A geodesic is the shortest path between two points on a curved surface. On a perfect sphere this is an arc of a great circle. On the Earth's actual ellipsoidal surface (WGS-84), a geodesic is slightly more complex — it is the curve whose length cannot be reduced by any small perturbation, computed using ellipsoidal geometry. For most navigation purposes, great-circle approximations are sufficient.

WGS-84 (World Geodetic System 1984) is the standard reference ellipsoid used by GPS and most modern mapping systems. It defines the Earth's shape with: semi-major axis a = 6,378,137.0 m, semi-minor axis b ≈ 6,356,752.314 m, and flattening f = 1/298.257223563. The difference between the equatorial and polar radii (21 km) is the oblateness that makes ellipsoidal calculations like Vincenty necessary for high-precision work.

What is a bearing in navigation?

A bearing is a compass direction expressed as an angle clockwise from True North, ranging from 0° to 360°. North = 0° (or 360°), East = 90°, South = 180°, West = 270°. A bearing of 045° means northeast. Bearings are used in navigation, surveying, aviation (where they are called headings), and maritime applications. They differ from azimuths in some contexts (azimuths sometimes measure from South in astronomy) but the terms are often used interchangeably in geodesy.

How do I convert degrees-minutes-seconds (DMS) to decimal degrees?

Decimal Degrees = Degrees + (Minutes ÷ 60) + (Seconds ÷ 3600). For example, 40° 42′ 46″ N = 40 + (42/60) + (46/3600) = 40 + 0.7000 + 0.01278 = 40.71278°. For South latitudes and West longitudes, apply a negative sign: 34° 3′ 8″ S = −34.05222°. This tool accepts decimal degrees directly.

Why does the great-circle path look curved on a flat map?

When a sphere's surface is projected onto a flat map (e.g., using the Mercator projection), straight lines on the map do not correspond to straight lines on the sphere. The shortest path on the sphere — the great circle — appears curved on most flat projections. A rhumb line (constant compass bearing, used in Mercator navigation) looks straight on a Mercator map but is actually longer than the great-circle path.

What is a rhumb line and how does it differ from a great circle?

A rhumb line (or loxodrome) is a path that crosses every meridian at the same fixed angle — meaning you maintain a constant compass bearing throughout the journey. It is convenient for navigation because the heading never changes, but it is longer than the great-circle (geodesic) path except for east-west travel along the equator. A great circle is always the shortest path on a sphere; a rhumb line is the constant-bearing path.

What is the reduced latitude in Vincenty's formula?

The reduced latitude (also called parametric latitude) U is a transformation of the geodetic latitude φ that accounts for the Earth's flattening: tan(U) = (1 − f) · tan(φ), where f is the flattening (1/298.257223563 for WGS-84). This transformation maps the ellipsoid to a sphere, simplifying the iterative calculation of geodesic distance and bearings in the Vincenty algorithm.

How accurate is the Vincenty formula for bearings?

The Vincenty formula is accurate to approximately 0.5 mm on the WGS-84 ellipsoid for distance, and the corresponding angular accuracy of bearings is within a fraction of an arcsecond. For most practical applications — aviation flight planning, marine navigation, surveying, and GIS — this level of accuracy far exceeds what is needed. The only scenario where it fails is for nearly antipodal points (diametrically opposite points on the globe), where convergence can be slow or fail entirely.

Geodesic Bearing Calculator — Initial & Final Bearing with Vincenty Formula

Geodesy Navigation Great-Circle Vincenty Formula WGS-84 Haversine

Whether you're planning a transoceanic flight path, aiming a satellite dish, working on a GIS project, or simply satisfying your curiosity about how far apart two cities are, the HeLovesMath Geodesic Bearing Calculator gives you precise answers. Enter any two points on the Earth in decimal degrees and instantly receive the initial bearing, final bearing, and great-circle distance — all calculated using rigorous mathematical models based on the WGS-84 ellipsoid.

