🥎Non-Euclidean Geometry Unit 10 – Spherical Geometry: Concepts and Uses

Key Concepts in Spherical Geometry

• Spherical geometry studies geometric properties on the surface of a sphere
• Lines in spherical geometry are great circles, the largest possible circles on the sphere's surface
• Angles in spherical geometry are defined by the angle between two great circles at their point of intersection
• Spherical triangles are formed by three points connected by great circle arcs
  • The angles of a spherical triangle always sum to more than 180°
• Spherical polygons are regions on the sphere bounded by great circle arcs
• Spherical distance is measured along the shortest path on the surface of the sphere (great circle) and is typically expressed in angular units (radians or degrees)
• Spherical coordinates $(r, \theta, \phi)$ uniquely identify points on a sphere using radial distance $r$, polar angle $\theta$, and azimuthal angle $\phi$

Differences from Euclidean Geometry

• In Euclidean geometry, parallel lines never intersect, while in spherical geometry, all great circles intersect at two antipodal points
• The sum of the angles in a spherical triangle is always greater than 180°, unlike in Euclidean geometry, where it is exactly 180°
• There are no similar triangles in spherical geometry, as all spherical triangles with the same angle measures are congruent
• The area of a spherical triangle is proportional to the excess of the sum of its angles over 180°, known as the spherical excess
• In spherical geometry, the shortest path between two points is along a great circle, while in Euclidean geometry, it is a straight line
• Spherical geometry has no concept of similarity, as there is no way to scale figures on the surface of a sphere without distorting their shape
• The Pythagorean theorem does not hold in spherical geometry; instead, the spherical law of cosines relates the sides and angles of spherical triangles

Spherical Trigonometry Basics

• Spherical trigonometry deals with the relationships between the sides and angles of spherical triangles
• The spherical law of cosines relates the cosine of a side to the cosines and sines of the other sides and the cosine of the opposite angle: $\cos a = \cos b \cos c + \sin b \sin c \cos A$
• The spherical law of sines states that the ratio of the sine of a side to the sine of the opposite angle is constant for all sides and angles in a spherical triangle: $\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B} = \frac{\sin c}{\sin C}$
• Napier's rules provide a mnemonic device for remembering the relationships between the parts of a right-angled spherical triangle
  • Napier's rules involve circular permutations of the parts of the triangle and their complements
• The spherical excess $E$ of a spherical triangle is given by $E = A + B + C - \pi$, where $A$, $B$, and $C$ are the angles of the triangle in radians
• The area of a spherical triangle with spherical excess $E$ on a sphere of radius $r$ is given by $A = Er^2$
• Spherical trigonometry has applications in navigation, astronomy, and geodesy

Theorems and Proofs on the Sphere

• Girard's theorem states that the area of a spherical triangle is proportional to its spherical excess: $A = Er^2$, where $E$ is the spherical excess and $r$ is the radius of the sphere
• Lexell's theorem states that the locus of points on a sphere equidistant from two given points is a great circle perpendicular to the great circle passing through the two points
• The spherical Pythagorean theorem for right-angled spherical triangles: $\cos c = \cos a \cos b$
• The spherical Menelaus' theorem relates the lengths of the sides of a spherical triangle and the great circle segments formed by extending the sides
• The spherical Ceva's theorem states that three great circle arcs connecting the vertices of a spherical triangle to points on the opposite sides concur at a single point if and only if a specific ratio condition holds
• The spherical law of cosines for angles: $\cos A = -\cos B \cos C + \sin B \sin C \cos a$
• The spherical Euler line theorem states that the centroid, circumcenter, and orthocenter of a spherical triangle lie on a single great circle

Applications in Navigation and Cartography

• Great circle navigation involves finding the shortest path between two points on the Earth's surface, which is a great circle route
• Rhumb lines, or loxodromes, are curves on the sphere that maintain a constant angle with meridians, making them useful for navigation with a compass
• Spherical trigonometry is used to solve problems in celestial navigation, such as determining a ship's position based on the altitudes of celestial bodies
• Map projections, like the Mercator projection, aim to represent the Earth's spherical surface on a flat plane, each with its own set of distortions and preserved properties
  • The Mercator projection preserves angles and shapes but distorts areas, especially near the poles
• The gnomonic projection maps great circles to straight lines, making it useful for navigation and finding the shortest path between points
• The stereographic projection preserves angles and maps circles on the sphere to circles on the plane, with applications in cartography and crystallography
• The azimuthal equidistant projection preserves distances and directions from a central point, making it useful for radio and seismic wave propagation studies

Connections to Other Non-Euclidean Geometries

• Spherical geometry is a type of non-Euclidean geometry, along with hyperbolic geometry and elliptic geometry
• Hyperbolic geometry has a constant negative curvature, while spherical geometry has a constant positive curvature
• In elliptic geometry, lines are defined as great circles on the sphere, and there are no parallel lines, similar to spherical geometry
• The Poincaré disk and upper half-plane models provide ways to visualize hyperbolic geometry using Euclidean constructs
• The Klein model of hyperbolic geometry uses straight lines to represent hyperbolic lines, similar to how great circles represent lines in spherical geometry
• The Beltrami-Klein model of projective geometry is related to the gnomonic projection of the sphere, highlighting the connections between spherical and projective geometries
• Non-Euclidean geometries have applications in physics, particularly in the theory of general relativity, where spacetime is modeled using curved geometries

Problem-Solving Techniques

• Use the spherical laws of cosines and sines to solve for unknown sides and angles in spherical triangles
• Apply Napier's rules to solve problems involving right-angled spherical triangles
• Utilize the spherical excess formula and Girard's theorem to calculate the areas of spherical triangles and polygons
• Convert between spherical coordinates and Cartesian coordinates to solve problems involving points on the sphere
• Apply theorems like Lexell's theorem and the spherical Menelaus' theorem to solve problems involving great circle arcs and spherical triangle properties
• Use the properties of map projections, such as the Mercator and gnomonic projections, to solve navigation and cartography problems
• Break down complex spherical geometry problems into simpler sub-problems, such as solving for right-angled spherical triangles or applying known theorems
• Visualize problems using 3D models or diagrams to better understand the relationships between points, lines, and angles on the sphere

Real-World Uses and Modern Applications

• Spherical geometry is used in navigation systems, including GPS and inertial navigation, to calculate routes and distances on the Earth's surface
• Satellite and space mission planning rely on spherical trigonometry to determine orbits, coverage areas, and communication paths
• Geodesy, the science of measuring the Earth's shape and size, uses spherical geometry to model the Earth's surface and gravitational field
• Climate modeling and weather forecasting employ spherical geometry to analyze global atmospheric and oceanic circulation patterns
• Computer graphics and virtual reality systems use spherical coordinates and projections to create immersive 3D environments and map textures onto spherical objects
• Crystallography and materials science use spherical geometry to describe the arrangement of atoms and molecules in crystals and to analyze diffraction patterns
• Spherical geometry is applied in the study of geometric optics, particularly in the design of wide-angle lenses and the analysis of spherical mirror and lens systems
• Astrophysics and cosmology use spherical geometry to model the large-scale structure of the universe and to study the properties of celestial objects like planets, stars, and galaxies