Trott Curve

Author

Eric W. Weisstein
April 22, 2013

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

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

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

Definition

eqn=12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81;

Plot

V5

<<Graphics`ImplicitPlot`

ImplicitPlot[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81==0,{x,-10,10},PlotPoints->50,Ticks->None,PlotStyle->Red]

V6

ContourPlot[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81==0,{x,-1,1},{y,-1,1},ContourStyle->Red,ImageSize->200]

Tangents

Boundary

Maximize[{x,eqn==0},{x,y}]

Maximize[{y,eqn==0&&0<y<x},{x,y}]//FullSimplify

Properties

Area

Numerical

NIntegrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,-∞,∞},{x,-∞,∞}]//Timing

8NIntegrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,0,1},{x,0,y}]

8Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,0,1},{x,0,y}]//Timing

8Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x,0,1},{y,0,x}]//Timing

V5

<<Graphics`InequalityGraphics`

InequalityPlot[eqn<0,{x},{y}]

<<Calculus`Integration`

NIntegrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x},{y}]

V6

Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,-∞,∞},{x,-∞,∞}]//Timing

Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x,-∞,∞},{y,-∞,∞}]//Timing

V8

Area[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0,{y,x}]//N//Chop

Area[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0,{x,y}]//N//Chop

Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x,-∞,∞},{y,-∞,∞}]//Timing

8Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,0,1},{x,0,y}]//Timing

N[%]

FunctionExpand[-Root[81-225 (y^2)+144 (y^4)+(-225+350 (y^2)) ((#1)^2)+144 ((#1)^4)&,3]+Root[81-225 (y^2)+144 (y^4)+(-225+350 (y^2)) ((#1)^2)+144 ((#1)^4)&,4],Root[3969-27900 ((#1)^2)+39556 ((#1)^4)&,4]<y<3/4]

V9

8NIntegrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,0,1},{x,0,y}]//Timing

8Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,0,1},{x,0,y}]

8NIntegrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x,0,1},{y,0,x}]//Timing

8Integrate[Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x,0,1},{y,0,x}]//Timing

FunctionExpand[Root[81-225 (x^2)+144 (x^4)+(-225+350 (x^2)) ((#1)^2)+144 ((#1)^4)&,2],3/4<x<1]

FunctionExpand[Root[81-225 (y^2)+144 (y^4)+(-225+350 (y^2)) ((#1)^2)+144 ((#1)^4)&,2],3/4<y<1]

Plot[Root[81-225 (y^2)+144 (y^4)+(-225+350 (y^2)) ((#1)^2)+144 ((#1)^4)&,2],{y,3/4,1}]

FullSimplify[Solve[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81==0,y]]

Solve[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])==0,x]

Factor[3969-27900 (x^2)+39556 (x^4)]

Plot[{((Sqrt[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])])/(12 (Sqrt[2]))),((Sqrt[225-350 (x^2)+(Sqrt[3969-27900 (x^2)+39556 (x^4)])])/(12 (Sqrt[2])))},{x,-.5,.5},Epilog->{PointSize[.02],Point[{(5/2) (Sqrt[((3 (372-7 (Sqrt[806])))/19778)]),3 (Sqrt[((775+12 (Sqrt[806]))/19778)])}]}]

FullSimplify[D[((Sqrt[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])])/(12 (Sqrt[2]))),x]]

Solve[x (-175+((31 (225-638 (x^2)))/(Sqrt[3969-27900 (x^2)+39556 (x^4)])))==0,x]//FullSimplify

N[%]

FullSimplify[((Sqrt[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])])/(12 (Sqrt[2])))/.x->(5/2) (Sqrt[((3 (372-7 (Sqrt[806])))/19778)])]

(1/(3(Sqrt[2])))NIntegrate[(Sqrt[225-350 (x^2)+(Sqrt[3969-27900 (x^2)+39556 (x^4)])])-(Sqrt[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])]),{x,-3 Root[49-3100 ((#1)^2)+39556 ((#1)^4)&,3],3 Root[49-3100 ((#1)^2)+39556 ((#1)^4)&,3]}]

(2/(3(Sqrt[2])))NIntegrate[(Sqrt[225-350 (x^2)+(Sqrt[3969-27900 (x^2)+39556 (x^4)])])-(Sqrt[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])]),{x,0,3 Root[49-3100 ((#1)^2)+39556 ((#1)^4)&,3]},WorkingPrecision->30]

(2/(3(Sqrt[2])))Integrate[(Sqrt[225-350 (x^2)+(Sqrt[3969-27900 (x^2)+39556 (x^4)])])-(Sqrt[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])]),{x,0,3 Root[49-3100 ((#1)^2)+39556 ((#1)^4)&,3]}]