This guide explains every concept from scratch: what a geodesic is, how the Vincenty and Haversine formulas work (with fully rendered mathematical expressions), why initial and final bearings differ, what WGS-84 means, and how all of this is used in the real world. By the end, you'll understand not just how to use the calculator but why each result is what it is.

Free Online Geodesic Bearing Calculator

📍 Point 1 — Origin

Latitude 1 (decimal °) Longitude 1 (decimal °)

🏁 Point 2 — Destination

Latitude 2 (decimal °) Longitude 2 (decimal °)

Initial Bearing (α₁)

—

Final Bearing (α₂)

—

Distance (km)

—

Distance (miles)

—

Calculator ready. Enter coordinates for two points and click Calculate.

What Is a Geodesic? Spheres vs. Ellipsoids

In mathematics and physics, a geodesic is the generalisation of a straight line to curved surfaces — it is the locally shortest path between two points on a surface. On a flat plane, a geodesic is an ordinary straight line. On a sphere, it is an arc of a great circle — any circle whose plane passes through the sphere's centre.

The Earth is not a flat plane, and it is not a perfect sphere either. More precisely, it is an oblate spheroid — a sphere that is slightly flattened at the poles and bulging at the equator, due to its rotation. The equatorial radius (semi-major axis, a) is approximately 6,378 km, while the polar radius (semi-minor axis, b) is approximately 6,357 km — a difference of about 21 km or 0.33%.

✦ Ellipsoidal Flattening

\[f = \frac{a - b}{a}\]

a = semi-major axis (equatorial radius) | b = semi-minor axis (polar radius) | For WGS-84: \(f = 1/298.257223563 \approx 0.003353\)

This small but non-zero flattening means that truly accurate geodesic calculations — for aviation, satellite positioning, surveying, and military applications — require ellipsoidal geometry rather than simple spherical geometry. This is why the Vincenty formula (which models the Earth as a WGS-84 ellipsoid) is preferred over the simpler Haversine formula (which assumes a perfect sphere) when high accuracy is needed.

🔵 Perfect Sphere Model

Assumes uniform radius R ≈ 6,371 km. Simple, fast formulas (Haversine). Maximum distance error ≈ 0.5%. Acceptable for casual use, rough estimates, and distances under a few hundred km.

🌍 Oblate Spheroid (Ellipsoid)

Uses semi-major axis a and flattening f. Requires iterative algorithms (Vincenty, Karney). Distance accurate to 0.5 mm. Bearings accurate to sub-arcsecond. Required for aviation, GPS, and precise surveying.

🗺️ Geoid (Physical Reality)

The actual equipotential surface of Earth's gravity field — lumpy and irregular. Used in precise vertical datums (heights above mean sea level). For horizontal distance and bearing calculations, the WGS-84 ellipsoid is an excellent approximation.

Bearings, Azimuths & Compass Directions Explained

A bearing is the horizontal angle between the direction to a target and True North, measured clockwise from North. Bearings range from 0° (due North) through 90° (East), 180° (South), 270° (West), and back to 360° (North again). They are the fundamental output of this calculator.

✦ Bearing Convention

\[\theta \in [0°, 360°), \quad \text{measured clockwise from True North}\]

N = 0°, NE = 45°, E = 90°, SE = 135°, S = 180°, SW = 225°, W = 270°, NW = 315°

Initial Bearing vs. Final Bearing

On a flat surface, the bearing from A to B is constant — you just point and walk. But on a sphere or ellipsoid, the shortest path (geodesic) continuously curves relative to the local meridians. This means:

Initial bearing (α₁): The compass direction you face at Point 1 to set off on the geodesic toward Point 2.
Final bearing (α₂): The compass direction at which the geodesic arrives at Point 2 from Point 1's direction.

For example, a great-circle flight from London (51.5°N, 0°W) to Tokyo (35.7°N, 139.7°E) departs at approximately 040° (northeast) but arrives from approximately 325° (northwest) — a difference of nearly 105°. Without understanding this, a navigator who maintained a constant course of 040° would miss Tokyo entirely, flying a longer rhumb-line path instead of the geodesic.

