Ochoa Curve

Author

Eric W. Weisstein
February 20, 2012

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

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

©2012 Wolfram Research, Inc. except for portions noted otherwise

Integer Solutions

ochoa=y^2==x^3-440067x+106074110;

Reduce[ochoa,{x,y},Integers]

(pts=Sort[{x,y}/.{ToRules[Reduce[ochoa&&Abs[x]<10^6&&Abs[y]<10^6,{x,y},Integers]]}])//Timing

Length[pts]

Union[pts,SameTest->(#1[[1]]==#2[[1]]&&Abs[#1[[2]]]==Abs[#2[[2]]]&)]

Length[%]

Developer`SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->10^4}];

(pts=Sort[{x,y}/.{ToRules[Reduce[ochoa&&Abs[x]<10^6&&Abs[y]<10^6,{x,y},Integers]]}])//Timing

Union[pts,SameTest->(#1[[1]]==#2[[1]]&&Abs[#1[[2]]]==Abs[#2[[2]]]&)]

Length[%]

solns=Sort[{{-761,504},{-745,4520},{-557,13356},{-446,14616},{-17,10656},{91,8172},{227,4228},{247,3528},{271,2592},{455,200},{499,3276},{523,4356},{530,4660},{599,7576},{751,14112},{1003,25956},{1862,75778},{3511,204552},{5287,381528},{23527,3607272},{64507,16382772},{100102,31670478},{1657891,2134685628}}]

Length[solns]

Select[solns,Max[Abs[#]]<10^6&]//Length

ochoa/.Thread[{x,y}->#]&/@solns

Plot

plot=ContourPlot[Evaluate[ochoa],{x,-1000,2000},{y,-10^5,10^5},PlotPoints->100,MaxRecursion->5,ContourStyle->Red,Epilog->{Point[Join@@({{1,-1}#,#}&/@solns)]},ImageSize->400,Axes->True,AxesLabel->{x,y}]

Properties

Area of Closed Portion

RegionPlot[Evaluate[Less@@ochoa],{x,-1000,400},{y,-10^5,10^5}]

NIntegrate[Boole[Less@@ochoa],{x,-1000,400},{y,-10^5,10^5}]

(i=Integrate[Boole[Less@@ochoa],{x,-1000,400},{y,-10^5,10^5}])//Timing

N[%]//ReleaseHold

NIntegrate[Boole[(y^2)<106074110-440067 x+(x^3)],{x,-1000,400},{y,-10^5,10^5}]

ReleaseHold[i]

