Hypercycle (geometry)

showing the hypercycle that is determined by the straight line (termed straight because it cuts the horizon at right angles) and point In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line and a point not on , one can construct a hypercycle by taking all points on the same side of as , with perpendicular distance to equal to that of . The line is called the axis, center, or base line of the hypercycle. The lines perpendicular to , which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

showing the hypercycle that is determined by the straight line (termed straight because it cuts the horizon at right angles) and point ]] In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line and a point not on , one can construct a hypercycle by taking all points on the same side of as , with perpendicular distance to equal to that of . The line is called the axis, center, or base line of the hypercycle. The lines perpendicular to , which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle. The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity. The congruence classes of Steiner parabolas in the hyperbolic plane are in one-to-one correspondence with the hypercycles in a given half-plane of a given axis. In an incidence geometry, the Steiner conic at a point produced by a collineation is the locus of intersections for all lines through . This is the analogue of Steiner's definition of a conic in the projective plane over a field. The congruence classes of Steiner conics in the hyperbolic plane are determined by the distance between and and the angle of rotation induced by about . Each Steiner parabola is the locus of points whose distance from a focus is equal to the distance to a hypercycle directrix that is not a line. Assuming a common axis for the hypercycles, the location of is determined by as follows. Fixing , the classes of parabolas are in one-to-one correspondence with . In the conformal disk model, each point is a complex number with . Let the common axis be the real line and assume the hypercycles are in the half-plane with . Then the vertex of each parabola will be in , and the parabola is symmetric about the line through the vertex perpendicular to the axis. If the hypercycle is at distance from the axis, with \tanh d = \tan\tfrac{\phi}{2}, then F = \left(\frac{1-\tan\phi}{1+\tan\phi}\right)i. In particular, when . In this case, the focus is on the axis; equivalently, inversion in the corresponding hypercycle leaves invariant. This is the harmonic case, that is, the representation of the parabola in any inversive model of the hyperbolic plane is a harmonic, genus 1 curve. ==References== , in a Poincaré disk model, can be seen with edge sequences that follow hypercycles.]] Martin Gardner, Non-Euclidean Geometry, Chapter 4 of The Colossal Book of Mathematics, W. W. Norton & Company, 2001, M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 3rd edition, W. H. Freeman, 1994. George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975. J. G. Ratcliffe, Foundation of Hyperbolic Manifolds, Springer, New York, 1994. David C. Royster, Hypercycles and Horocycles. J. Sarli, Conics in the hyperbolic plane intrinsic to the collineation group, J. Geom. 103: 131-138 (2012)

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