Hyperbolic geometry By Loheshwar Jayakumar
INTRODUCTION TO HYPERBOLIC GEOMETRY Hyperbolic geometry, also known as non-Euclidean geometry, is a fascinating branch of mathematics that explores the properties of geometric spaces where the parallel postulate of Euclidean geometry does not hold. Here's a basic introduction to hyperbolic geometry: --> Hyperbolic Space: In hyperbolic geometry, space is negatively curved, unlike the flat space of Euclidean geometry. This negative curvature leads to some intriguing properties, such as the existence of multiple parallel lines through a point not on a given line and the non-existence of similar triangles with congruent angles. --> Hyperbolic Models: Several models help us visualize hyperbolic space. The most common are the Poincaré disk model, the Poincaré half-plane model, and the hyperboloid model. Each model provides a different perspective on hyperbolic space and aids in understanding its properties and relationships. --> Distance and Angles: Hyperbolic distance grows exponentially as you move away from a point, and the sum of angles in a triangle is always less than 180 degrees, contrary to Euclidean geometry. --> Applications: Hyperbolic geometry has applications in various fields, including physics, cosmology, art, and computer science.
HYPERBOLIC PARALLEL POSTULATE Hyperbolic Parallel Postulate, also known as the Lobachevskian or Saccheri-Legendre Axiom, is a fundamental principle in hyperbolic geometry. There exists a line and a point not on that line such that there are two lines on that point that are parallel to the original line. For any line and any point not on that line, there are two lines on that point that are parallel to the original line. These two axioms are equivalent; i.e., one is true if and only if the other is true. Proof: Obviously, the second implies the first so there is nothing to prove in this direction. Conversely, assume the local form is true, say line ℓℓ and point PP with m� and n� distinct lines on P parallel to ℓℓ. First we prove a special case, multiple parallels at any point P′�′ on the line on P� perpendicular to ℓℓ, say PF, where F� is the foot of the perpendicular from P.
PROPERTIES OF HYPERBOLIC TRIANGLES Angle Sum: In hyperbolic geometry, the sum of the interior angles of a triangle is always less than 180 degrees, in contrast toEuclidean geometry where it equals 180 degrees. The angle sum depends on the area of the triangle and can be expressedas π−A where A is the area of the triangle measured in square units of hyperbolic space. Side Lengths: In hyperbolic geometry, side lengths are hyperbolic segments. The lengths of sides are often measured interms of hyperbolic distance or hyperbolic length, which are different from the lengths measured in Euclidean geometry. Hyperbolic distances grow exponentially as you move away from a given point. Congruence: Two hyperbolic triangles are congruent if their corresponding sides and angles are equal. However, hyperboliccongruence is more restrictive than in Euclidean geometry due to the non-Euclidean nature of space. Similarity: Similarity in hyperbolic triangles is defined similarly to Euclidean triangles, where corresponding angles areequaland corresponding sides are proportional. However, the ratios of side lengths may differ due to the non-Euclidean natureofhyperbolic space. Trigonometric Relationships: Hyperbolic trigonometry involves hyperbolic functions such as hyperbolic sine, cosine, andtangent. These functions govern the relationships between the sides and angles of hyperbolic triangles. Hyperbolic Area: The area of a hyperbolic triangle is given by its angle excess, which is the difference between the sumof itsangles and �π (the angle sum of a Euclidean triangle). The area can be calculated using various formulas, including theGauss-Bonnet formula for hyperbolic polygons. Hyperbolic Parallel Postulate: Unlike in Euclidean geometry, in hyperbolic geometry, there exist infinitely many lines parallel to a given line through a point not on the given line. This property affects the construction and properties of hyperbolic triangles. EAS T CORDAL E S C HOOL
Non-existence of rectangles In hyperbolic geometry, rectangles as defined in Euclidean geometry do not exist. In Euclidean geometry, a rectangle is a quadrilateral with four right angles. However, in hyperbolic geometry, due to the non-Euclidean nature of space, the sum of angles in a quadrilateral can be less than 360 degrees. Here's why rectangles don't exist in hyperbolic geometry: Angle Sum: In hyperbolic geometry, the sum of angles in a quadrilateral is always less than 360 degrees. This property is a consequence of the Hyperbolic Parallel Postulate, which allows for the existence of multiple parallel lines through a point not on a given line. As a result, the angles of a quadrilateral in hyperbolic geometry can be smaller than those of a Euclidean rectangle. No Right Angles: Since the sum of angles in a hyperbolic quadrilateral is less than 360 degrees, it's impossible for all four angles to be right angles (90 degrees) simultaneously. Thus, by definition, rectangles do not exist in hyperbolic geometry.
Lambert and Saccheri quadrilaterals Lambert Quadrilateral: A Lambert quadrilateral is a quadrilateral in hyperbolic geometry with three right angles and one acute angle. The acute angle is called the "Lambert angle. " This type of quadrilateral was first studied by Johann Heinrich Lambert in the 18th century. Properties of Lambert quadrilaterals: Three of its angles are right angles. One angle (the Lambert angle) is acute. The sum of its angles is less than 360 degrees.
Saccheri Quadrilateral: A Saccheri quadrilateral is a quadrilateral with two equal sides (congruent legs) and two right angles. The two equal sides are opposite each other. Saccheri quadrilaterals were extensively studied by Giovanni Girolamo Saccheri in the 18th century as part of his investigation into the properties of parallel lines in hyperbolic geometry. Properties of Saccheri quadrilaterals: Two opposite angles are right angles. The other two angles are not necessarily equal but are congruent in hyperbolic space. The sum of its angles is less than 360 degrees.
ANALYSIS OF STATEMENTS RELATED TO HYPERBOLIC GEOMETRY In hyperbolic geometry, the sum of the angles in a triangle is less than 180 degrees." Analysis: This statement is true. In hyperbolic geometry, the sum of the angles in a triangle is inversely related to the triangle's area. As the area of the triangle increases, the sum of its angles decreases. This is a fundamental property of hyperbolic geometry and stands in contrast to Euclidean geometry, where the sum of the angles in a triangle is always 180 degrees. "Hyperbolic geometry exhibits negative curvature." Analysis: This statement is true. In hyperbolic geometry, the curvature of space is negative. This negative curvature is evident in properties such as the existence of multiple parallel lines through a point not on a given line and the non-existence of similar triangles with congruent angles
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