Cassinian Oval

Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2004 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["Graphics`ImplicitPlot`"]

<< Graphics`ContourPlot3D`

Cassinian Oval as a torus cut, proof

problem

prove that cassinian oval Sqrt[(x - a)^2 + y^2]*Sqrt[(x + a)^2 + y^2]==b^2 is the intersection of a plane and a torus.

solution

The formula of torus with tube radius c and its center to torus center d is

(Sqrt[x^2 + y^2] - d)^2 + z^2 == c^2

ContourPlot3D[(Sqrt[x^2 + y^2] - d)^2 + z^2 - c^2 /. {c -> 1, 
          d -> 2}, {x, -3, 3}, {y, -3, 3}, {z, -3, 3}, 
      Contours -> {0}, PlotPoints -> {1, 1, 1} 5]

we want to find the intersection of this torus with a plane x==c

eliminate x by replacing it with c we have:

cutTorus  = {(Sqrt[x^2 + y^2] - d)^2 + z^2, c^2} /. 
        x -> c

minus c^2 on both sides and expand

cutTorus = Expand[cutTorus - c^2]

regroup and isolate the term with square root

cutTorus = cutTorus - cutTorus[[1, 3]]

square both sides. (we know that the expression under the square root is positive.

cutTorus = cutTorus^2

regroup and expand

cutTorus = Expand[cutTorus - cutTorus[[2]]]

We want to show that the above equation has the Cassinian oval form defined as
Sqrt[(x - a)^2 + y^2]*Sqrt[(x + a)^2 + y^2] == b^2
(possibly up to scale, rotation, and translation.)

first we manipulate the Cassinian oval equation

cassini={Sqrt[(x - a)^2 + y^2]*Sqrt[(x + a)^2 + y^2],b^2}

square both sides (the radicands are positive in our context since they are distances), regroup, and expand we have

cassini = Expand[cassini^2 - b^4]

change the variable in the cut torus

cutTorus = cutTorus /. {z -> y, y -> x}

We see that the two are similar. Now we just let d=a and c=b^2/(Sqrt[4]*a)

(cutTorus /. {d -> a, c -> b^2/(Sqrt[4]*a)}) - 
      cassini

end of proof.