Hyperbolic Geometry
Hyperbolic Geometry is geometry in which Euclid’s fifth postulate is replaced with the following:
Given a line l, and a point A, not lying on l, there exists at least two lines through A that are parallel to line l.
Hyperbolic geometry takes place on a curved 2-dimensional surface called hyperbolic space.
In hyperbolic space, every point looks like a saddle.
Terms used in the modules will be defined as follows:
Line segment: The segment AB, consists of the points A and B and all the points on line AB that are between A and B
Circle: The set of all points, P, that are a fixed distance from a fixed point, O, called the center of the circle.
Parallel lines: Two lines, l and m are parallel if they do not intersect.
Polygons: A sequence of points and geodesic segments joining those points
Given a line l, and a point A, not lying on l, there exists at least two lines through A that are parallel to line l.
Hyperbolic geometry takes place on a curved 2-dimensional surface called hyperbolic space.
In hyperbolic space, every point looks like a saddle.
Terms used in the modules will be defined as follows:
Line segment: The segment AB, consists of the points A and B and all the points on line AB that are between A and B
Circle: The set of all points, P, that are a fixed distance from a fixed point, O, called the center of the circle.
Parallel lines: Two lines, l and m are parallel if they do not intersect.
Polygons: A sequence of points and geodesic segments joining those points
Angle Measures: The measure of an angle is the radian measure of the angle formed by
• The tangent rays at a point of intersection of two arcs, or
• An ordinary ray and the tangent ray at a point of intersection of the arc and the ordinary ray
• The tangent rays at a point of intersection of two arcs, or
• An ordinary ray and the tangent ray at a point of intersection of the arc and the ordinary ray
Models for Studying Hyperbolic Geometry.
Models are useful for visualizing and exploring the properties of geometry. A number of models exist for exploring the geometric properties of the hyperbolic plane. It should be pointed out however, that these models do not “look like” the hyperbolic plane. The models merely serve as a means of exploring the properties of the geometry
• The Beltrami-Klein Model for Studying Hyperbolic Geometry.
• The Poincaré Half Plane Model for Studying Hyperbolic Geometry.
• The Poincaré Disk Model for Studying Hyperbolic Geometry. (This is the model we will be using)
The Poincaré Disk Model for Studying Hyperbolic Geometry.
Henri Poincaré (1854 – 1912) developed a disk model that represents points in the hyperbolic plane as points in the interior of a Euclidean circle. In this model, lines are not straight as the student is used to seeing them on the Euclidean plane. Instead, lines are represented by arcs of circles that are orthogonal to the circle defining the disk. In this model therefore, the only lines that appear to be straight in the Euclidean sense are diameters of the disk. In addition, the boundary of the circle does not really exist, and distances become distorted in this model. All the points in the interior of the circle are part of the hyperbolic plane. In this plane, two points lie on a “line” if the “line” forms an arc of a circle orthogonal to C. The only hyperbolic lines that are straight in the Euclidean sense are those that are diameters of the circle.
Models are useful for visualizing and exploring the properties of geometry. A number of models exist for exploring the geometric properties of the hyperbolic plane. It should be pointed out however, that these models do not “look like” the hyperbolic plane. The models merely serve as a means of exploring the properties of the geometry
• The Beltrami-Klein Model for Studying Hyperbolic Geometry.
• The Poincaré Half Plane Model for Studying Hyperbolic Geometry.
• The Poincaré Disk Model for Studying Hyperbolic Geometry. (This is the model we will be using)
The Poincaré Disk Model for Studying Hyperbolic Geometry.
Henri Poincaré (1854 – 1912) developed a disk model that represents points in the hyperbolic plane as points in the interior of a Euclidean circle. In this model, lines are not straight as the student is used to seeing them on the Euclidean plane. Instead, lines are represented by arcs of circles that are orthogonal to the circle defining the disk. In this model therefore, the only lines that appear to be straight in the Euclidean sense are diameters of the disk. In addition, the boundary of the circle does not really exist, and distances become distorted in this model. All the points in the interior of the circle are part of the hyperbolic plane. In this plane, two points lie on a “line” if the “line” forms an arc of a circle orthogonal to C. The only hyperbolic lines that are straight in the Euclidean sense are those that are diameters of the circle.
This model satisfies all the axioms of incidence, betweenness, congruence, continuity, and the hyperbolic axiom of parallelism. The angle between two lines is the measure of the Euclidean angle between the tangents drawn to the lines at their points of intersection.
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