True North vs. Magnetic North: All bearings in this calculator are True North bearings, referenced to the geographic North Pole (Earth's rotation axis). Compass needles point to Magnetic North, which differs from True North by an angle called magnetic declination — ranging from a few degrees to over 20° in some locations. Navigation instruments require a declination correction to convert between them.

Standard Compass Points

Direction	Abbreviation	Bearing (°)	Quadrant
North	N	0° / 360°	—
North-Northeast	NNE	22.5°	NE
Northeast	NE	45°	NE
East	E	90°	—
Southeast	SE	135°	SE
South	S	180°	—
Southwest	SW	225°	SW
West	W	270°	—
Northwest	NW	315°	NW

Haversine Formula — Great-Circle Distance on a Sphere

The Haversine formula is the classic method for computing the great-circle distance between two points expressed as latitude–longitude pairs, under the assumption that the Earth is a perfect sphere. It is named after the haversine (half-versed sine) trigonometric function.

✦ Definition — Haversine Function

\[\operatorname{hav}(\theta) = \sin^2\!\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}\]

The haversine function maps any angle θ to a value in [0, 1] and is numerically stable for small angles, unlike formulas based on the raw cosine rule.

✦ Haversine Formula — Great-Circle Distance

\[a = \sin^2\!\left(\frac{\Delta\varphi}{2}\right) + \cos\varphi_1 \cdot \cos\varphi_2 \cdot \sin^2\!\left(\frac{\Delta\lambda}{2}\right)\] \[c = 2\,\arctan2\!\left(\sqrt{a},\,\sqrt{1-a}\right)\] \[d = R \cdot c\]

φ₁, φ₂ = latitudes of Point 1 and Point 2 (in radians) | Δφ = φ₂ − φ₁ = difference in latitudes | Δλ = λ₂ − λ₁ = difference in longitudes | R = Earth's mean radius ≈ 6,371 km | d = great-circle distance | Note: atan2 is used instead of arcsin for numerical stability.

The Haversine formula's maximum error is approximately 0.5% because the Earth is not a perfect sphere. For the 3,940 km New York–Los Angeles route, Haversine gives ≈ 3,940 km while the true ellipsoidal distance is ≈ 3,944 km — a difference of only 4 km. For most practical purposes this is entirely acceptable.

Bearing Calculation on a Sphere

The initial bearing between two points on a sphere is:

✦ Initial Bearing (Spherical)

\[\theta = \arctan2\!\bigl(\sin\Delta\lambda\cdot\cos\varphi_2,\; \cos\varphi_1\cdot\sin\varphi_2 - \sin\varphi_1\cdot\cos\varphi_2\cdot\cos\Delta\lambda\bigr)\] \[\alpha_1 = (\theta \times 180/\pi + 360) \bmod 360\]

The modulo operation normalises the result to the [0°, 360°) range. The atan2 function preserves the correct quadrant sign.

Vincenty Formula — Ellipsoidal Bearings & Distance

The Vincenty formula was developed by Polish-American geodesist Thaddeus Vincenty in 1975 and published in the journal Survey Review. It accounts for the Earth's ellipsoidal shape (via WGS-84 parameters) and iteratively solves for the geodesic distance and bearings to sub-millimetre accuracy. This is the algorithm powering the bearing calculations in this tool.

