Generalized conic

In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of some defining property of the classical 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. Since an ellipse is the equidistant set of two circles, where one circle is inside the other, the equidistant set of two arbitrary sets of points in a plane can be viewed as a generalized conic. In rectangular Cartesian coordinates, the equation y = x² represents a parabola. The generalized equation y = x^r, for r ≠ 0 and r ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications in approximation theory and optimization theory.^[1]

Among the several possible ways in which the concept of a conic can be generalized, the most widely used approach is to define it as a generalization of the ellipse. The starting point for this approach is to look upon an ellipse as a curve satisfying the 'two-focus property': an ellipse is a curve that is the locus of points the sum of whose distances from two given points is constant. The two points are the foci of the ellipse. 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. This can be further generalized to curves which are obtained as the loci of points such that some weighted sum of the distances from a finite set of points is a constant. A still further generalization is possible by assuming that the weights attached to the distances can be of arbitrary sign, namely, plus or minus. Finally, the restriction that the set of fixed points, called the set of foci of the generalized conic, be finite may also be removed. The set may be assumed to be finite or infinite. In the infinite case, the weighted arithmetic mean has to be replaced by an appropriate integral. Generalized conics in this sense are also called polyellipses, egglipses, or, generalized ellipses. 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]

Change the definition of the average. In the formulation, the average was interpreted as the arithmetic mean. This may be replaced by other notions of averages like geometric mean of the distances. If the geometric mean is used to specify the average, the resulting curves turn out to be . "Lemniscates are sets all of whose points have the same geometric mean of the distances (i.e. their product is constant). Lemniscates play a central role in the theory of approximation. The polynomial approximation of a holomorphic function can be interpreted as the approximation of the level curves with lemniscates. The product of distances corresponds to the absolute value of the root-decomposition of polynomials in the complex plane."^[6]

lemniscates

Change the of the focal set. Modify the definition so that the definition can be applied even in the case where the focal set infinite. This possibility was first introduced by C. Gross and T.-K. Strempel [2] and they posed the problem whether which results (of the classical case) can be extended to the case of infinitely many focal points or to continuous set of foci.^[7]

cardinality

Change the dimension of the underlying space. The points may be assumed to lie in some d-dimensional space.

Change the definition of the distance. Traditionally euclidean definitions are employed. in its place, other notions of distance like , may be used.^[6]^[8] Generalized conics with this notion of distance have found applications in geometric tomography.^[6]^[9]

taxicab distance

In the special case when k = 1, the generalized conic reduces to an ordinary conic.

In the special case when k > 1, there is a simple geometrical method for the generation of the corresponding generalized conic.

[11]

For a detailed discussion of generalized conics from the viewpoint of differential geometry, see the chapter on generalized conics in the book Convex Geometry by Csaba Vincze available online.