Bearing Calculator
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About Bearing Calculator
Understanding Bearings
Bearings, a fundamental concept in navigation, originated in ancient civilizations and evolved through maritime exploration. From celestial navigation using the North Star to modern GPS systems, bearing calculations remain crucial for human exploration and navigation. Persian mathematician Al-Biruni's 11th-century work on spherical trigonometry revolutionized geographical coordinate determination and bearing calculations across Earth's curved surface.
During the Age of Exploration (15th-17th centuries), Portuguese navigators refined maritime bearing calculations, leading to more accurate ocean navigation. These methods evolved with improved instruments and understanding of magnetic variation.
Mathematical Foundation
θ = atan2(sin(Δλ)×cos(φ₂), cos(φ₁)×sin(φ₂) - sin(φ₁)×cos(φ₂)×cos(Δλ))
d = acos(sin(φ₁)×sin(φ₂) + cos(φ₁)×cos(φ₂)×cos(Δλ))×R
Δσ = atan2(√[(cos(φ₂)×sin(Δλ))² + (cos(φ₁)×sin(φ₂) - sin(φ₁)×cos(φ₂)×cos(Δλ))²], sin(φ₁)×sin(φ₂) + cos(φ₁)×cos(φ₂)×cos(Δλ))
Key Variables
- θ (theta): Initial bearing angle
- φ (phi): Latitude in radians
- λ (lambda): Longitude difference
- R: Earth's mean radius (6,371 km)
- d: Great circle distance
- Δσ: Angular distance on sphere
Important Notes
- All angles must be in radians
- Accounts for Earth's spherical shape
- Uses Haversine formula principles
- Considers great circle paths
Reference Systems
True Bearings
- References true north
- Uses 000° to 359° notation
- Essential for GPS systems
- Standard in aviation
Magnetic Bearings
- References magnetic north
- Includes declination
- Changes annually
- Used in basic navigation
Grid Bearings
- Based on map grid
- Uses grid north
- Varies by projection
- Used in land surveys
Cardinal Directions
Primary Points
- North (N): 000°
- East (E): 090°
- South (S): 180°
- West (W): 270°
Intercardinal Points
- Northeast (NE): 045°
- Southeast (SE): 135°
- Southwest (SW): 225°
- Northwest (NW): 315°
Frequently Asked Questions
What is a bearing angle?
A bearing angle represents the direction from one point to another, measured clockwise from true north (0°). It's commonly used in navigation, surveying, and mapping. The angle ranges from 0° to 360°, where 0° is North, 90° is East, 180° is South, and 270° is West.
Why are my coordinates not working?
Ensure your coordinates are in decimal degrees format (e.g., 51.5074 for 51°30'26.6"N). Latitude must be between -90° and 90°, while longitude must be between -180° and 180°. Also, the same point for both start and end coordinates will not yield a valid bearing.
What's the difference between true and magnetic bearings?
True bearings are measured from true north (the Earth's rotational axis), while magnetic bearings are measured from magnetic north (where a compass points). The difference between them is called magnetic declination, which varies by location and time. This calculator provides true bearings based on geographic coordinates.
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