Download
1 / 7
70 likes | 188 Vues
This text explores the fundamental concepts of hyperbolic geometry, detailing key axioms including the Hyperbolic Axiom and the Universal Hyperbolic Theorem, which establish the nature of parallel lines and their interactions with points not on them. It also presents important theorems that describe the unique properties of triangles and quadrilaterals, such as their angle sums and congruence criteria. Additional insights into parallel lines and perpendicular segments further emphasize the distinctive characteristics of this non-Euclidean geometric space.
E N D
Hyperbolic Geometry Axiom • Hyperbolic Axiom (HA): In hyperbolic geometry there exist a line l and point P not on l such that at least two distinct lines parallel to l pass through P. • Lemma 6.1: Rectangles do not exist.
Hyperbolic Geometry UHT • Universal Hyperbolic Theorem (UHT): In hyperbolic geometry, for every line l and every point P not on l there pass through P at least two distinct parallels to l. • Cor: In hyperbolic geometry, for every line l and every point P not on l, there are infinitely many parallels to l through P.
Hyperbolic Geometry Theorems 1- 3 • Thm 6.1: In hyperbolic geometry, all triangles have angle sum less than 180. • Cor: In hyperbolic geometry, all convex quadrilaterals have angle sum less than 360. • Thm 6.2: In hyperbolic geometry, if two triangles are similar, they are congruent. • Thm 6.3: In hyperbolic geometry, if l and l' are any distinct parallel lines, then any set of points on l equidistant from l' has at most two points in it.
Hyperbolic Geometry Theorem 4 • Thm 6.4: In hyperbolic geometry, if l and l' are parallel lines for which there exists a pair of points A and B on l equidistant from l', then l and l' have a common perpendicular segment that is also the shortest segment between l and l' • L 2: The segment joining the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both the base and the summit, and this segment is shorter than the sides.
Hyperbolic Geometry Theorem 5 • Thm 6.5: In hyperbolic geometry, if lines l and l' have a common perpendicular segment MM' then they are parallel and MM' is unique. Moreover, if A and B are any points on l such that M is the midpoint of segment AB, then A and B are equidis-tant from l'.
Hyperbolic Geometry Theorem 6 • Thm 6.6: For every line l and every point P not on l, let Q be the foot of the perpendicular from P to l. Then there are two unique nonopposite rays . and on opposite sides of that do not meet l and have the property that a ray emanating from P meets l if and only if it is between and . . Moreover these limiting rays are situated symmetrically about in the sense that XPQ X'PQ. (See following figure.)
