Spherical Geometry
Spherical geometry is the study of figures on the surface of a sphere, where the analogue of a straight line is a great circle (the intersection of the sphere with a plane through its centre). Unlike Euclidean geometry, the angles of a spherical triangle sum to more than 180°, parallel lines do not exist, and the shortest path between two points is along a great-circle arc. It is essential in navigation (great-circle routes), astronomy (celestial coordinates), geodesy (Earth's shape and GPS calculations), and general relativity.
Key Formula
D = R · arccos(sin(φ₁)sin(φ₂) + cos(φ₁)cos(φ₂)cos(Δλ))
LaTeX: D = R \arccos(\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos\Delta\lambda)
| Symbol | Meaning | Unit |
|---|---|---|
| D | great-circle distance between two points on the sphere | km or m |
| R | radius of the sphere (Earth: 6371 km) | km |
| φ₁, φ₂ | latitudes of the two points | radians |
| Δλ | difference in longitudes of the two points (λ₂ − λ₁) | radians |
Worked Example
Problem
Calculate the great-circle distance between Mumbai (19.08°N, 72.88°E) and Delhi (28.61°N, 77.21°E). Use R = 6371 km.
Solution
Step 1: Convert to radians: φ₁ = 19.08° × π/180 = 0.3331 rad; φ₂ = 28.61° × π/180 = 0.4994 rad. Δλ = (77.21 − 72.88)° × π/180 = 4.33° × π/180 = 0.07557 rad. Step 2: Compute: sin(φ₁)sin(φ₂) = sin(0.3331)×sin(0.4994) = 0.3268 × 0.4790 = 0.1565. cos(φ₁)cos(φ₂)cos(Δλ) = cos(0.3331)×cos(0.4994)×cos(0.07557) = 0.9450 × 0.8777 × 0.9971 = 0.8269. Step 3: Sum = 0.1565 + 0.8269 = 0.9834. Step 4: D = 6371 × arccos(0.9834) = 6371 × 0.1829 rad = 1165 km.
Answer
Great-circle distance from Mumbai to Delhi ≈ 1165 km.
Euclidean vs Spherical Geometry — Key Differences
| Property | Euclidean Geometry | Spherical Geometry |
|---|---|---|
| Straight line analogue | Infinite straight line | Great circle (finite, closed) |
| Triangle angle sum | Exactly 180° | Between 180° and 540° (exclusive) |
| Parallel lines | Exactly one through any point | None — all great circles intersect |
| Area of triangle | Depends on base and height | Depends on angular excess (E = A+B+C−π)·R² |
| Shortest path | Straight line segment | Great-circle arc |
| Pythagoras theorem | a² + b² = c² | cos(c/R) = cos(a/R)·cos(b/R) |
Interactive Tools
GeoGebra 3D Calculator
Construct spherical triangles and great-circle paths on an interactive 3D sphere.
Open ToolWolfram Alpha
Compute great-circle distances, spherical triangle properties, and geodesic calculations.
Open ToolKhan Academy — Geometry
Foundational Euclidean geometry to contextualise how spherical geometry differs.
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Polar Coordinates
Polar coordinates are a two-dimensional coordinate system in which each point in the plane is specified by a radial distance r from a fixed origin (pole) and an angle θ measured from a fixed reference direction (polar axis), written as the ordered pair (r, θ). Unlike Cartesian coordinates that use perpendicular axes, polar coordinates are natural for describing curves with rotational symmetry such as circles, spirals, roses, and limaçons. They are widely used in physics (orbital mechanics, wave interference), engineering (antenna patterns), and complex number representation.
Law of Sines
The Law of Sines (also called the Sine Rule) states that in any triangle, the ratio of the length of each side to the sine of the opposite angle is the same constant, equal to the diameter of the triangle's circumscribed circle. This law applies to all triangles, not just right triangles, making it a powerful tool for solving oblique triangles when two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA) are known. It is fundamental in surveying, navigation, and geodesy.
Parabola (geometry)
A parabola is a symmetric open curve formed by the intersection of a cone with a plane parallel to one of its sides, defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). It is a conic section with eccentricity exactly equal to 1, placing it between the ellipse and the hyperbola. Parabolas appear extensively in physics (projectile paths, parabolic mirrors, antenna reflectors) and engineering design.
From the Greek sphaira (σφαῖρα, "ball, globe") + Latin geometria (from Greek geōmetria, "earth measurement"). The formal study of spherical geometry was developed by Menelaus of Alexandria (c. 100 AD) in his Sphaerica, building on earlier Babylonian and Greek astronomical work. The term "spherical geometry" gained currency in 17th–19th century European mathematics.