# Hyperbolic geometry

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.

• 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:
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.