ATPL General Navigation

Great Circles & Rhumb Lines

Exit lesson
ATPL General Navigation55 min

Great Circles & Rhumb Lines

Shortest route versus constant bearing

Two important paths can connect positions on Earth. A great-circle route follows the shortest path over the spherical Earth, but its true track usually changes continuously. A rhumb line crosses all meridians at a constant angle, making it easier to steer, but it is normally longer.

Learning objectives

By the end of this chapter, you should be able to:

  • 01Define a great circle
  • 02Define a rhumb line
  • 03Explain why a great circle is the shortest route
  • 04Explain why great-circle track normally changes
  • 05Explain why a rhumb line has constant true track
  • 06Identify great circles on Earth
  • 07Identify special cases where a rhumb line is also a great circle
  • 08Compare both routes on a globe
  • 09Compare both routes on a Mercator projection
  • 10Recognise the practical navigation advantages of each route
  • 11Answer basic ATPL exam questions on route geometry
01Section 1

Sphere and Route Fundamentals

Both routes join the same two positions, yet they take different paths across the sphere. Everything in this chapter builds on the reference lines shown here.

Interactive educational diagram

Both routes on one sphere

Hover, tap or tab through a legend item to isolate that element on the globe.

Great circle and rhumb line between two positions on a spherical EarthA perfectly round Earth showing the North and South Poles, the Equator, several meridians, departure point A in the North Atlantic and destination point B in southern Europe. A solid red great-circle arc runs between A and B, passing north of the dashed purple rhumb line, which crosses every meridian at the same angle.Earth’s centreN poleAB

A 48°00′N 060°00′W

B 40°00′N 040°00′E

Both paths join the same two positions. The red arc is a section of a circle whose plane cuts Earth’s centre; the purple path holds one steady angle against every meridian it crosses.

02Section 2

Great Circles — The Shortest Path

A great circle is formed by the intersection of Earth with a plane that passes through Earth's centre.
A plane passing through Earth’s centre creating a great circleCross-section of a spherical Earth. A flat plane passes exactly through the centre of Earth. Its intersection with the surface is a circle that divides Earth into two equal hemispheres. The upper hemisphere and lower hemisphere are shaded differently and labelled as equal halves.RPlane passes through Earth’s centreCircular intersection = great circleEqual hemisphereEqual hemisphereCentre
The cutting plane passes exactly through Earth’s centre, so the intersection is a great circle and the two halves are equal hemispheres.
A plane that misses Earth’s centre creating a small circleCross-section of a spherical Earth with a flat plane above the centre. The intersection is a smaller circle and the two parts of Earth either side of it are unequal, so this is a small circle, not a great circle.Plane does not pass through the centreOffsetSmall circle — e.g. 50° N parallelCentreUnequal parts — not hemispheres
Offset the plane and the intersection shrinks to a small circle — which is why every parallel except the Equator fails the test.

Key facts

  • 01A great circle divides Earth into two equal hemispheres.
  • 02The centre of the great-circle plane is the centre of Earth.
  • 03The shortest route between two points on a sphere lies along the minor arc of a great circle.
  • 04The Equator is a great circle.
  • 05Every meridian and its opposite meridian form a great circle.
  • 06Most parallels are not great circles.
  • 07A great-circle route normally changes true track continuously.
  • 08The initial and final great-circle tracks are usually different.
  • 09The highest latitude reached by the route is called the vertex.
  • 10A great circle has two vertices, opposite each other.
  • 11The route normally curves toward the nearer pole when shown on a Mercator chart.
03Section 3

Explore a Great-Circle Route

Set any two positions, or pick a published city pair, and read the distance, tracks and vertex straight off the spherical calculation.

Interactive educational diagram

Explore a great-circle route

Drag any slider, or pick a preset city pair. Distance uses the haversine central angle; tracks use spherical trigonometry.

Great-circle route between the selected departure and destinationA perfectly round Earth centred near 46° N. The solid red minor arc runs from departure 51°29′N 000°28′W to destination 40°38′N 073°47′W, with an arrow showing the direction of travel. Initial track 288°, final track 231°, distance 2,991 NM. The vertex of this great circle lies at 53.7° N.N poleABVertex 54° N

Preset city pairs

Classic mid-latitude North Atlantic route: the great circle passes well north of the rhumb line.

