The Earth is Spherical, Why Doesn't the Prayer Direction "Deviate"?
earth curvature and prayer direction, spherical trigonometry calculation, Islamic geography
Experience the precise legacy of thousand-year wisdom - use our prayer direction calculation tool developed based on classical algorithms and modern geographic information systems!
I. Spherical Geometry Principles
Great Circle concept:
- Shortest path between two points on a sphere
- All longitude lines are standard great circles
- Connection lines between Mecca and all places form unique great circle routes
Constant direction principle:
- The determination direction of each place is the tangent of the great circle from that place to Mecca
- Mathematical expression:
tan(α) = sin(Δλ) / [cos(φ₁)tan(φ₂) - sin(φ₁)cos(Δλ)]
(α: azimuth angle, φ: latitude, λ: longitude, Δλ=λ₂-λ₁)
Practical calculation case:
- Beijing (39.9°N, 116.4°E) → Mecca (21.4°N, 39.8°E):
Azimuth angle: 263.7° (West-South)
Distance: 6,817 km
At error <0.1°, deviation only 1.7 meters per 1000 km
II. Analysis of Common Misconceptions
Flat projection fallacy:
- Direction illusion caused by Mercator projection map
- Comparison of "straight line" differences in different projection methods:
Mercator projection: appears curved
Spherical projection: actual great circle route
Dynamic compensation mechanism:
- Earth curvature was already integrated into classical Islamic mathematics
- Al-Biruni (11th century) solved this problem in "Determination of Mecca Coordinates"
- Modern GPS automatically performs ellipsoid correction (WGS84 model)
III. Special Geographic Situations
Polar regions:
- North Pole: All southern directions are determination direction
- Antarctica: Must calculate great circle route to nearest north
- Case: Zhongshan Station, Antarctica (69.4°S, 76.4°E) → Mecca azimuth: 342.3°
Antipodes regions:
- Theoretically farthest from Mecca (Pacific near Tahiti)
- All directions are correct (special legal provision)
Mecca same-longitude cities:
- Direct north or south direction (like Jeddah, St. Petersburg)
- No curvature compensation calculation required