Cassini oval
In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant.
This may be contrasted with an ellipse, for which the sum of the distances is constant, rather than the product.
Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in the late 17th century.
[1] Cassini believed that a planet orbiting around another body traveled on one of these ovals, with the body it orbited around at one focus of the oval.
are called the foci of the Cassini oval.
If the foci are (a, 0) and (−a, 0), then the equation of the curve is When expanded this becomes The equivalent polar equation is The curve depends, up to similarity, on e = b/a.
When e < 1, the curve consists of two disconnected loops, each of which contains a focus.
When e = 1, the curve is the lemniscate of Bernoulli having the shape of a sideways figure eight with a double point (specifically, a crunode) at the origin.
[3][4] When e > 1, the curve is a single, connected loop enclosing both foci.
[6] The curve has double points at the circular points at infinity, in other words the curve is bicircular.
When e ≠ 1 the curve has class eight, which implies that there should be a total of eight real foci.
Six of these have been accounted for in the two triple foci and the remaining two are at
So the additional foci are on the x-axis when the curve has two loops and on the y-axis when the curve has a single loop.
For Cassini ovals one has: Proof: For simplicity one chooses
These conic sections have no points with the y-axis in common and intersect the x-axis at
Their discriminants show that these curves are hyperbolas.
A more detailed investigation reveals that the hyperbolas are rectangular.
In order to get normals, which are independent from parameter
Hence the Cassini ovals and the hyperbolas intersect orthogonally.
Remark: The image depicting the Cassini ovals and the hyperbolas looks like the equipotential curves of two equal point charges together with the lines of the generated electrical field.
But for the potential of two equal point charges one has
Instead these curves actually correspond to the (plane sections of) equipotential sets of two infinite wires with equal constant line charge density, or alternatively, to the level sets of the sums of the Green’s functions for the Laplacian in two dimensions centered at the foci.
The single-loop and double loop Cassini curves can be represented as the orthogonal trajectories of each other when each family is coaxal but not confocal.
Each curve, up to similarity, appears twice in the image, which now resembles the field lines and potential curves for four equal point charges, located at
Further, the portion of this image in the upper half-plane depicts the following situation: The double-loops are a reduced set of congruence classes for the central Steiner conics in the hyperbolic plane produced by direct collineations;[10] and each single-loop is the locus of points
The second lemniscate of the Mandelbrot set is a Cassini oval defined by the equation
Cassini ovals appear as planar sections of tori, but only when the cutting plane is parallel to the axis of the torus and its distance to the axis equals the radius of the generating circle (see picture).
The intersection of the torus with equation and the plane
yields After partially resolving the first bracket one gets the equation which is the equation of a Cassini oval with parameters
Cassini's method is easy to generalize to curves and surfaces with an arbitrarily many defining points: describes in the planar case an implicit curve and in 3-space an implicit surface.