Step 1: Reduced (Parametric) Latitude

First, geodetic latitudes φ are transformed into reduced latitudes U, which project the ellipsoid onto an auxiliary sphere:

✦ Reduced Latitude

\[\tan U = (1 - f)\,\tan\varphi\] \[\cos U = \frac{1}{\sqrt{1 + \tan^2 U}}, \qquad \sin U = \tan U \cdot \cos U\]

f = flattening = 1/298.257223563 (WGS-84) | φ = geodetic latitude (radians) | U = reduced (parametric) latitude (radians)

Step 2: Iterative Solution for λ

The longitude difference on the auxiliary sphere, λ, is found iteratively. Start with λ = L (the actual longitude difference) and iterate:

✦ Vincenty Iteration (core equations)

\[\sin\sigma = \sqrt{(\cos U_2\,\sin\lambda)^2 + (\cos U_1\sin U_2 - \sin U_1\cos U_2\cos\lambda)^2}\] \[\cos\sigma = \sin U_1\sin U_2 + \cos U_1\cos U_2\cos\lambda\] \[\sigma = \arctan2(\sin\sigma,\,\cos\sigma)\] \[\sin\alpha = \frac{\cos U_1\cos U_2\sin\lambda}{\sin\sigma}\] \[\cos^2\alpha = 1 - \sin^2\alpha\] \[\cos 2\sigma_m = \cos\sigma - \frac{2\sin U_1\sin U_2}{\cos^2\alpha}\] \[C = \frac{f}{16}\cos^2\alpha\bigl(4 + f(4 - 3\cos^2\alpha)\bigr)\] \[\lambda' = L + (1-C)f\sin\alpha\!\left(\sigma + C\sin\sigma\!\left(\cos 2\sigma_m + C\cos\sigma(-1 + 2\cos^2 2\sigma_m)\right)\right)\]

Iterate until |λ′ − λ| < 10⁻¹² (typically 2–3 iterations for non-antipodal points). σ is the angular distance on the sphere; α is the geodesic azimuth at the equator.

Step 3: Bearings from Converged Values

✦ Initial and Final Bearings (Vincenty)

\[\alpha_1 = \arctan2(\cos U_2\,\sin\lambda,\;\cos U_1\sin U_2 - \sin U_1\cos U_2\cos\lambda)\] \[\alpha_2 = \arctan2(\cos U_1\,\sin\lambda,\;-\sin U_1\cos U_2 + \cos U_1\sin U_2\cos\lambda)\]

Both results in radians — convert to degrees and apply mod 360 to obtain compass bearings in [0°, 360°).

✦ Vincenty Distance (ellipsoidal)

\[u^2 = \cos^2\alpha\,\frac{a^2 - b^2}{b^2}\] \[A_v = 1 + \frac{u^2}{16384}\!\left(4096 + u^2(-768 + u^2(320 - 175u^2))\right)\] \[B_v = \frac{u^2}{1024}\!\left(256 + u^2(-128 + u^2(74 - 47u^2))\right)\] \[\Delta\sigma = B_v\sin\sigma\!\left(\cos 2\sigma_m + \tfrac{B_v}{4}\!\left(\cos\sigma(-1+2\cos^2 2\sigma_m) - \tfrac{B_v}{6}\cos 2\sigma_m(-3+4\sin^2\sigma)(-3+4\cos^2 2\sigma_m)\right)\right)\] \[s = b\,A_v\,(\sigma - \Delta\sigma)\]

s = geodesic distance in metres | b = 6,356,752.314 m (WGS-84 semi-minor axis) | A_v and B_v are series expansion coefficients for the ellipsoidal correction. Note: This calculator uses Haversine for displayed distance (fast, <0.5% error); the Vincenty iteration above governs bearing accuracy.

💡 Vincenty Limitation: The algorithm can fail to converge for nearly antipodal points — points nearly diametrically opposite on the globe (e.g., London and its antipode near the Auckland Islands in the southern ocean). For such edge cases, the Karney (2013) algorithm provides unconditional convergence. For all other point pairs, Vincenty converges in ≤ 3 iterations.

WGS-84 — The World Geodetic System Explained

WGS-84 (World Geodetic System 1984) is the standard reference frame adopted by GPS and used by virtually all modern navigation, mapping, and GIS systems. It defines the Earth's shape, orientation, and origin so that positions anywhere on Earth can be described unambiguously and shared between systems.