51.5° N
-89.589.5
000°28′W
-180180
40.6° N
-89.589.5
073°47′W
-180180
Great-circle distance
2,991 NM
Δσ = 49.82°
Central angle × 60
2989 NM
1° of arc ≈ 60 NM
Initial true track
288° (WNW)
Final true track
231° (SW)
Vertex latitude
53.7° N
Northern vertex lies on the route
Direction of travel
Westbound
How it is calculated

Central angle from the haversine form, worked in radians:

Δσ = 2 asin( √( sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2) ) )

Distance = R · Δσ  with R = 3440.065 NM (mean Earth radius)

Distance = 60 × Δσ°  (the ATPL rule of thumb)

Initial track, then normalised to 000°–359°:

θ = atan2( sin Δλ · cos φ₂ , cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ )

The final track is the reverse initial track plus 180°. Clairaut’s equation gives the vertex:

cos φ_vertex = | sin θ · cos φ₁ |

04Section 4

Why Does the Track Change?

The aircraft follows one fixed great-circle plane, but the direction of local meridians changes as the aircraft moves across Earth. The angle between the route and local north therefore changes.

Interactive educational diagram

Fly the route and watch the track change

Move the route-progress slider with the pointer or the arrow keys. The aircraft symbol turns to follow the local tangent.

Aircraft position along a great-circle route with the local meridian and tangent shownThe aircraft is 0% along the route, at 51°29′N 000°28′W. Current true track 288°. Distance flown 0 NM, distance remaining 2,991 NM. A green line shows the local meridian and a dashed line shows the local tangent to the route.N polelocal N288°Initial pointVertexFinal point
0%
0100
Current position
51°29′N 000°28′W
Current true track
288° (WNW)
Distance flown
0 NM
Distance remaining
2,991 NM

Track comparison

  • Initial track288°
  • Current track288°
  • Final track231°
  • Total change, initial to final−56.6°

The route is fixed, but the reference direction of local north changes.

Try the same route on another leg

Classic mid-latitude North Atlantic route: the great circle passes well north of the rhumb line.

Local north is drawn at 51.5° N, meridian angle reference 0.0175 radians per degree.

05Section 5

The Vertex

The vertex is the point where a great circle reaches its maximum latitude in one hemisphere.

Interactive educational diagram

Vertices of the great circle

Switch between the complete great circle and the segment actually flown to see whether the vertex is crossed.

Northern and southern vertices of a great circleThe complete great circle through the two positions reaches a maximum latitude of 53.7° N at its northern vertex, near 22.9° W, and the same latitude in the south at the antipodal southern vertex. At each vertex the track is 090 or 270 degrees. The segment flown between A and B is drawn as a thicker line and does include a vertex.N poleABN vertex 54° N · 090°/270°S vertex 54° S
Vertex latitude
53.7° N
Northern vertex longitude
22.9° W
Track at the vertex
090° or 270°
Crossed on this leg?
Yes
The aircraft actually reaches the maximum latitude.
Highest latitude on the leg
53.7° N
Midpoint latitude
52.2° N
The midpoint is not the vertex.

Vertex facts

  • 01At the vertex, the great-circle track is 090° or 270°.
  • 02A great circle has a northern and a southern vertex.
  • 03The two vertices are antipodal.
  • 04The vertex may lie outside the route segment being flown.
  • 05Not every route reaches the vertex between departure and destination.
  • 06On a Mercator chart, the route appears to curve toward the vertex.

Preset city pairs

Classic mid-latitude North Atlantic route: the great circle passes well north of the rhumb line.

06Section 6

Rhumb Lines — Constant True Track

A rhumb line, or loxodrome, crosses every meridian at the same angle. It therefore maintains a constant true track.

Interactive educational diagram

One constant angle with every meridian

Each amber wedge measures the angle between the route and local north. Every wedge is the same size — that is what makes this a rhumb line.

A rhumb line crossing several meridians at the same angleA rhumb line runs from 08 North 070 West to 58 North 020 East on a spherical Earth, holding a constant true track of 055°. At each of 4 labelled meridian crossings the angle between the route and local north is drawn and is identical. When extended, the route spirals towards the North Pole without reaching it.N pole055°055°055°055°Same angle with each meridianAB
Constant true track
055°
Meridian crossings shown
4
Angle at every crossing
055°
Identical at each meridian
Longitude range
70° W → 20° E

Because the angle never changes, a navigator could hold one compass course for the whole leg. Extended indefinitely the track keeps turning across new meridians and winds ever tighter around the pole — approaching it, but never arriving.