39556((((x^2)-(27900/(2 39556))))^2)+(3969-((27900^2)/(4 39556)))//Expand

-(3969-((27900^2)/(4 39556)))(((-39556)/(3969-((27900^2)/(4 39556))))((((x^2)-(27900/(2 39556))))^2)-1)//Expand

{-(3969-((27900^2)/(4 39556))),((-39556)/(3969-((27900^2)/(4 39556))))}//N

Solve[(Sqrt[((-39556)/(3969-((27900^2)/(4 39556))))])(xx-(27900/(2 39556)))==Cosh[t],xx]

N[%]

FullSimplify[(Sqrt[3969-27900 (x^2)+39556 (x^4)])/.x->(Sqrt[((9 (25 (Sqrt[806])+312 Cosh[t]))/(638 (Sqrt[806])))])]

FullSimplify[(Sqrt[225-350 (x^2)+(Sqrt[3969-27900 (x^2)+39556 (x^4)])])-(Sqrt[225-350 (x^2)-(Sqrt[3969-27900 (x^2)+39556 (x^4)])])/.x->(Sqrt[((9 (25 (Sqrt[806])+312 Cosh[t]))/(638 (Sqrt[806])))])]

D[(Sqrt[((9 (25 (Sqrt[806])+312 Cosh[t]))/(638 (Sqrt[806])))]),t]

Solve[0==((9 (25 (Sqrt[806])+312 Cosh[t]))/(638 (Sqrt[806]))),t]//FullSimplify

%//N

Solve[3 Root[49-3100 ((#1)^2)+39556 ((#1)^4)&,3]==((9 (25 (Sqrt[806])+312 Cosh[t]))/(638 (Sqrt[806]))),t]

%//N

(2/(3(Sqrt[2])))NIntegrate[6 (Sqrt[(3/9889)]) (-(Sqrt[9300-175 (Sqrt[806]) Cosh[t]-31 (Sqrt[8294]) (Sqrt[((Sinh[t])^2)])])+(Sqrt[9300-175 (Sqrt[806]) Cosh[t]+31 (Sqrt[8294]) (Sqrt[((Sinh[t])^2)])]))((18 ((((2/31)))^(1/4)) (13^(3/4)) Sinh[t])/((Sqrt[319]) (Sqrt[25 (Sqrt[806])+312 Cosh[t]]))),{t,ArcCosh[-((25 (Sqrt[(31/26)]))/12)],ArcCosh[-(1/36) (Sqrt[(31/26)]) (-75+638 Root[49-3100 ((#1)^2)+39556 ((#1)^4)&,3])]},WorkingPrecision->30]

0.6881590087844519510700860788897635185394570392798525764078`30.

{t,ArcCosh[-((25 (Sqrt[(31/26)]))/12)],ArcCosh[-(1/36) (Sqrt[(31/26)]) (-75+638 Root[49-3100 ((#1)^2)+39556 ((#1)^4)&,3])]}//N

FullSimplify[6 (Sqrt[(3/9889)]) (-(Sqrt[9300-175 (Sqrt[806]) Cosh[t]-31 (Sqrt[8294]) (Sqrt[((Sinh[t])^2)])])+(Sqrt[9300-175 (Sqrt[806]) Cosh[t]+31 (Sqrt[8294]) (Sqrt[((Sinh[t])^2)])]))((18 ((((2/31)))^(1/4)) (13^(3/4)) Sinh[t])/((Sqrt[319]) (Sqrt[25 (Sqrt[806])+312 Cosh[t]]))),t∈Reals]

Centroid

Numerical

NIntegrate[{x,y}Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x,-∞,∞},{y,-∞,∞}]//Timing

V8

Integrate[{x,y}Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[{x,y}Boole[12^2(x^4+y^4)-15^2(x^2+y^2)+350x^2y^2+81<0],{y,-∞,∞},{x,-∞,∞}]//Timing

Curvature

FullSimplify[ImplicitCurvature[eqn,{x,y}],(x|y)∈Reals]

SortBy[FullSimplify[Curvature[eqn,{x,y}],(x|y)∈Reals],ByteCount]

ByteCount/@%

Inflection Points

pts=InflectionPoints[eqn,{x,y}]//RootReduce

N[pts]

ContourPlot[eqn==0,{x,-2,2},{y,-2,2},Epilog->{Point[pts]}]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

FullSimplify[Slope[12^2 (x^4+y^4)-15^2 (x^2+y^2)+350 x^2 y^2+81==0,{x,y}]]

FullSimplify[Slope[12^2 (x^4+y^4)-15^2 (x^2+y^2)+350 x^2 y^2+81==0,{x,y}]]