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Concept
Hypercycle (geometry)
Summary
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.
The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.
Hypercycles in hyperbolic geometry have some properties similar to those of lines in Euclidean geometry:
In a plane, given a line and a point not on it, there is only one hypercycle of that of the given line (compare with Playfair's axiom for Euclidean geometry).
No three points of a hypercycle are on a circle.
A hypercycle is symmetrical to each line perpendicular to it. (Reflecting a hypercycle in a line perpendicular to the hypercycle results in the same hypercycle.)
Hypercycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry:
A line perpendicular to a chord of a hypercycle at its midpoint is a radius and it bisects the arc subtended by the chord.
Let AB be the chord and M its middle point.
By symmetry the line R through M perpendicular to AB must be orthogonal to the axis L.
Therefore R is a radius.
Also by symmetry, R will bisect the arc AB.
The axis and distance of a hypercycle are uniquely determined.
Let us assume that a hypercycle C has two different axes L1 and L2.
Using the previous property twice with different chords we can determine two distinct radii R1 and R2. R1 and R2 will then have to be perpendicular to both L1 and L2, giving us a rectangle.
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