Models of Hyperbolic and Euclidean Geometry

The goal of this handout is to discuss models of Hyperbolic and Euclidean Geometry, and the consistency of Hyperbolic Geometry. 1. Some concepts from Euclidean Geometry We will use circles in Euclidean Geometry to build up models for Hyperbolic Geometry. So we will review some basic facts about circles. (Mostly without proofs). In this section assume that we are working in Euclidean Geometry. Definition 1 (Chord). Let γ be a circle. If A and B are distinct points on γ, then the set {C | A * C * B} is called an open chord of γ. The points A and B are called endpoints of the chord (but A and B are not themselves in the open chord). Definition 2 (Diameter). Let γ be a circle. An open chord containing the center of γ is called an open diameter. Definition 3 (Tangent). Let γ be a circle. A line that intersects γ in exactly one point is called a tangent to γ. Proposition 1. If A is a point on a circle γ, then there is exactly one tangent l to γ passing through A. Lemma 1. Let γ be a circle, and let r be an open chord with endpoints A and B. Let l be the tangent containing A and let m be the tangent containing B. If r is not an open diameter, then l and m intersect at a point P outside γ. Definition 4 (Pole). Let r and P be as in the above lemma. Then P is called the pole of r. Lemma 2. Let r and s be two chords of a circle γ that do not intersect. Suppose the endpoints of r and s are distinct. Let P be the pole of r and let Q be the pole of s. Then the line ← → P Q intersects the circle γ in two points, and intersects both r and s. Thus ← → P Q contains a chord that intersects both r and s. Definition 5 (Orthogonal Circles). Let γ and β be two circles intersecting in exactly two points P 1 and P 2. Suppose the tangent to γ containing P 1 and the tangent to β containing P 1 are perpendicular. Suppose this condition holds for P 2 as well. Then we say that γ and β are orthogonal. The following (whose proof we skip) shows the existence …