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Post Number: 45 Posted on Friday, 22 March, 2013 - 04:48 pm:
I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this?

  1. Show that there exists a tangent hyperbolic straight line at every point on a hyperbolic circle, horocycle, or hypercycle. Show that this tangent is perpendicular to the diameter at the point.

  2. Prove that a hyperbolic circle is the locus of points that are a fixed distance from its center.

  3. Let C be a hypercycle, and let L be the hyperbolic straight line that shares the same ideal points as C. Prove that the perpendicular distance from C to L is the same at every point of C.

Thanks in advance

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What's that with the first line? – joriki Mar 23 '13 at 9:59
These are all true. Some are easier in the Poincare disc model, some in the upper half plane. All done in existing books. – Will Jagy Mar 23 '13 at 18:59
What is your definition of a hyperbolic circle, if it is not 2.? If you define it algebraically, what model do you use? – MvG Mar 23 '13 at 22:35
For the first question, i think that i choose a point on the hyperbolic circle and map it into 0 so that the whole hyperbolic circle is map to the x-axis accordingly so that i prove there is tangent at every point of hyperbolic circle?? Am i right ? – charles1118 Mar 24 '13 at 4:57
is there any book suggest any clue for that?? – charles1118 Mar 24 '13 at 5:05

Probably you are searching for $\tau$ and $\sigma$ iso-surfaces:

They are circles through two foci in the plane and their orthogonal trajectories are Apollonius circles.

Hyperbolic Circles in Bipolar Coordinates

Text Book reference (Differential Geometry by H.W. Guggenheimer, Dover, 1977 ) for classic hyperbolic circles:

Chapter on Inner Geometry of Surfaces

Theorem 11-15 Plane hyperbolic geometry is the geometry of the group $SL_2$ of projectivities of $ P^1$ acting as a group of motions in a plane.

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