Characteristic Postulate / Hyperbolic parallel postulate Let C be a point not on a given line l. Then there are at least two lines that passes through C and does not intersect l. This is called Characteristic postulate of hyperbolic geometry Boundary Parallels Let C be a point not on a given line l. Then there are at least two lines that passes through C and does not intersect l. Let such parallel lines are named as l1and l2. If these are first line through C in either direction that does not meet l, then these two lines l1and l2 are called the boundary parallels.

Parallel lines

This completes the proof. Right hand and left hand parallel These are first two lines through C that does not meet l in either direction and makes least angle with the perpendicular line from C to l.

Left hand parallel

Right hand parallel

Note: The boundary parallels makes the angle of parallel in either direction. These boundary parallels are simply called parallel lines. Angle of parallelism (or Angle of Parallel) Let C be a point not on a given line l. Then by Characteristic postulate of hyperbolic geometry, there are two lines at C which are parallel to l. Let such parallel lines are named as l1and l2. If we draw, CE perpendicular to l, then the angle formed by l1 and l2 with CE is called angle of parallel. This angle of parallel, is the least angle formed by parallel lines. This angle of parallel is indirectly proportional to the length of perpendicular. For example, if k is the distance of perpendicular line segment CE, then angle of parallel is denoted by Π(𝑘) and have the following relations. lim Π(𝑘) = 0 and lim Π(𝑘) = 90 𝑘→∞

𝑘→0

The angle determined by either the right or left hand parallel with the perpendicular line drawn from C to l is called the angle of parallelism. Non Intersecting lines Let C be a point not on a given line l. Then by Characteristic postulate of hyperbolic geometry, there are two lines at C which are parallel to l. Let such parallel lines are named as l1 and l2.

Others all lines which passes through C but do not meet l are called nonintersecting lines. These nonintersecting lines are also called hyper parallels. Such non intersecting lines inside the boundary of parallel lines. There are infinitely many nonintersecting lines through C. . .}Non intersecting lines .

Intersecting lines Let C be a point not on a given line l. Then by Characteristic postulate of hyperbolic geometry, there are infinitely many lines at C which do not intersect l. Such parallel lines are named as parallel lines and non-intersecting lines. Others all lines which passes through C and do intersect l are called intersecting lines. Such intersecting lines outside the boundary of parallel lines. There are infinitely many intersecting lines through C.

. .}Intersecting lines .

Theorem 1 Through a given point C, not on a given line l, an infinite number of lines pass from C and not intersecting the given linel. Proof Let C be a point not on a given line l. Then by Characteristic postulate of hyperbolic geometry, there are two lines at C which are parallel to l.

Let such parallel lines are named as l1 and l2. Then, there are an infinite number of lines through C in the interior of l1 and l2 . Suppose that one of these lines, say l3 meets l at a point D. Now, drop a perpendicular from C to l, meeting at E. Then, the line l2 enters triangle CED, and by Pasch's Axiom it must meet side ED (= l ) This contradicts our supposition that l and l2 are parallel.

Theorem 2 The two angles of parallelism for the same distance are congruent. Proof Let C be a point not on a given line l. Then by Characteristic postulate of hyperbolic geometry, there are two lines at C which are parallel to l.

Suppose that1 and 2 are the angles of parallelism for CD, where CD is perpendicular to l. If possible, suppose that 1(=∠PCD) and2 (= ∠QCD) are not congruent. Say, 1 is the larger angle. Then, there exists apoint E on l so that 1 is congruent to 3. Now, Construct point F on l so that DE = DF. Then by SAS, Δ𝐶𝐷𝐸 ≅△ 𝐶𝐷𝐹. Thus, ∠FCD=∠ECD (Corresponding angle of congruent triangle) Also, ∠1 =∠3 (=∠ECD) , Assumption. This implies that, ∠PCD =∠FCD, Contradiction. Therefore, the two angles of parallelism must be congruent.

Summary

Ideal Point Let C be a point not on a given line l. Then by Characteristic postulate of hyperbolic geometry, there are two lines at C which are parallel to l.

Now, The intersection point of parallel lines l and l2 is called ideal point. or simply we can say that two parallel lines intersect at an ideal point. In the figure above, l and l2 are two parallel lines, therefore the intersection point of l and l2 is called ideal point. The ideal point is denoted by Ω (omega). Thus ideal point is also called omega point.

Ultra Ideal Point Let C be a point not on a given line l. Then by Characteristic postulate of hyperbolic geometry, there are two lines at C which are parallel to l. Let such parallel lines are named as l1and l2. Also, there are infinitely many non-intersecting lines which passes through C and lies between l1 and l2 . Now, The intersection point of such nonintersecting line and l is called ultra ideal point. or simply we can say that non-intersecting lines intersect at an ultra ideal point.

In the figure above, l and m are two non intersecting lines, therefore the intersection point of l and m is called ultra ideal point. The ultra ideal point is denoted by Γ (gamma). Thus ultra ideal point is also called gamma point.