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]