Cassinian Oval

cassinian Oval
Cassinian Ovals.

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Studied by Giovanni Domenico (1680) in relation to motions of earth and sun.


Cassinian oval describe a family of curves. Cassinian Oval is defined as follows: Given fixed points F1 and F2. Given a constant c. The locus of points such that distance[P,F1] * distance[P,F2] == c is cassinian oval.

Cassinian oval is analogous to the definition of ellipse, where sum of two distances is replace by product.

definition .

The fixed points F1 and F2 are called foci. Let the foci be {a,0} and {-a,0}. Let the constant c be b^2. The distance from a point {x,y} to another {m,n} is Sqrt[(x-m)^2+(y-n)^2]. Thus the equation for cassinian oval is Sqrt[(x-a)^2+y^2]*Sqrt[(x+a)^2+y^2]==b^2

When the square root of the constant c is less than half the distance between the foci, then there are two branches of the curve. In other words, if b < a, then there are two ovals. When a==b, the curve is called lemniscate of Bernoulli .

cassinian oval
Cassinian Ovals with a == 1 and different values of b.



Cassinian Oval as a Surface

cassinian oval
The Cassinian Oval plotted as a surface of the function Sqrt[(x-a)^2+y^2]*Sqrt[(x+a)^2+y^2]-b^2 Cassinian Oval as a surface

Torus cut

Cassinian ovals are the intersection of a torus and a plane in certain positions.

Let c be the radius of the torus tube. Let d be the distance from the center of the tube to axis of the torus. The intersection of a plane c distant from the torus's axis is a Cassinian oval, with a = d and b^2 = Sqrt[4]*c*d, where a is half of the distance between foci, and b^2 is the product constant.

One thing we realize is that for Cassinian oval with large constant b^2, the curve approches a circle, and the corresponding torus is one such that the tube radius is larger than the center to directrix. That is, a self-intersecting torus without the hole. This surface also approaches a sphere.

Note that the toris in the figure below are not identical. Arbitrary vertical slice of a torus are NOT Cassinian ovals, they are called Spiric Sections .

cassinian oval cassinian oval
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Proof outline: start with a torus equation (Sqrt[x^2 + y^2] - d)^2 + z^2 == c^2. Eliminate the square root and regroup to one side. Replace d=a and c = b^2/(Sqrt[4]*a). Now do the same with cassian oval implicit equation Sqrt[(x-a)^2+y^2]*Sqrt[(x+a)^2+y^2]==b^2. Luckly, one sees that the two equations match without further algebra considering scale and rotation of the curve. (detailed proof)

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