MTES3223 GEOMETRY HYPERBOLIC GEOMETRY HYPERBOLIC GEOMETRY TOPIC 2: NON-EUCLIDEAN GEOMETRY SHUVANISRI RAMACHANDRAN
CONTENTS 3. 4. 5. 1. INTRODUCTION TO HYPERBOLIC GEOMETRY 2. HYPERBOLIC PARALLEL POSTULATE PROPERTIES OF HYPERBLOIC TRIANGLES NON-EXISTENCE OF RECTANGLES LAMBERT AND SACCHERI QUADRILATERALS 6. ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY
INTRODUCTION TO HYPERBOLIC GEOMETRY Hyperbolic geometry is an area of mathematics which has an interesting history. Let start with basic geometry we study in high school, know as Euclidean geometry. The postulates stated by Euclid are the foundation of this geometry, which we enlist here, 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the Parallel Postulate.
INTRODUCTION TO HYPERBOLIC GEOMETRY In the mid 18th century, Lobachevsky, Gauss, and some other mathematicians, in an attempt of eliminating Euclid’s fifth postulate, realized that the first four axioms of Euclid could give rise to a separate geometry. Gauss claimed to have made the discovery of this new geometry in his unpublished work “Non- Euclidean Geometry ” , which he had mentioned in a letter he had sent to Franz, Taurinus, another mathematician who was then studying the same geometry. Finally, Nikolai Lobachevsky published the complete system of hyperbolic geometry around 1830, where he had altered the parallel postulate by stating the existence of infinitely many lines passing through a point which are parallel to a given line (see [5]). In the 19th century, mathematicians started studying hyperbolic geometry extensively, and it is still being actively studied by researchers around the world. Hyperbolic geometry has close connections with a number of different fields, which include Abstract Algebra, Number theory, Differential geometry, and Low-dimensional Topology.
HYPERBOLIC PARALLEL POSTULATE For any line and any point not on that line, there are two lines on that point that are parallel to the original line. For more specificity, let ℓ be the line and let P be the point not on the line and let m and n be two lines on P that are parallel to ℓ.
PROPERTIES OF HYPERBOLIC TRIANGLES Hyperbolic triangles have some properties that are analogous to those of triangles in Euclidean geometry: Each hyperbolic triangle has an inscribed circle but not every hyperbolic triangle has a circumscribed circle (see below). Its vertices can lie on a horocycle or hypercycle. Hyperbolic triangles have some properties that are analogous to those of triangles in spherical or elliptic geometry: Two triangles with the same angle sum are equal in area. There is an upper bound for the area of triangles. There is an upper bound for radius of the inscribed circle. Two triangles are congruent if and only if they correspond under a finite product of line reflections. Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).
PROPERTIES OF HYPERBOLIC TRIANGLES Hyperbolic triangles have some properties that are the opposite of the properties of triangles in spherical or elliptic geometry: The angle sum of a triangle is less than 180°. The area of a triangle is proportional to the deficit of its angle sum from 180°. Hyperbolic triangles also have some properties that are not found in other geometries: Some hyperbolic triangles have no circumscribed circle, this is the case when at least one of its vertices is an ideal point or when all of its vertices lie on a horocycle or on a one sided hypercycle. Hyperbolic triangles are thin, there is a maximum distance δ from a point on an edge to one of the other two edges. This principle gave rise to δ-hyperbolic space.
NON-EXISTENCE OF RECTANGLES NON-EXISTENCE OF RECTANGLES 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. 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.
Saccheri' s Quadrilateral is a Quadrilateral with: a pair of parallel lines that are congruent, a base that is perpendicular to both parallel lines, and a summit connected to the top of the parallel lines. LAMBERT AND SACCHERI QUADRILATERALS LAMBERT AND SACCHERI QUADRILATERALS A Lambert Quadrilateral is on such that at least three Angles are Right Angles.
Theorem 3.1: The diagonals of a Saccheri quadrilateral are congruent. Theorem 3.2: The summit angles of a Saccheri quadrilateral are congruent. Theorem 3.3: The summit angles of a Saccheri quadrilateral are not obtuse and thus are both acute or both right. Theorem 3.4: The line joining the midpoints of both the summit and the base of a Saccheri quadrilateral is perpendicular to both. Theorem 3.5: The summit and base of a Saccheri quadrilateral are parallel. LAMBERT AND SACCHERI QUADRILATERALS LAMBERT AND SACCHERI QUADRILATERALS
Theorem 3.6: In any Sacherri quadrilateral the length of the summit is greater than or equal to the length of the base. Theorem 3.7: The fourth angle of a Lambert quadrilateral is not obtuse and thus is acute or right. Theorem 3.8: The measure of the line joining the midpoints of the base and the summit of a Saccheri quadrilateral is less than or equal to the measure of its sides. LAMBERT AND SACCHERI QUADRILATERALS LAMBERT AND SACCHERI QUADRILATERALS
ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY I'll analyze a few statements related to hyperbolic geometry: "In hyperbolic geometry, parallel lines intersect." Analysis: This statement is true in hyperbolic geometry. In hyperbolic space, parallel lines do not remain equidistant; instead, they converge. As a result, they eventually intersect, in contrast to Euclidean geometry where parallel lines never meet. "Hyperbolic geometry is a model of curved space." Analysis: True. Hyperbolic geometry provides a mathematical model for spaces with constant negative curvature. It' s an example of a nonEuclidean geometry, in contrast to Euclidean geometry, which assumes a flat or zero-curvature space.
ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY The sum of angles in a hyperbolic triangle is less than 180 degrees." Analysis: True. In hyperbolic geometry, the sum of angles in a triangle is always less than 180 degrees. This is a departure from Euclidean geometry where the sum of angles in a triangle is always 180 degrees. "Hyperbolic geometry has no practical applications in the real world." Analysis: This statement is not entirely accurate. While hyperbolic geometry might not be as directly applicable in everyday situations as Euclidean geometry, it finds applications in certain areas, including physics, especially in the study of spacetime in general relativity, and in some areas of art and design
ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY "Hyperbolic geometry is consistent with the parallel postulate." Analysis: False. Hyperbolic geometry is a nonEuclidean geometry precisely because it rejects or modifies the parallel postulate. In hyperbolic geometry, there are multiple parallel lines through a given point that do not intersect with a given line. "Circles in hyperbolic geometry are similar to circles in Euclidean geometry." Analysis: Somewhat true. In hyperbolic geometry, circles exhibit different properties compared to Euclidean circles. While they share some similarities, such as having a center and a radius, hyperbolic circles also have distinct characteristics, such as non-constant curvature.
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