Cartesian Ovals

Author

Eric W. Weisstein
February 17, 2012

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

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

©2011 Wolfram Research, Inc. except for portions noted otherwise

Degree

eqn[m_,n_,a_,k_][x_,y_]:=m(Sqrt[(((x-a))^2)+(y^2)])+n(Sqrt[(((x+a))^2)+(y^2)])-(k^2)

AlgebraicDegree[eqn[a,b][x,y],{x,y}]

Equation

Cartesian

Plus

FullSimplify/@{FullSimplify[(((m(Sqrt[(((x-a))^2)+(y^2)])))^2)],Expand[(((k-n(Sqrt[(((x+a))^2)+(y^2)])))^2)]}

expr1=FullSimplify[(((m(Sqrt[(((x-a))^2)+(y^2)])))^2)-Expand[(((k-n(Sqrt[(((x+a))^2)+(y^2)])))^2)]]

expr2=(x^2+y^2+a^2)(m^2-n^2)-2a x(m^2+n^2)-(k^2)+2k n(Sqrt[(x+a)^2+y^2]);

expr1-expr2//Simplify

((x^2+y^2+a^2)(m^2-n^2)-2a x(m^2+n^2)-(k^2))^2-(((2k n(Sqrt[(x+a)^2+y^2])))^2)

Simplify[%]

eqn=-4 (k^2) (n^2) ((((a+x))^2)+(y^2))+((((k^2)+2 a ((m^2)+(n^2)) x-((m^2)-(n^2)) ((a^2)+(x^2)+(y^2))))^2);

FullSimplify[eqn]

AlgebraicDegree[%215,{x,y}]

Minus

expr1=FullSimplify[(((m(Sqrt[(((x-a))^2)+(y^2)])))^2)-Expand[(((k+n(Sqrt[(((x+a))^2)+(y^2)])))^2)]]

expr2=(x^2+y^2+a^2)(m^2-n^2)-2a x(m^2+n^2)-(k^2)-2k n(Sqrt[(x+a)^2+y^2]);

expr1-expr2//Simplify

Change (m, n) to (b, c)

Solve[{b==m^2-n^2,c==m^2+n^2},{m,n}]

(((-((Sqrt[-b+c])/(Sqrt[2]))))^2)

eqn/.{m->(Sqrt[((c+b)/2)]),n->(Sqrt[((c-b)/2)])}//FullSimplify

e1=FullSimplify[%/.{a->1,k->1}]

e1//Expand

Lawrence's expression

e2=2c((x^2)+(y^2)+1)-4b x-1-(((b((x^2)+(y^2)+1)-2c x))^2);

e1+e2//Simplify

Special Cases

Hyperbola

FullSimplify[eqn/.n->m]

eqn

Central Conic

FullSimplify[eqn/.m->1]

Plot

V5

<<Graphics`ImplicitPlot`

ImplicitPlot[(b(x^2+y^2+1)-2c x)^2==2 c(x^2+y^2+1)-4b x-1/.{b->1.,c->4.},{x,-1.,11.},{y,-6.,6.},Ticks->None,AspectRatio->Automatic,PlotPoints->100,
PlotStyle->Red,
AxesOrigin->{0,0}]

ImplicitPlot[(b(x^2+y^2+1)-2c x)^2==2 c(x^2+y^2+1)-4b x-1/.{b->1.,c->4.},{x,-1.,11.},Ticks->None,AspectRatio->Automatic,
PlotPoints->50,
PlotStyle->Red,
AxesOrigin->{0,0}]

V6

-4 k^2 n^2 ((a+x)^2+y^2)+(k^2+2 a (m^2+n^2) x-(m^2-n^2n) (a^2+x^2+y^2))^2

-4 (k^2) (n^2) ((((a+x))^2)+(y^2))+((((k^2)+2 a ((m^2)+(n^2)) x-((m^2)-(n^2)) ((a^2)+(x^2)+(y^2))))^2)

eqn

(((((#2)^2)+2 #1(((#3)^2)+((#4)^2)) x-(((#3)^2)-((#4)^2)) (((#1)^2)+(x^2)+(y^2))))^2)-4 ((#2)^2) ((#4)^2) ((((#1+x))^2)+(y^2))==0&@@@{{.5,1,8,6},{-.41,9,3,10},{-.41,9,6,10},{0,9,7,5},{1,4,1,1},{1.5,7,4,4}}

ContourPlot[
(((((#2)^2)+2 #1(((#3)^2)+((#4)^2)) x-(((#3)^2)-((#4)^2)) (((#1)^2)+(x^2)+(y^2))))^2)-4 ((#2)^2) ((#4)^2) ((((#1+x))^2)+(y^2))==0,
{x,-5,5},{y,-5,5},MaxRecursion->4]&@@@{{.5,1,8,6},{-.41,9,3,10},{-.41,9,6,10},{0,9,7,5},{1,4,1,1},{1.5,7,4,4}}

Manipulate[ContourPlot[
((((k^2)+2 a ((m^2)+(n^2)) x-((m^2)-(n^2)) ((a^2)+(x^2)+(y^2))))^2)==4 (k^2) (n^2) ((((a+x))^2)+(y^2)),
{x,-2.5,5},{y,-3,3},AspectRatio->Automatic],
{{a,1},-5,5},
{{k,1},1,10,1},
{{m,1},1,10,1},
{{n,1},1,10,1}
]

Properties

Curvature

FullSimplify[Curvature[eqn,{x,y}],a∈Reals&&(m|n)∈Integers&&m>0&&n>0&&k>0]

ByteCount/@%

%318[[-1]]

Slope

Slope[-4 k^2 n^2 ((a+x)^2+y^2)+(k^2+2 a (m^2+n^2) x-(m^2-n^2) (a^2+x^2+y^2))^2==0,{x,y}]//FullSimplify

Slope[-4 k^2 n^2 ((a+x)^2+y^2)+(k^2+2 a (m^2+n^2) x-(m^2-n^2) (a^2+x^2+y^2))^2==0,{x,y}]//FullSimplify