i=-(4/5) (Identity[3 I (Sqrt[((Root[106074110-440067 #1+((#1)^3)&,2])/53037055)]) (EllipticF[ArcSin[(1/(Sqrt[1-((Root[106074110-440067 #1+((#1)^3)&,2])/(Root[106074110-440067 #1+((#1)^3)&,3]))]))] (Sqrt[1-((Root[106074110-440067 #1+((#1)^3)&,2])/(Root[106074110-440067 #1+((#1)^3)&,3]))]) Root[-11251716812292100+101265451310628900 ((#1)^2)-319838885716734837 ((#1)^4)+400051544640975522 ((#1)^6)-107287461892396881 ((#1)^8)-111285972513709056 ((#1)^10)+37095324171236352 ((#1)^12)&,10],((Root[106074110-440067 #1+((#1)^3)&,2]-Root[106074110-440067 #1+((#1)^3)&,3])/(Root[106074110-440067 #1+((#1)^3)&,1]-Root[106074110-440067 #1+((#1)^3)&,3]))] (-53037055+146689 Root[106074110-440067 #1+((#1)^3)&,1])-146689 EllipticE[ArcSin[(1/(Sqrt[1-((Root[106074110-440067 #1+((#1)^3)&,2])/(Root[106074110-440067 #1+((#1)^3)&,3]))]))] (Sqrt[1-((Root[106074110-440067 #1+((#1)^3)&,2])/(Root[106074110-440067 #1+((#1)^3)&,3]))]) Root[-11251716812292100+101265451310628900 ((#1)^2)-319838885716734837 ((#1)^4)+400051544640975522 ((#1)^6)-107287461892396881 ((#1)^8)-111285972513709056 ((#1)^10)+37095324171236352 ((#1)^12)&,10],((Root[106074110-440067 #1+((#1)^3)&,2]-Root[106074110-440067 #1+((#1)^3)&,3])/(Root[106074110-440067 #1+((#1)^3)&,1]-Root[106074110-440067 #1+((#1)^3)&,3]))] (Root[106074110-440067 #1+((#1)^3)&,1]-Root[106074110-440067 #1+((#1)^3)&,3])) Root[848592880-1760268 ((#1)^2)+((#1)^6)&,4] Root[-11251716812292100+101265451310628900 ((#1)^2)-319838885716734837 ((#1)^4)+400051544640975522 ((#1)^6)-107287461892396881 ((#1)^8)-111285972513709056 ((#1)^10)+37095324171236352 ((#1)^12)&,8]]+EllipticF[ArcSin[Root[-11251716812292100+101265451310628900 ((#1)^2)-319838885716734837 ((#1)^4)+400051544640975522 ((#1)^6)-107287461892396881 ((#1)^8)-111285972513709056 ((#1)^10)+37095324171236352 ((#1)^12)&,9]],Root[50885218341888-152655655025664 #1-2851093116829441 ((#1)^2)+5956612325368322 ((#1)^3)-2851093116829441 ((#1)^4)-152655655025664 ((#1)^5)+50885218341888 ((#1)^6)&,4]] Root[18216914336228627953920-1030425671423232 #1-159111165 ((#1)^2)+((#1)^3)&,1] Root[-62208+126739296 ((#1)^4)-64552988163 ((#1)^8)+1060108715456 ((#1)^12)&,3]+EllipticE[ArcSin[Root[-11251716812292100+101265451310628900 ((#1)^2)-319838885716734837 ((#1)^4)+400051544640975522 ((#1)^6)-107287461892396881 ((#1)^8)-111285972513709056 ((#1)^10)+37095324171236352 ((#1)^12)&,9]],Root[50885218341888-152655655025664 #1-2851093116829441 ((#1)^2)+5956612325368322 ((#1)^3)-2851093116829441 ((#1)^4)-152655655025664 ((#1)^5)+50885218341888 ((#1)^6)&,4]] Root[-369576593651290480400063796373669100761727235072+806994001388633781139978304540241 ((#1)^2)-56815279683853842 ((#1)^4)+((#1)^6)&,6] Root[-62208+126739296 ((#1)^4)-64552988163 ((#1)^8)+1060108715456 ((#1)^12)&,3]-EllipticK[Root[50885218341888-152655655025664 #1-2851093116829441 ((#1)^2)+5956612325368322 ((#1)^3)-2851093116829441 ((#1)^4)-152655655025664 ((#1)^5)+50885218341888 ((#1)^6)&,3]] Root[-18216914336228627953920-1030425671423232 #1+159111165 ((#1)^2)+((#1)^3)&,1] Root[-62208+126739296 ((#1)^4)-64552988163 ((#1)^8)+1060108715456 ((#1)^12)&,8]-EllipticE[Root[50885218341888-152655655025664 #1-2851093116829441 ((#1)^2)+5956612325368322 ((#1)^3)-2851093116829441 ((#1)^4)-152655655025664 ((#1)^5)+50885218341888 ((#1)^6)&,3]] Root[-369576593651290480400063796373669100761727235072+806994001388633781139978304540241 ((#1)^2)-56815279683853842 ((#1)^4)+((#1)^6)&,6] Root[-62208+126739296 ((#1)^4)-64552988163 ((#1)^8)+1060108715456 ((#1)^12)&,8]);

N[i]

i/.{el:(_EllipticF|_EllipticE):>RootReduce[el],s_Sqrt:>RootReduce[s]}//ReleaseHold

Curvature

eqn=x^3-440067x+106074110-y^2;

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

ByteCount/@%

Inflection Points

InflectionPoints[eqn,{x,y}]

pts=RootReduce[%]

Show[plot,Graphics[{Blue,PointSize[.015],Point[pts]}]]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

Slope[y^2==x^3-440067 x+106074110,{x,y}]//FullSimplify