🥎Non-Euclidean Geometry Unit 8 – Elliptic Geometry: Triangles & Trigonometry

Key Concepts

• Elliptic geometry is a non-Euclidean geometry that studies the properties of figures on the surface of a sphere
• Assumes all lines are "great circles" on the sphere, which are the largest possible circles that can be drawn on the surface
• Parallel lines do not exist in elliptic geometry since all lines eventually intersect
• The sum of the angles in a triangle is always greater than 180 degrees
  • Depends on the size of the triangle relative to the sphere
• Trigonometric functions and identities have modified forms in elliptic geometry
• Has applications in fields such as cartography, navigation, and cosmology
• Contrasts with Euclidean geometry, where parallel lines exist and the sum of angles in a triangle is always 180 degrees

Historical Context

• Elliptic geometry emerged in the early 19th century as mathematicians explored alternatives to Euclidean geometry
• Developed independently by Bernhard Riemann and Ludwig Schläfli
  • Riemann introduced the concept of curved spaces in his 1854 lecture "On the Hypotheses Which Lie at the Foundations of Geometry"
  • Schläfli studied the properties of spherical triangles and polygons in his 1852 work "Theorie der vielfachen Kontinuität"
• Motivated by the desire to prove Euclid's parallel postulate from his other axioms
  • Attempts to do so led to the realization that consistent geometries could be constructed without the parallel postulate
• Influenced by earlier work on spherical geometry by mathematicians such as Leonhard Euler and Adrien-Marie Legendre
• Played a crucial role in the development of Riemannian geometry and the theory of general relativity in the early 20th century

Axioms and Foundations

• Elliptic geometry is based on a modified set of axioms compared to Euclidean geometry
• Retains most of Euclid's axioms, such as the existence of straight lines and the ability to construct circles
• Replaces the parallel postulate with the axiom that any two lines must intersect
  • In Euclidean geometry, the parallel postulate states that given a line and a point not on the line, there is exactly one line through the point that does not intersect the given line
• Assumes that space is finite and unbounded, meaning that there are no edges or boundaries, but the total area is finite
• Can be modeled on the surface of a sphere, where lines are great circles and points are represented by pairs of antipodal points
• The distance between two points is the length of the shorter arc of the great circle connecting them
• Angle measures are defined using the dihedral angle between the planes containing the great circles that form the angle

Elliptic Geometry Basics

• In elliptic geometry, the shortest path between two points is along the great circle that connects them
• The distance between two points is proportional to the angle subtended by the arc connecting them at the center of the sphere
• Circles in elliptic geometry have modified properties compared to Euclidean circles
  • The ratio of a circle's circumference to its radius is always less than $2\pi$
  • The area of a circle is proportional to the square of its radius, but with a smaller constant of proportionality than in Euclidean geometry
• Elliptic geometry has no concept of similarity, as there is an absolute scale determined by the size of the sphere
• Angles in elliptic geometry are measured in radians, with a complete rotation corresponding to $2\pi$ radians
• The area of a triangle in elliptic geometry is proportional to the excess of its angle sum over $\pi$ radians (180 degrees)

Triangles in Elliptic Space

• Triangles in elliptic geometry have angle sums greater than 180 degrees
• The excess of the angle sum over 180 degrees is proportional to the area of the triangle
  • Larger triangles have greater angle sums, up to a maximum of 540 degrees for a triangle covering an entire hemisphere
• The side lengths of a triangle determine its angles, and vice versa, unlike in Euclidean geometry where similar triangles can have different sizes
• The trigonometric laws for triangles, such as the sine and cosine laws, have modified forms in elliptic geometry
  • Involves using the spherical excess and the spherical functions (sin, cos, tan) instead of their planar counterparts
• Special types of triangles, such as right triangles and equilateral triangles, have unique properties in elliptic space
  • The angles of a right triangle satisfy different relationships compared to Euclidean right triangles
  • Equilateral triangles have angle sums greater than 180 degrees and cannot be arbitrarily large

Trigonometric Relationships

• Trigonometric functions in elliptic geometry are defined using the great circles on the sphere
• The spherical sine, cosine, and tangent functions are used to relate the sides and angles of spherical triangles
  • Defined in terms of the angles and distances along the surface of the sphere, rather than in terms of ratios of side lengths
• The spherical law of sines relates the sides and angles of a spherical triangle
  • $\frac{\sin(a)}{\sin(A)} = \frac{\sin(b)}{\sin(B)} = \frac{\sin(c)}{\sin(C)}$, where $a$, $b$, $c$ are the side lengths and $A$, $B$, $C$ are the opposite angles
• The spherical law of cosines relates the cosine of an angle to the cosines of the sides
  • $\cos(A) = -\cos(B)\cos(C) + \sin(B)\sin(C)\cos(a)$
• Spherical trigonometric identities, such as the half-angle and addition formulas, have modified forms compared to their Euclidean counterparts
• Napier's rules provide a mnemonic device for remembering the spherical trigonometric identities
  • Involves arranging the parts of a right spherical triangle in a circle and using the relationships between them

Practical Applications

• Elliptic geometry has numerous applications in fields that involve modeling phenomena on spherical surfaces
• Used in cartography and navigation to accurately represent distances and directions on the Earth's surface
  • Enables the creation of maps that minimize distortion and preserve important properties (Mercator projection, gnomonic projection)
  • Allows for precise calculations of shortest paths and great circle routes for ships and aircraft
• Employed in astronomy and cosmology to study the large-scale structure and geometry of the universe
  • Some cosmological models posit that the universe has a spherical or elliptic geometry on the largest scales
  • Helps analyze the properties of cosmic microwave background radiation and test theories of cosmic inflation
• Relevant to the design of certain types of antennas and optical systems that involve spherical or hemispherical surfaces
• Used in computer graphics and virtual reality to create immersive environments and render realistic scenes on spherical displays
• Helps in understanding the behavior of physical systems confined to spherical surfaces, such as the motion of particles on a sphere

Comparisons with Euclidean Geometry

• Elliptic geometry is a non-Euclidean geometry that differs from Euclidean geometry in several key aspects
• In Euclidean geometry, parallel lines exist and the sum of angles in a triangle is always 180 degrees
  • Elliptic geometry has no parallel lines and triangle angle sums are greater than 180 degrees
• Euclidean geometry extends infinitely in all directions, while elliptic geometry is finite and unbounded
  • Elliptic geometry has a maximum distance between points, equal to half the circumference of the sphere
• The Pythagorean theorem holds in Euclidean geometry but takes a modified form in elliptic geometry
  • Involves the spherical excess and the spherical trigonometric functions
• Euclidean geometry has a concept of similarity, where figures can be scaled up or down while preserving their shape
  • Elliptic geometry lacks similarity, as the size of figures is constrained by the size of the sphere
• Many formulas and relationships in Euclidean geometry, such as the circumference and area of a circle, have different forms in elliptic geometry
• Despite these differences, elliptic geometry shares some common axioms with Euclidean geometry and can be seen as a generalization of spherical geometry