Generalized conic

For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed points – the foci – in the plane is a constant.

The curve obtained when the set of two fixed points is replaced by an arbitrary, but fixed, finite set of points in the plane is called an n–ellipse and can be thought of as a generalized ellipse.

In rectangular Cartesian coordinates, the equation y = x2 represents a parabola.

The curve obtained by replacing the set of two fixed points by an arbitrary, but fixed, finite set of points in the plane can be thought of as a generalized ellipse.

Generalized conics with three foci are called trifocal ellipses.

A still further generalization is possible by assuming that the weights attached to the distances can be of arbitrary sign, namely, plus or minus.

Since such curves were considered by the German mathematician Ehrenfried Walther von Tschirnhaus (1651 – 1708) they are also known as Tschirnhaus'sche Eikurve.

[2] Also such generalizations have been discussed by René Descartes[3] and by James Clerk Maxwell.

[4] René Descartes (1596–1650), father of analytical geometry, in his La Geometrie published in 1637, set apart a section of about 15 pages to discuss what he had called bifocal ellipses.

A bifocal oval was defined there as the locus of a point P which moves in a plane such that

where A and B are fixed points in the plane and λ and c are constants which may be positive or negative.

Descartes had also recognized these ovals as generalizations of central conics, because for certain values of λ these ovals reduce to the familiar central conics, namely, the circle, the ellipse or the hyperbola.

[3] Multifocal ovals were rediscovered by James Clerk Maxwell (1831–1879) while he was still a school student.

At the young age of 15, Maxwell wrote a scientific paper on these ovals with the title "Observations on circumscribed figures having a plurality of foci, and radii of various proportions" and got it presented by Professor J. D. Forbes in a meeting of the Royal Society of Edinburgh in 1846.

Professor J. D. Forbes also published an account of the paper in the Proceedings of the Royal Society of Edinburgh.

A multifocal oval is a curve which is defined as the locus of a point moving such that where A1, A2, .

illustrates the general approach adopted by Maxwell for drawing such curves.

The curve traced by the pencil is the locus of P. His ingenuity is more visible in his description of the method for drawing a trifocal oval defined by an equation of the form

In the two years after his paper was presented to the Royal Society of Edinburgh, Maxwell systematically developed the geometrical and optical properties of these ovals.

[5] As a special case of Maxwell's approach, consider the n-ellipse—the locus of a point which moves such that the following condition is satisfied: Dividing by n and replacing c/n by c, this defining condition can be stated as This suggests a simple interpretation: the generalised conic is a curve such that the average distance of every point P on the curve from the set {A1, A2, .

This formulation of the concept of a generalized conic has been further generalised in several different ways.

The formulation of the definition of the generalized conic in the most general case when the cardinality of the focal set is infinite involves the notions of measurable sets and Lebesgue integration.

All these have been employed by different authors and the resulting curves have been studied with special emphasis on applications.

Tom M. Apostol and Mamikon A. Mnatsakanian in their study of curves drawn on the surfaces of right circular cones introduced a new class of curves which they called generalized conics.

For constants r0 ≥ 0, λ ≥ 0 and real k, a plane curve described by the polar equation is called a generalized conic.

In 1996, Ruibin Qu introduced a new notion of generalized conic as a tool for generating approximations to curves.

A similar approach considers a generalization of the focus/directrix/eccentricity interpretation of conics, by retaining a single point F for the focus, any differentiable curve d serving as the directrix, and e > 0, the eccentricity.

Let X be a variable point on d. The resultant generalized conic is the set of points P (each lying on a normal to d through X) for which the distances PF and PX satisfy the ratio PF/PX = e. Norman[14] and Poplin[15] referred to these curves as pseudoconics and the constraint that the distance from P to the directrix be minimal has been discarded.

If one retains the minimality requirement, then the set of points P satisfying this requirement are considered to be the primary pseudoconic, and the remainder of the curve is the secondary branch of the pseudoconic.

Similar examples of generalized parabolas can be found in Joseph et al..[16]

Construction of the oval defined by AP + 2 BP = c using pins, pencil and string as described by James Clerk Maxwell.
Construction of the oval defined by AP + BP + CP = c using pins, pencil and string as described by James Clerk Maxwell.
Figure shows the initial position of right circular cone, together with a plane section, before it is unwrapped onto a plane.
Figure shows an arbitrary position a right circular cone, together with a plane section, while the cone is being unwrapped onto a plane. The figure also shows the generalized conic (dotted curve in the plane) to which the conic section on the cone is unwrapped into the plane.
r 0 = 5, λ = 0.6, k = 1.5
r 0 = 5, λ = 0.22, k = 5.5
r 0 = 5, λ = 1, k = 1.5
r 0 = 5, λ = 1, k = 1.15
r 0 = 5, λ = 1.6, k = 1.5
r 0 = 5, λ = 0.8, k = 0.5
r 0 = 5, λ = 1.0, k = 0.5
r 0 = 5, λ = 1.5, k = 0.5
Animation showing the generation of an ellipse as the equidistant set of two circles.
Pseudoparabolas with parabolic and cubic directrices, respectively