page_title: Generalized conic
section_title: Examples
hierarchical_section_title: Generalized conic / Generalized conics via polar equations / Examples
caption_attribution_description: English: Example of generalized conic in the sense in which the term was used by Tom M. Apostol and Mamikon A. Mnatsakanian in their study of curves drawn on the surfaces of right circular cones. Parameter values: r_0=5, lambda=1, k=1.5
context_page_description: 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, 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.
  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.
caption: r0 = 5, λ = 1, k = 1.5
exif: {}