Trigonometry
Spherical Trigonometry
Цели урока
- Apply the spherical law of cosines and sines to solve spherical triangles
- Compute great-circle distances using the haversine formula
- Interpret spherical excess as the area of a spherical triangle
- Distinguish great-circle routes from rhumb lines and explain when each is used
Предварительные знания
- Plane trigonometry
- Vector dot product
- Coordinate systems
Why do transatlantic flights pass over Greenland instead of following the latitude line?
- Aviation: great-circle routing saves 12% fuel on long-haul flights
- GPS: all coordinate transformations and distance computations on WGS-84
- Astronomy: sky coordinates, spherical cap queries in survey databases
- Computer graphics: environment mapping and skybox projections use spherical geometry
History: from ancient navigation to modern GPS
Spherical trigonometry was developed by Arab mathematicians (al-Battani, 10th c.) for astronomical calculations and later formalized by Napier in 1614 with his mnemonics for right spherical triangles. The haversine formula appeared in navigation tables in the 19th century. Gauss and Bonnet connected spherical area to angle excess in 1827 - a result that generalizes to all curved surfaces. Today every GPS device solves spherical triangles hundreds of times per second.
Spherical Triangles and Core Theorems
The GPS aviation navigation algorithm uses spherical trigonometry for great-circle routing: the Moscow to New York route at 9,761 km passes over Greenland rather than along the latitude circle, saving 12% of fuel. The Boeing 787 recomputes its route every 30 seconds.
Haversine formula for great-circle distance
Numerically stable formula for geographic distances
For points (lat1, lon1) and (lat2, lon2) in radians: a = sin^2((lat2-lat1)/2) + cos(lat1)*cos(lat2)*sin^2((lon2-lon1)/2); distance = 2*R*arcsin(sqrt(a)). This avoids numerical cancellation near antipodal points.
What is the spherical excess E of a triangle with angles A, B, C on the unit sphere?
Spherical excess E = A+B+C-pi > 0. By Gauss-Bonnet, area = E*R^2 on a sphere of radius R. On the unit sphere area = E exactly.
Spherical Coordinates and Solid Angles
The James Webb Space Telescope maps the sky in spherical coordinates: right ascension and declination. Every catalog lookup, every cross-matching of two surveys, uses spherical trigonometry. A query returning all sources within 0.5 arcseconds of a given position is a spherical cap intersection - exactly the type of computation at the core of astroquery libraries.
| Formula | Plane version | Spherical version |
|---|---|---|
| Law of cosines | c^2 = a^2 + b^2 - 2ab cos C | cos c = cos a cos b + sin a sin b cos C |
| Law of sines | a/sin A = b/sin B | sin A/sin a = sin B/sin b |
| Angle sum | A + B + C = pi | A + B + C = pi + E, E > 0 |
| Area | (1/2) ab sin C | E * R^2 (spherical excess) |
What is the total solid angle of a sphere?
Integrating sin(theta) d(theta) d(phi) over theta in [0, pi] and phi in [0, 2*pi] gives 4*pi. This is why the full sky covers 4*pi steradians, and the full-sky CMB maps from Planck cover exactly that.
Navigation, Aviation, and Geodesy
Modern aviation uses spherical trigonometry constantly. Every departure procedure, every en-route waypoint, every STAR (Standard Terminal Arrival Route) is defined in terms of great-circle tracks and rhumb lines. The Mercator projection preserves angles (rhumb lines become straight) at the cost of distorting areas - Greenland looks as large as Africa, though it is 14 times smaller.
For distances under 100 km, flat-Earth approximations introduce errors under 0.1%. Beyond that, spherical trigonometry is mandatory. GPS uses the WGS-84 ellipsoid, not a perfect sphere.
Why does a great-circle route from London to Tokyo pass over northern regions rather than straight east?
On a sphere, the shortest path between two points is a great circle. For routes crossing large differences in longitude, great circles arc toward the poles. On a flat Mercator map they appear curved, though they are geodesics on the sphere.
Connections to other topics
Spherical trigonometry connects geometry, navigation, and differential geometry
- Gauss-Bonnet theorem — Related topic
- Geodesy — Related topic
- Spherical harmonics — Related topic
- Mercator projection — Related topic
Итоги
- Spherical triangles satisfy the law of cosines cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C) and the law of sines sin(A)/sin(a) = sin(B)/sin(b)
- Spherical excess E = A+B+C-pi > 0 and equals the area of the triangle on the unit sphere (Gauss-Bonnet)
- Great-circle distance: d = R*arccos(sin(phi1)sin(phi2) + cos(phi1)cos(phi2)cos(delta_lambda)); haversine is numerically stable
- Great circles (minimum distance) vs. rhumb lines (constant bearing): aviation uses great circles, traditional compass navigation used rhumb lines
Вопросы для размышления
- Why does the angle sum exceed pi in a spherical triangle, and what does the excess measure?
- When is a flat-Earth approximation adequate and when does spherical geometry become mandatory?
- How does the Mercator projection preserve angles but distort areas - and what are the mathematical consequences?