Parameter	Symbol	WGS-84 Value	Meaning
Semi-major axis	a	6,378,137.0 m	Equatorial radius
Flattening	f	1 / 298.257223563	Degree of polar flattening
Semi-minor axis	b	6,356,752.3142 m	Polar radius (derived: b = a(1−f))
First eccentricity²	e²	6.69438 × 10⁻³	Measure of ellipse departure from circle
Geocentric gravitational constant	GM	3.986004418 × 10¹⁴ m³/s²	Earth's mass × grav. constant
Angular velocity	ω	7.292115 × 10⁻⁵ rad/s	Earth's rotation rate

✦ First Eccentricity of the WGS-84 Ellipsoid

\[e^2 = \frac{a^2 - b^2}{a^2} = 2f - f^2 \approx 0.00669438\]

The eccentricity e quantifies how much the ellipsoid deviates from a sphere. For a perfect sphere, e = 0. The WGS-84 value of e ≈ 0.0818 (first eccentricity) confirms the Earth is very close to spherical but not exactly.

Degrees–Minutes–Seconds ↔ Decimal Degrees Conversion

Geographic coordinates are traditionally expressed in degrees–minutes–seconds (DMS) format (e.g., 40° 42′ 46″ N). This calculator requires decimal degrees (DD). Here are the conversion formulas:

✦ DMS → Decimal Degrees

\[\text{DD} = d + \frac{m}{60} + \frac{s}{3600}\]

d = whole degrees | m = minutes (0–59) | s = seconds (0–59.999…) | For S latitude or W longitude, apply a negative sign to the final result.

✦ Decimal Degrees → DMS

\[d = \lfloor\text{DD}\rfloor, \quad m = \lfloor(\text{DD} - d)\times 60\rfloor, \quad s = \bigl((\text{DD} - d)\times 60 - m\bigr)\times 60\]

DMS → DD Example: 51° 30′ 26″ N, 0° 7′ 39″ W (London)

Latitude: \(51 + \frac{30}{60} + \frac{26}{3600} = 51 + 0.5 + 0.007222 = 51.5072°\ \text{N}\)

Longitude: \(-(0 + \frac{7}{60} + \frac{39}{3600}) = -(0 + 0.11667 + 0.010833) = -0.1275°\)

✅ London: Lat = 51.5072°, Lon = −0.1275°

Rhumb Lines vs. Great Circles — A Critical Navigation Distinction

The two most important types of navigational paths are great circles (geodesics on a sphere) and rhumb lines (loxodromes). Understanding their difference is essential for practical navigation.

🌐 Great Circle — Shortest Path

The great-circle path between two points is the shortest possible route on a sphere. Every great circle divides the Earth into two equal hemispheres. All meridians and the equator are great circles.

Advantage: Minimum distance — saves time and fuel for long-haul aviation and shipping.

Disadvantage: Bearing constantly changes, requiring continuous course corrections.

📏 Rhumb Line — Constant Bearing

A rhumb line crosses every meridian at the same angle — so a ship or aircraft can maintain a constant compass heading throughout the journey.

Advantage: Simple to navigate — set a fixed compass bearing and hold it.

Disadvantage: Longer path than the great circle (except along the equator or pure N/S routes).

✦ Rhumb Line Distance Formula

\[\Delta\psi = \ln\!\left(\tan\!\left(\frac{\pi}{4}+\frac{\varphi_2}{2}\right)\bigg/\tan\!\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)\right)\] \[q = \begin{cases}\cos\varphi_1 & \text{if } |\Delta\psi| < 10^{-10}\\ \Delta\varphi/\Delta\psi & \text{otherwise}\end{cases}\] \[d_R = R\,\sqrt{\Delta\varphi^2 + q^2\,\Delta\lambda^2}\]

Δψ = Mercator projection latitude difference (isometric latitude) | Δφ = latitude difference in radians | Δλ = longitude difference in radians | R = Earth radius ≈ 6,371 km

💡 Practical Rule: For long distances (trans-oceanic), always use a great-circle route to minimise distance. For short navigation legs where simplicity matters more than the tiny extra distance, a rhumb-line course is often preferred. Commercial aviation exclusively uses great-circle routing, adjusted for winds.

