+((b k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)]))),Sin[t] (b+((a k)/(Sqrt[(b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)])))},{t,0,2π},Assumptions->a>0&&b>0&&k∈Reals]

Centroid

Centroid[par,{t,0,2π}]

Curvature

FullSimplify[Curvature[par,t],a>0&&b>0&&(k|t)∈Reals]

(κ=Curvature[((a^8) (b^4)-2 (a^6) (b^6)+(a^4) (b^8)-2 (a^8) (b^2) (k^2)+2 (a^6) (b^4) (k^2)+2 (a^4) (b^6) (k^2)-2 (a^2) (b^8) (k^2)+(a^8) (k^4)+2 (a^6) (b^2) (k^4)-6 (a^4) (b^4) (k^4)+2 (a^2) (b^6) (k^4)+(b^8) (k^4)-2 (a^6) (k^6)+2 (a^4) (b^2) (k^6)+2 (a^2) (b^4) (k^6)-2 (b^6) (k^6)+(a^4) (k^8)-2 (a^2) (b^2) (k^8)+(b^4) (k^8)-4 (a^6) (b^4) (x^2)+6 (a^4) (b^6) (x^2)-2 (a^2) (b^8) (x^2)+6 (a^6) (b^2) (k^2) (x^2)-8 (a^4) (b^4) (k^2) (x^2)+4 (a^2) (b^6) (k^2) (x^2)-2 (b^8) (k^2) (x^2)-2 (a^6) (k^4) (x^2)+4 (a^4) (b^2) (k^4) (x^2)-8 (a^2) (b^4) (k^4) (x^2)+6 (b^6) (k^4) (x^2)-2 (a^4) (k^6) (x^2)+6 (a^2) (b^2) (k^6) (x^2)-4 (b^4) (k^6) (x^2)+6 (a^4) (b^4) (x^4)-6 (a^2) (b^6) (x^4)+(b^8) (x^4)-6 (a^4) (b^2) (k^2) (x^4)+10 (a^2) (b^4) (k^2) (x^4)-6 (b^6) (k^2) (x^4)+(a^4) (k^4) (x^4)-6 (a^2) (b^2) (k^4) (x^4)+6 (b^4) (k^4) (x^4)-4 (a^2) (b^4) (x^6)+2 (b^6) (x^6)+2 (a^2) (b^2) (k^2) (x^6)-4 (b^4) (k^2) (x^6)+(b^4) (x^8)-2 (a^8) (b^2) (y^2)+6 (a^6) (b^4) (y^2)-4 (a^4) (b^6) (y^2)-2 (a^8) (k^2) (y^2)+4 (a^6) (b^2) (k^2) (y^2)-8 (a^4) (b^4) (k^2) (y^2)+6 (a^2) (b^6) (k^2) (y^2)+6 (a^6) (k^4) (y^2)-8 (a^4) (b^2) (k^4) (y^2)+4 (a^2) (b^4) (k^4) (y^2)-2 (b^6) (k^4) (y^2)-4 (a^4) (k^6) (y^2)+6 (a^2) (b^2) (k^6) (y^2)-2 (b^4) (k^6) (y^2)+6 (a^6) (b^2) (x^2) (y^2)-10 (a^4) (b^4) (x^2) (y^2)+6 (a^2) (b^6) (x^2) (y^2)+4 (a^6) (k^2) (x^2) (y^2)-6 (a^4) (b^2) (k^2) (x^2) (y^2)-6 (a^2) (b^4) (k^2) (x^2) (y^2)+4 (b^6) (k^2) (x^2) (y^2)+6 (a^4) (k^4) (x^2) (y^2)-10 (a^2) (b^2) (k^4) (x^2) (y^2)+6 (b^4) (k^4) (x^2) (y^2)-6 (a^4) (b^2) (x^4) (y^2)+2 (a^2) (b^4) (x^4) (y^2)-2 (b^6) (x^4) (y^2)-2 (a^4) (k^2) (x^4) (y^2)+2 (a^2) (b^2) (k^2) (x^4) (y^2)-6 (b^4) (k^2) (x^4) (y^2)+2 (a^2) (b^2) (x^6) (y^2)+2 (b^4) (x^6) (y^2)+(a^8) (y^4)-6 (a^6) (b^2) (y^4)+6 (a^4) (b^4) (y^4)-6 (a^6) (k^2) (y^4)+10 (a^4) (b^2) (k^2) (y^4)-6 (a^2) (b^4) (k^2) (y^4)+6 (a^4) (k^4) (y^4)-6 (a^2) (b^2) (k^4) (y^4)+(b^4) (k^4) (y^4)-2 (a^6) (x^2) (y^4)+2 (a^4) (b^2) (x^2) (y^4)-6 (a^2) (b^4) (x^2) (y^4)-6 (a^4) (k^2) (x^2) (y^4)+2 (a^2) (b^2) (k^2) (x^2) (y^4)-2 (b^4) (k^2) (x^2) (y^4)+(a^4) (x^4) (y^4)+4 (a^2) (b^2) (x^4) (y^4)+(b^4) (x^4) (y^4)+2 (a^6) (y^6)-4 (a^4) (b^2) (y^6)-4 (a^4) (k^2) (y^6)+2 (a^2) (b^2) (k^2) (y^6)+2 (a^4) (x^2) (y^6)+2 (a^2) (b^2) (x^2) (y^6)+(a^4) (y^8)),{x,y}])//Timing

(κ2=Assuming[a>0&&b>0&&(k|t)∈Reals,Simplify@κ])//Timing

ByteCount/@κ2

(κ3=Assuming[a>0&&b>0&&(k|t)∈Reals,FullSimplify[κ2[[1]]]])//Timing

Slope

Slope[{Cos[t] (a+(b k)/Sqrt[b^2 Cos[t]^2+a^2 Sin[t]^2]),Sin[t] (b+(a k)/Sqrt[b^2 Cos[t]^2+a^2 Sin[t]^2])},t]//FullSimplify

SpecialAffineCurvature

(sac=Assuming[a>0&&b>0&&(k|t)∈Reals,Simplify@SpecialAffineCurvature[par,t]])//Timing

Assuming[a>0&&b>0&&(k|t)∈Reals,FullSimplify[sac]]//Timing

TangentialAngle

FullSimplify[TangentialAngle[par,t],a>0&&b>0&&(k|t)∈Reals]

Vector Properties

parm={Cos[t] (a+(b k)/Sqrt[b^2 Cos[t]^2+a^2 Sin[t]^2]),Sin[t] (b+(a k)/Sqrt[b^2 Cos[t]^2+a^2 Sin[t]^2])};
assum=a>0&&b>0&&0<t<2π&&k∈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]]