Key facts

  • 01A rhumb line maintains a constant true track.
  • 02It crosses meridians at a constant angle.
  • 03It appears as a straight line on a Mercator chart.
  • 04It is normally longer than the corresponding great-circle route.
  • 05It spirals toward a pole when extended indefinitely, except in special cases.
  • 06A rhumb line does not normally divide Earth into equal hemispheres.
  • 07It was historically convenient because a navigator could maintain one compass course.
07Section 7

When Is a Rhumb Line Also a Great Circle?

Only two families of route satisfy both definitions at once. Everything else is a rhumb line only.

The Equator highlighted on a globeA spherical Earth with the Equator drawn as a bold highlighted circle around the widest part of the globe.

The Equator

The only parallel whose plane passes through Earth's centre, so it satisfies both definitions at once.

Rhumb line?
Yes
Great circle?
Yes
Constant track?
Yes — 090° or 270°
Shortest route?
Yes
A meridian highlighted on a globeA spherical Earth with one meridian drawn boldly from the North Pole to the South Pole, together with its anti-meridian completing the circle.

A meridian

A meridian and its anti-meridian together form a complete great circle, flown on a constant track.

Rhumb line?
Yes
Great circle?
Yes
Constant track?
Yes — 000° or 180°
Shortest route?
Yes
The 50 degrees North parallel highlighted on a globeA spherical Earth with the 50 degrees North parallel drawn boldly as a small circle well above the Equator, which is shown for comparison.

A non-equatorial parallel

Its plane is parallel to the Equator but does not contain Earth's centre, so it is a small circle.

Rhumb line?
Yes
Great circle?
No
Constant track?
Yes — 090° or 270°
Shortest route?
No
An oblique rhumb line on a globeA spherical Earth with an oblique rhumb line running from the southern tropics to northern mid latitudes, crossing every meridian at the same angle.

An oblique rhumb line

Any other constant-track route. Extended indefinitely it spirals towards a pole without reaching it.

Rhumb line?
Yes
Great circle?
No
Constant track?
Yes — but oblique
Shortest route?
No
08Section 8

Explore a Rhumb-Line Route

The same positions as the great-circle tool, now solved with the loxodromic equations.

Interactive educational diagram

Explore a rhumb-line route

Set any departure and destination. The great circle is shown faintly for reference — notice the rhumb line never wanders from its single track.

Rhumb-line route between the selected departure and destinationA perfectly round Earth showing a purple rhumb line from 51°29′N 000°28′W to 40°38′N 073°47′W on a constant true track of 258°, covering 3,109 NM. The corresponding great circle is shown as a faint red line for comparison.N poleAB

Preset city pairs

Classic mid-latitude North Atlantic route: the great circle passes well north of the rhumb line.

51.5° N
-89.589.5
000°28′W
-180180
40.6° N
-89.589.5
073°47′W
-180180
Constant true track
258° (WSW)
Rhumb-line distance
3,109 NM
Change of longitude
73.3° W
Change of latitude
10.8° S
Route sense
Westbound
Hemispheres
Northern throughout
How it is calculated

Worked in radians, with Δλ taken as the shorter difference in longitude:

Δψ = ln[ tan(π/4 + φ₂/2) / tan(π/4 + φ₁/2) ]

q = Δφ / Δψ  (q = cos φ₁ when |Δψ| is negligible, e.g. along a parallel)

d = R · √( Δφ² + q²·Δλ² )

θ = atan2( Δλ , Δψ ) , normalised to 000°–359°

The Δψ term is the Mercator latitude difference, which is exactly why a rhumb line plots straight on a Mercator chart.

09Section 9

Great Circle and Rhumb Line on a Mercator Chart

Change the projection and the two routes swap appearances: the rhumb line becomes straight and the great circle bends towards the nearer pole.

Interactive educational diagram

Mercator chart comparison

Toggle each route on and off. The rhumb line is drawn from the same loxodromic equations as the globe view — it plots straight because Mercator preserves angles.

Mercator chart showing the great circle curved and the rhumb line straightA mathematical Mercator graticule from 78 degrees North to 78 degrees South. The rhumb line between the two positions plots as a straight line on a constant track of 258° and measures 3,109 NM. The great circle plots as a curve concave to the Equator, reaching a higher latitude, and measures 2,991 NM.180°W150°W120°W90°W60°W30°W000°30°E60°E90°E120°E150°E180°E75° S60° S45° S30° S15° SEquator 0°15° N30° N45° N60° N75° NABGreat circle · 2,991 NMRhumb line · 3,109 NMInitial 288°Final 231°Constant 258°
Chart layers
Great circle
2,991 NM
288° → 231°
Rhumb line
3,109 NM
Constant 258°

Preset city pairs

Classic mid-latitude North Atlantic route: the great circle passes well north of the rhumb line.