Real-World Applications of Geodesic Bearing Calculations

✈️ Aviation Flight Planning

Long-haul routes (e.g., London–Tokyo, New York–Singapore) are planned along great-circle paths to minimise flight time and fuel cost. The initial bearing tells pilots the departure heading; autopilot systems then continuously update the heading along the geodesic.

🛰️ Satellite Dish Alignment

To align a satellite antenna at a ground station toward a geostationary satellite, engineers compute both azimuth (bearing) and elevation angle from the station's coordinates to the satellite's orbital slot over the equator. Bearing accuracy of <0.05° is required.

🚢 Maritime Navigation

Ocean-going vessels plan great-circle routes to reduce voyage time and fuel consumption. A voyage from Los Angeles to Yokohama follows a great circle that passes through the northern Pacific — appearing curved on a Mercator chart but being the shortest path.

🗺️ GIS & Spatial Analysis

Geographic information systems use geodesic calculations for proximity searches ("find all hospitals within 50 km"), geofencing, buffer zone creation, and visualising coverage areas — all requiring accurate great-circle distances and bearings.

📡 Radio & Microwave Links

Point-to-point microwave links, terrestrial radio relays, and radar installations must be aimed with precise bearings between transmitter and receiver sites. Errors of even 0.1° can cause signal degradation over long links.

🔭 Astronomy & VLBI

Very Long Baseline Interferometry (VLBI) combines signals from radio telescopes on different continents to create a virtual telescope the size of the Earth. Precise geodesic baselines between telescopes are needed to correlate observations and image distant radio sources.

Formula Comparison — Haversine vs. Vincenty vs. Karney

Property	Haversine (Sphere)	Vincenty (Ellipsoid)	Karney (2013)
Earth model	Perfect sphere	WGS-84 ellipsoid	WGS-84 ellipsoid
Distance accuracy	±0.5% (max ~30 km)	±0.5 mm	±15 nm (nanometres)
Bearing accuracy	~0.3°	sub-arcsecond	sub-arcsecond
Computation speed	Instant (closed form)	Fast (2–3 iterations)	Moderate (series expansion)
Antipodal points	Handles	May not converge	Always converges
Complexity	Low	Medium	High
Best use case	Quick estimates, <500 km	General navigation, GIS	Surveying, space geodesy

Worked Examples — Step-by-Step Calculations

Example 1 — Haversine Distance: New York City to London

NYC: φ₁ = 40.7128°, λ₁ = −74.0060°. London: φ₂ = 51.5074°, λ₂ = −0.1278°. Convert all to radians: φ₁ = 0.7104 rad, λ₁ = −1.2920 rad, φ₂ = 0.8991 rad, λ₂ = −0.00223 rad.

Δφ = 0.8991 − 0.7104 = 0.1887 rad. Δλ = −0.00223 − (−1.2920) = 1.2898 rad.

\(a = \sin^2(0.09435) + \cos(0.7104)\cdot\cos(0.8991)\cdot\sin^2(0.6449)\) \(\approx 0.00890 + 0.7603\times 0.6267\times 0.36216 \approx 0.00890 + 0.17245 = 0.18135\)

\(c = 2\arctan2(\sqrt{0.18135}, \sqrt{0.81865}) = 2\times 0.43694 = 0.87388\ \text{rad}\)

\(d = 6371 \times 0.87388 \approx 5,565\ \text{km}\). (True geodesic: 5,570 km — 5 km difference ≈ 0.09%)

✅ NYC to London ≈ 5,565 km (Haversine) / 5,570 km (Vincenty)

Example 2 — Initial Bearing: New York to London

Use the spherical bearing formula with Δλ = 1.2898 rad, φ₁ = 0.7104 rad, φ₂ = 0.8991 rad.

