7.1 Axioms and basic properties of Elliptic Geometry
2 min read•july 22, 2024
challenges our Euclidean assumptions. It's a world where parallel don't exist, and all lines eventually meet. Picture a sphere where act as "straight" lines, always intersecting at two points.
In this geometric realm, triangles have angle sums exceeding 180°, and their areas relate directly to this excess. Rectangles and squares? They're impossible here. It's a mind-bending shift from our flat-plane thinking.
Axioms and Basic Properties of Elliptic Geometry
Axioms of elliptic geometry
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- Elliptic geometry founded on axioms differing from Euclidean geometry
- replaced by axiom stating given a line and a point not on , no lines through parallel to
- Other axioms remain the same such as existence of through two distinct points
- Sphere surface serves as model for elliptic geometry
- Lines represented by great circles on the sphere (equator, meridians)
- Two lines always intersect at exactly one point on the sphere
Proofs in elliptic geometry
- always exceeds in elliptic geometry
- Proven using axioms and properties of great circles on a sphere
- Triangle area proportional to excess of angle sum over
- derived from axioms relates area to angle sum
- Rectangles and squares do not exist in elliptic geometry
- Follows from triangle angle sum always greater than
Properties of lines and angles
- Lines are great circles on a sphere
- Great circles formed by intersection of sphere with plane through its center (slicing sphere in half)
- Any two distinct points on sphere determine unique great circle
- measured by dihedral angle between planes forming great circles
- Angle between lines equals angle between tangent vectors at intersection point
- Parallel lines nonexistent in elliptic geometry
- Any two distinct great circles on sphere intersect at exactly one point
Antipodal points on spheres
- are diametrically opposite pairs on a sphere
- For point on sphere, antipodal point obtained by drawing line through sphere center and , finding intersection on opposite side
- Antipodal points identified as single point in elliptic geometry
- Results in lines (great circles) closing back on themselves to form loops
- Guarantees any two lines intersect at exactly one point
- Identifying antipodal points impacts properties of figures
- Triangles have maximum area equal to one-eighth of sphere surface area
Key Terms to Review (15)
Angle of elevation: The angle of elevation is the angle formed between the horizontal line from the observer's eye and the line of sight to an object above that line. It is an important concept in both elliptic geometry and spherical trigonometry, as it helps in understanding how angles interact with curved surfaces and how they relate to distances and heights in these non-Euclidean contexts.
Angles: Angles are formed when two rays originate from a common point, known as the vertex, and they are fundamental in understanding geometric relationships. In non-Euclidean geometries, such as elliptic geometry, angles behave differently compared to Euclidean geometry, leading to unique properties and implications. The sum of angles in triangles and their relationships to curvature are essential concepts that connect angles to broader geometric principles.
Antipodal Points: Antipodal points are pairs of points on a surface that are located directly opposite each other, such that a straight line connecting them would pass through the center of the surface. This concept is particularly significant in non-Euclidean geometries, where it helps in understanding the structure and properties of different surfaces, especially in spherical and elliptic geometries, as well as in projective models.
Congruence: Congruence refers to the concept of two geometric figures being identical in shape and size. In the context of elliptic geometry, this concept takes on a unique perspective, as it is influenced by the properties of curved space. Understanding congruence involves exploring how figures relate to each other in a non-Euclidean setting, which emphasizes the differences in relationships between angles and lengths compared to traditional Euclidean geometry.
Distance on a sphere: Distance on a sphere refers to the measurement of the shortest path between two points on the surface of a sphere, typically calculated along the great circle that connects them. This concept is crucial in understanding the unique properties of spherical geometry, particularly in elliptic geometry where all lines (great circles) eventually intersect and parallel lines do not exist. Grasping this notion helps clarify how distances differ from those in Euclidean space, where straight lines are the norm.
Elliptic Geometry: Elliptic geometry is a type of non-Euclidean geometry where the parallel postulate does not hold, and there are no parallel lines—any two lines will eventually intersect. This geometry describes a curved surface, like that of a sphere, where the usual rules of Euclidean geometry are altered, impacting our understanding of concepts such as distance and angle.
Gauss-Bonnet Formula: The Gauss-Bonnet Formula is a fundamental result in differential geometry that connects the topology of a surface with its geometric properties, specifically relating the total Gaussian curvature of a surface to its Euler characteristic. This formula illustrates how the intrinsic geometry of a surface, through curvature, correlates with topological features, such as the number of holes. In the context of elliptic geometry, it emphasizes that closed surfaces, like spheres, have a positive curvature and their Euler characteristic is always positive.
Great Circles: Great circles are the largest circles that can be drawn on a sphere, defined as the intersection of the sphere with a plane that passes through the center of the sphere. They play a crucial role in various types of geometry, especially in understanding the unique properties of non-Euclidean spaces like elliptic geometry, where lines are represented as great circles. These circles help in examining angles, distances, and navigation on spherical surfaces.
Lines: In geometry, a line is defined as a straight one-dimensional figure that extends infinitely in both directions, having no endpoints. Lines are fundamental elements in geometry, serving as the basis for various constructions and theorems. The concept of lines is crucial in understanding geometric relationships, especially in the context of both Euclidean and non-Euclidean geometries.
Parallel Postulate: The Parallel Postulate is a foundational statement in Euclidean geometry which asserts that if a line is drawn parallel to one side of a triangle, it will not intersect the other two sides. This postulate underpins many concepts in geometry, influencing our understanding of space, the development of non-Euclidean geometries, and the philosophical discussions surrounding the nature of mathematical truth.
Positive Curvature: Positive curvature refers to a geometric property where the surface curves outward, resembling the shape of a sphere. This concept is central to understanding the nature of spaces in which parallel lines eventually converge, leading to unique properties such as the inability to have more than one parallel line through a point not on a given line. In this context, positive curvature helps define the foundational principles and behaviors of geometries that operate under these rules.
Similarity in Spherical Geometry: Similarity in spherical geometry refers to the property where two shapes can be transformed into one another through a series of rotations and dilations without altering their angular relationships. This concept is crucial in understanding how shapes behave on the surface of a sphere, where traditional notions of similarity from Euclidean geometry do not directly apply, leading to unique properties and definitions specific to spherical contexts.
Spherical triangles: Spherical triangles are the triangles formed on the surface of a sphere, where the vertices are points on the sphere and the sides are arcs of great circles. Unlike Euclidean triangles, spherical triangles have unique properties, such as the sum of their angles exceeding 180 degrees and varying relationships between their angles and sides, reflecting the curvature of the sphere. These properties connect deeply to the principles of elliptic geometry, offering insights into how traditional Euclidean concepts adapt in non-Euclidean contexts.
Triangle Angle Sum: The Triangle Angle Sum refers to the fundamental property that states the sum of the angles in a triangle is always 180 degrees in Euclidean geometry. However, in Non-Euclidean geometries, this property changes. For instance, in elliptic geometry, the sum of the angles in a triangle exceeds 180 degrees, highlighting the distinctions between different geometrical frameworks and their axiomatic bases.
Unique Line: A unique line is a fundamental concept in geometry that states that through any two distinct points, there exists exactly one line that connects them. This idea is pivotal in understanding different geometric systems, where the nature of lines can vary significantly. In certain geometries, such as elliptic and projective, this definition can take on deeper meanings and implications, particularly regarding the relationships between points and lines.