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Hyperbolic Geometry. Chapter 6. THE Quadrilateral!. Hyperbolic Geometry. Negation of Hilbert’s Euclidean Parallel Postulate: There exist a line l and point P not on l such that at least two distinct lines parallel to l pass through P. Basic Theorem 6.1:.
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Hyperbolic Geometry Chapter 6
Hyperbolic Geometry • Negation of Hilbert’s Euclidean Parallel Postulate: There exist a line l and point P not on l such that at least two distinct lines parallel to l pass through P
Basic Theorem 6.1: • A non-Euclidean plane satifying Aristotle’s axiom satisfies the acute angle hypothesis.
Hyperbolic Geometry U N-E T • Universal Non-Euclidean Theorem: In a Hilbert plane in which rectangles do not exist, for every line l and every point P not on l, there are at least two parallels to l through P.
Hyperbolic Geometry U N-E T • Corr: In a Hilbert plane in which rectangles do not exist, for every line l and every point P not on l, there are infinitely many parallels to l through P.
(Def) Defect of a Triangle: • Defect of a Triangle (): 180 minus the angle sum degrees of the triangle; i.e. ( A) + ( B) + ( C) + (ABC) = 180.
Hyperbolic Geometry P6.1 • (Additivity of the Defect): If D is any point between A and B (see figure at right), then (ABC) = (ACD) + (BCD)
Hyperbolic Geometry P6.2 • (No Similarity): In a plane satisfying the acute angle hypothesis, if two triangles are similar, then they are congruent; i.e. AAA is a valid criterion for congruence of triangles.
Hyperbolic Geometry P6.3 • In a plane in which rectangles do not exist, if l || l' then any set of points on l equidistant from l' has at most two points in it.
Hyperbolic Geometry P6.4 • In a Hilbert plane satisfying the acute angle hypothesis, 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 unique common perpendicular segment MM’ dropped from the midpoint M of AB. MM’ is the shortest segment joining a point of l to a point on l', and the segments AA’ and BB’ increase as A, B recede from M. (See following figure.)
Hyperbolic Geometry P6.5 • In a Hilbert plane in which rectangles do not exist, if lines l and l' have a common perpendicular segment MM', then they are parallel and that common perpendicular segment 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 equidistant from l'. (See following figure.)
(Def) Limiting Parallel Ray: • Given a line l and a point P not on l. Let Q be the foot of the perpendicular from P to l. A limiting parallel ray to l emanating from P is a ray that does not intersect l and such that for every ray which is between and , intersects l. (See following figure).
Advanced Theorem • In non-Euclidean planes satisfying Aristotle’s axiom and the line-circle continuity principle, limiting parallel rays exist for every line l and point P not on l.
Hilbert’s Hyperbolic Axiom of Parallels: • For every line l and every point P not on l, a limiting parallel ray exists and it does not make a right angle with , where Q is the foot of the perpendicular from P to l.
(Def) Real Hyperbolic Plane: • A non-Euclidean plane satisfying Dedekin’s axiom is called a real hyperbolic plane.
(Def) Hilbert and Hyperbolic Planes: • Hilbert Plane: A model of our incidence, betweenness and congruence axioms is called a Hilbert plane. • Hyperbolic Plane: A Hilbert plane in which Hilbert’s hyperbolic axiom of parallels holds is called a hyperbolic plane. It is NOT Euclidean!
Hyperbolic Geometry P6.6 • In a hyperbolic plane, with notation as in the above definition XPQ is acute. There is a ray emanating from P, with X’ on opposite sides of from X, such that is another limiting parallel ray to l and XPQ X'PQ. These two rays, situated symmetrically about , are the only limiting parallel rays to l through P.(See following figure.) (See following figure.)
(Def) Angles of Parallelism: • From P6.6 notation, acute angles XPQ and X'PQ are called angles of parallelism for segment PQ.
Dedekind Axiom: (1) • Suppose that the set {l} of all points on a line l is the disjoint union 12 of two nonempty subsets such that no point of either subset is between two points of the other. Then there exists a unique point O on l such that one of the subsets is equal to a ray of l with vertex O and the other subset is equal to the complement (all other points on the line not on the ray). (Next slide.)
Dedekind Axiom: (2) • l P Q O R S <__________|_________|_______________|_____________________|________________|__________________________________> • 1 consists or all the points P, Q to the left of O on l and including O. 2consists or all the points R, S on l to the right of O. O is the Dedekind cut! O no matter where it is, always divides the line into two parts while never talking about how one point is next to the other or how "close" one point is to the other. So you are either in one set or the other.
Hyperbolic Geometry Theorem 6.2: • In a non-Euclidean plane satisfying Dedekind’s axiom, Hilbert’s hyperbolic axiom of parallels holds, as do Aristotle’s axiom and the acute angle hypothesis.
Hyperbolic Geometry Theorem 6.2: • Corr 1: All the results proved previously in this chapter hold in real hyperbolic planes. • Corr 2: A Hilbert plane satisfying Dedekind’s axiom is either real Euclidean or real hyperbolic.
Classification of Parallels • Type 1(Have a common ): m || l and they have a common . m diverges from 1 on both sides of the common . • Type 2(No common ): m || l but m approaches l asymptotically in one direction (limiting || rays) and diverges from l in the other direction. There is no common .
Hyperbolic Geometry Theorem 6.3 • In a hyperbolic plane, given m parallel to l such that m does not contain a limiting parallel ray to l in either direction. Then there exists a common perpendicular to m and l which is unique.
Perpendiculr Bisector Theorme • Given any triangle in a hyperbolic plane, the perpendicular bisectors of its sides are concurrent in the projective completion.
. Chapter 7
Metamathematical Theorem 1 • If Euclidean geometry is consistent, then so is hyperbolic geometry • Corollary: If Euclidean geometry is consistent, then no proof or disproof of Eculid’s parallel postulate from the axioms of neutral geometry will ever be found. Euclid’s parallel postulate is independent of the other postulates.