\(y = \sin(1.2898)\cdot\cos(0.8991) = 0.9606\times 0.6267 = 0.6020\)

\(x = \cos(0.7104)\cdot\sin(0.8991) - \sin(0.7104)\cdot\cos(0.8991)\cdot\cos(1.2898)\)

\(x = 0.7603\times 0.7783 - 0.6496\times 0.6267\times(-0.2756) \approx 0.5920 + (0.1122) = 0.7042\) (corrected signs)

\(\theta = \arctan2(0.6020, 0.7042) \approx 40.5°\). Normalised: 040.5° (northeast). Vincenty gives ≈ 051.2°.

✅ Approximate initial bearing NYC → London ≈ 050°–051° (northeast). Use the calculator above for the exact Vincenty result.

Example 3 — DMS to Decimal: Sydney Airport (33° 56′ 46″ S, 151° 10′ 37″ E)

Latitude: \(33 + \frac{56}{60} + \frac{46}{3600} = 33 + 0.9333 + 0.01278 = 33.9461°\). South → negative: \(-33.9461°\)

Longitude: \(151 + \frac{10}{60} + \frac{37}{3600} = 151 + 0.1667 + 0.01028 = 151.1769°\). East → positive: \(+151.1769°\)

✅ Sydney Airport: Lat = −33.9461°, Lon = +151.1769°

Example 4 — Why Initial ≠ Final Bearing: Los Angeles to Tokyo

LA: 34.0522°N, 118.2437°W → (34.0522, −118.2437). Tokyo: 35.6762°N, 139.6503°E → (35.6762, +139.6503)

The great-circle path between LA and Tokyo arcs northward through the northern Pacific, passing near the Aleutian Islands at about 52°N.

Initial bearing from LA ≈ 296° (west-northwest). The path curves northward then southward across the Pacific.

Final bearing arriving at Tokyo ≈ 038° (northeast). The difference between initial (296°) and final (038°) is ~102° — reflecting the enormous curve of the geodesic.

✅ LA → Tokyo: Initial ≈ 296°, Final ≈ 038°, Distance ≈ 8,815 km. Enter coordinates above to verify with full precision.

Frequently Asked Questions

What is a geodesic?＋

A geodesic is the shortest curve connecting two points on a curved surface. On a perfect sphere, this is an arc of a great circle — any circle whose plane passes through the sphere's centre. On the Earth's actual ellipsoidal surface (modelled by WGS-84), a geodesic is slightly more complex and is computed using algorithms like Vincenty's or Karney's. For most navigation purposes, great-circle paths are used as the best practical approximation of geodesics.

What is the difference between initial bearing and final bearing?＋

The initial bearing (α₁) is the compass direction from Point 1 at which you must depart to follow the geodesic to Point 2. The final bearing (α₂) is the compass direction at which the same geodesic path arrives at Point 2. On a curved Earth, a geodesic path continuously changes compass direction, so α₁ ≠ α₂ for any point pair that is not due North/South or on the equator. For long intercontinental paths, the difference can exceed 100°.

What is the Vincenty formula?＋

Vincenty's formula (1975) is an iterative method for computing geodesic distance and bearings between two points on an oblate spheroid. It uses the WGS-84 ellipsoid parameters (a = 6,378,137 m; f = 1/298.257223563) and converges to better than 0.5 mm accuracy in 2–3 iterations for most point pairs. It is the industry standard for navigation software, GIS, and geodetic surveying. The only weakness is potential non-convergence for nearly antipodal points.

What is the Haversine formula?＋

The Haversine formula computes great-circle distance on a sphere using: \(a = \sin^2(\Delta\varphi/2) + \cos\varphi_1\cdot\cos\varphi_2\cdot\sin^2(\Delta\lambda/2)\), then \(d = 2R\arctan2(\sqrt{a},\sqrt{1-a})\). It assumes the Earth is a perfect sphere of radius R ≈ 6,371 km, giving a maximum distance error of ≈ 0.5% compared to the true ellipsoidal distance. It is fast, simple, and adequate for most applications not requiring sub-kilometre precision.

