Hypercycle (geometry)

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 $L$ and a point $P$ not on $L$ , one can construct a hypercycle by taking all points $Q$ on the same side of $L$ as $P$ , with perpendicular distance to $L$ equal to that of $P$ . The line $L$ is called the axis, center, or base line of the hypercycle. The lines perpendicular to $L$ , which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between $L$ and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle.^[1]

The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.

^ Martin, George E. (1986). The foundations of geometry and the non-euclidean plane (1., corr. Springer ed.). New York: Springer-Verlag. p. 371. ISBN 3-540-90694-0.

[1] Martin, George E. (1986). The foundations of geometry and the non-euclidean plane (1., corr. Springer ed.). New York: Springer-Verlag. p. 371. ISBN 3-540-90694-0.

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