Circle-Ellipse Intersection

Eric W. Weisstein
January 11, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Circle-EllipseIntersection.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Solve

PowerExpand[FullSimplify[Solve[{(x-x0)^2+(y-y0)^2==r^2,(x-x1)^2/a^2+(y-y1)^2/b^2==1},{x,y}]]]

PowerExpand[Simplify[Solve[{x^2+y^2==r^2,x^2/a^2+y^2/b^2==1},{x,y}]]]

CircleEllipsePoints[r_,{a_,b_}]:=Module[{
x=a Sqrt[(r^2-b^2)/(a^2-b^2)],
y=b Sqrt[(a^2-r^2)/(a^2-b^2)]
},
Map[Point,Table[{(-1)^i x,(-1)^j y},{i,2},{j,2}],{2}]
]

CircleEllipsePoints[1,{1.2,.5}]

Show[Graphics[{Circle[{0,0},1],Circle[{0,0},{1.2,.5}],
PointSize[.04],CircleEllipsePoints[1,{1.2,.5}]},
AspectRatio->Automatic]]