Cassinian Oval

cassinian Oval
Cassinian Ovals.

Graphics code.

History

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

Description

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 Family

Formula

Properties

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
graphics code

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)

cassinian_oval.pdf

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

Visual Complex Analysis by Tristan Needham. p 60 - 63. Buy at amazon He also mention curves formed by the locus of points whose product of distances to n points are constant.

If you have a question, put $5 at patreon and message me.

Plane Curves

Ancient

  1. Conic Sections
  2. Parabola
  3. Hyperbola
  4. Ellipse
  5. Cissoid
  6. Conchoid
  7. Quadratrix
  8. Archimedean Spiral
  9. Equiangular Spiral
  10. Lituus
  11. Cornu Spiral

Cyclodal

  1. Epitrochoid
  2. Hypotrochoid
  3. Epicycloid and Hypocycloid
  4. Rose Curve
  5. Astroid
  6. Deltoid
  7. Nephroid
  8. Cardioid
  9. Trochoid
  10. Cycloid

Calculus Era

  1. Cassinian Oval
  2. Cross Curve
  3. Folium of Descartes
  4. Piriform
  5. Semicubic Parabola
  6. Tractrix
  7. Trisectrix
  8. Trisectrix of Maclaurin
  9. Lemniscate of Bernoulli
  10. Lemniscate of Gerono
  11. Limacon Of Pascal
  12. Witch of Agnesi
  13. Sine Curve
  14. Catenary
  15. Bezier Curve

Methods

  1. Caustics
  2. Cissoid
  3. Conchoid
  4. Envelope
  5. Evolute
  6. Involute
  7. Geometric Inversion
  8. Orthoptic
  9. Parallel Curve
  10. Pedal Curve
  11. Radial Curve
  12. Roulette

Math of Curves

  1. Geometry: Coordinate Systems for Plane Curves
  2. Coordinate Transformation
  3. Vectors
  4. Naming and Classification of Curves
  1. Cusp
  2. Curvature