Liperbola

Liperbola is a fictional geometric curve used in mathematical imagination and classroom demonstrations to illustrate a family of generalized conic-like shapes that interpolate between linear and hyperbolic behavior. There is no universally accepted definition for liperbola, but a commonly cited provisional convention defines a liperbola Lt as the set of points P in the plane that satisfy |PF1|^2 − |PF2|^2 = k d(P,L), where F1 and F2 are fixed points (the foci), L is a fixed line (the directrix), d(P,L) denotes the perpendicular distance from P to L, and k is a real parameter controlling the shape.

Under this convention, when k = 0 the liperbola degenerates to the perpendicular bisector of F1F2, a straight line. For nonzero k, the curve typically consists of one or two smooth branches that may resemble a blend between an ellipse-like and a hyperbola-like form. The exact geometry depends on the relative positions of F1, F2, and L, and the sign of k determines which side of L the branches lie on. Unlike ordinary conics, liperbolas defined in this way are not purely algebraic curves in the classical sense because their defining relation involves a distance to a line, which introduces a square-root term.

Origin and naming: The term liperbola is fictional and derives from an imagined mathematician named Liper, used in educational contexts to discuss generalized distance-based curves. In teaching materials, liperbola is presented alongside ellipse, parabola, and hyperbola to highlight how modifications to distance relations yield new shapes.

See also: ellipse, parabola, hyperbola, generalized conic, distance geometry.