What is WGS-84?＋

WGS-84 (World Geodetic System 1984) is the international standard reference ellipsoid and coordinate system used by GPS and virtually all modern navigation and mapping systems. It defines Earth's shape with semi-major axis a = 6,378,137 m (equatorial radius) and flattening f = 1/298.257223563. GPS receivers output coordinates in WGS-84, so this is the coordinate system you should use with this calculator when working from a GPS fix.

How do I convert DMS (degrees-minutes-seconds) to decimal degrees?＋

Use the formula: \(\text{DD} = d + m/60 + s/3600\). For South latitudes or West longitudes, apply a negative sign. Example: 40° 42′ 46″ N = 40 + 42/60 + 46/3600 = 40.7128°. Example: 74° 0′ 21.6″ W = −(74 + 0 + 21.6/3600) = −74.006°. Most GPS devices can be set to display in decimal degrees (DD) directly.

Why does a great-circle path look curved on a map?＋

On a Mercator projection (the most common flat map), meridians appear as parallel vertical lines. However, on the actual sphere, meridians converge at the poles. A great-circle path between two mid-latitude points arcs poleward and then back — this curve is faithfully represented on a globe but appears as a curved arc on a Mercator map. Only rhumb lines (constant bearing paths) look straight on a Mercator projection, which is why Mercator was historically used for marine navigation.

What is a rhumb line and when should I use it?＋

A rhumb line (loxodrome) is a path that crosses every meridian at the same angle — meaning you maintain a constant compass heading throughout the journey. It is longer than a great-circle path (except for routes exactly along the equator or a meridian) but much simpler to navigate. Use a great-circle route for long-haul aviation and shipping where distance savings matter. Use a rhumb line for short legs where simplicity and ease of navigation outweigh the small distance penalty.

What is the reduced latitude in Vincenty's algorithm?＋

The reduced (parametric) latitude U is a transformation of the geodetic latitude φ: \(\tan U = (1-f)\tan\varphi\). It maps points on the ellipsoid to corresponding points on an auxiliary sphere, making the iterative geodesic calculation tractable. On the auxiliary sphere, geodesics are great circles, allowing spherical trigonometry to be used. The final results are then mapped back to the ellipsoid to give accurate ellipsoidal bearings and distances.

How accurate is this calculator?＋

Bearings are computed using the Vincenty algorithm on the WGS-84 ellipsoid, giving accuracy better than 0.001° (sub-arcsecond) for all non-antipodal point pairs. Distance is computed using the Haversine formula on a sphere of radius 6,371 km, giving accuracy better than 0.5% (typically <0.1% for most routes). For professional surveying or sub-metre navigation applications, use the Vincenty or Karney distance formula instead.

What are some real-world applications of geodesic bearing calculations?＋

Geodesic bearings are used in: (1) long-haul flight planning (great-circle routing saves thousands of km per transoceanic flight); (2) satellite dish alignment (azimuth and elevation to geostationary satellites); (3) maritime great-circle voyage planning; (4) military targeting and ballistics; (5) GIS proximity and buffer analysis; (6) telecommunications microwave link alignment; (7) earthquake seismology (locating epicentres); (8) astronomy (computing azimuth to celestial bodies); and (9) land surveying and boundary determination.

What causes Vincenty's formula to fail for antipodal points?＋

For nearly antipodal points (diametrically opposite, separated by ~20,000 km), Vincenty's iterative loop can oscillate without converging because the geodesic on an ellipsoid between antipodal points is not unique — infinitely many geodesics of the same length connect them. The algorithm detects this when the iteration limit (typically 100 iterations) is reached without the λ correction falling below 10⁻¹². For such cases, the Karney (2013) algorithm is recommended; it uses a different series expansion that always converges.