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Cycles are the portion of the Euclidean circle (or straight line) inside the unit disk,that are also not perpendicular to the unit disk. Cycles can be broken down into three different types:
Hyberbolic Circle
-the cycle is enitrely contained inside the unit circle. A hyperbolic circle can be translated in to Mobius terminology as a circle of Apollonius with respect to p and p*. In Euclidean terminology, it is a circle inside the unit disk. In hyperboic terms, a hyperbolic circle is a curve that is like a circle in Euclidean geometry
Horocycle
-the cycle is tangent to the unit circle. In Mobius geometry, a horocycle is a degenerate Steiner circle with respect to the point, p. A Euclidean description would be a circle tangent to the unit circle. In hyperbolic terminology, it is a unique curve unlike anything in Euclidean geometry.
Hypercycle
-the cycle intersects the unit circle, but the angle of intersection cannot be a right angle.In Mobius geometry, a hypercycle is a Steiner circle of the first kind with respect to the point, p and q. A Euclidean definition would be an arc of a circle not perpendicular or tangent to the unit circle. In hyperbolic terminology, it is a curve that stays a constant distance from a straight line.
Why are Cycles important?
Cycles are important because each type is closely connnected with one of the fundamental types of mobius transformation: elliptic, parabolic, and hyperbolic.