  • The rhumb line is straight because Mercator preserves angles.
  • The great circle usually appears curved.
  • The great-circle curve normally bends toward the nearer pole.
  • The Equator and meridians are special cases that appear straight and are great circles.
  • This chart deliberately shows a mathematical graticule only. No coastlines are drawn, so nothing on it is invented geography.
10Section 10

Compare Both Routes

Interactive educational diagram

Compare both routes

Both columns use the same departure and destination, so every difference you see comes from the geometry alone.

Great circle

solid line
Distance
2,991 NM
Δσ = 49.82°
Initial true track
288°
Final true track
231°
Vertex latitude
53.7° N
Crossed on this leg
Track changes?
Yes, 56.6° in total
Shortest route?
Yes — by definition on a sphere

Rhumb line

dashed line
Distance
3,109 NM
Constant true track
258°
Distance penalty
118 NM · 3.9%
Track changes?
No — one track throughout
Straight on Mercator?
Yes
Also a great circle?
No

The rhumb-line route is 118 NM longer, an increase of 3.9%.

Preset city pairs

Classic mid-latitude North Atlantic route: the great circle passes well north of the rhumb line.

51.5° N
-89.589.5
000°28′W
-180180
40.6° N
-89.589.5
073°47′W
-180180
How the penalty is calculated

Distance difference = rhumb-line distance − great-circle distance

Percentage penalty = [ (rhumb − great circle) / great circle ] × 100

Both distances use the same mean Earth radius, so the comparison is consistent. A percentage is only meaningful alongside the absolute figure: 2% of a 200 NM leg is 4 NM, which no flight plan would notice.

11Section 11

When Does the Difference Matter?

The geometric penalty depends on length, orientation and latitude — and geometry is only one input to a real flight plan.

  • Over short distances, the difference may be negligible.
  • Over long east-west routes at higher latitudes, the difference can become substantial.
  • Great-circle benefits are often more noticeable at higher latitudes.
  • Operational routing may still be influenced by airways, weather, restricted airspace, winds and traffic flow.

Short local route

Amsterdam → Brussels

Great circle
85 NM
Rhumb line
85 NM
Difference
< 1 NM · 0.0%

Negligible — well under 1 NM, far inside normal flight-planning tolerance.

Medium continental route

London → Istanbul

Great circle
1,344 NM
Rhumb line
1,352 NM
Difference
8 NM · 0.6%

Minor — 8 NM on this leg, rarely a routing driver on its own.

Long intercontinental route

Athens → Tokyo

Great circle
5,136 NM
Rhumb line
5,574 NM
Difference
438 NM · 8.5%

Substantial — 438 NM saved by flying the great circle, which is real fuel and time.

Shortest geometric distance does not automatically mean shortest flight time or lowest fuel burn: wind, airway structure, airspace restrictions and traffic flow all matter.

12Section 12

Worked Examples

Read each scenario, reason it through with the steps, then reveal the answer.

Two points joined along the EquatorA spherical Earth with two positions on the Equator joined by a bold arc along the Equator itself, which is both a great circle and a rhumb line.AB

Example 1

Two points on the Equator

Is the route along the Equator a great circle, a rhumb line, or both?

  1. S1The Equator's plane contains the centre of Earth, so it satisfies the definition of a great circle.
  2. S2The Equator also cuts every meridian at exactly 90°, which is a constant angle.
  3. S3A constant angle with every meridian is the definition of a rhumb line.
A route along one meridian from 30 North to 60 NorthA spherical Earth with a bold arc running north along one meridian from 30 degrees North to 60 degrees North, on a constant track of 000 degrees.30N60N

Example 2

Along one meridian, 30° N to 60° N

What kind of route is this, and what is the true track?

  1. S1A meridian together with its anti-meridian forms a complete circle whose plane contains Earth's centre.
  2. S2Flying along a meridian means the track is aligned with local north everywhere.
  3. S3The angle with every meridian crossed (only one is followed) is therefore constant.
A route along the 50 North parallel compared with the great circleA spherical Earth with a purple arc following the 50 degrees North parallel and a red great-circle arc between the same two positions passing north of it, showing that the parallel is longer.AB

Example 3

Along latitude 50° N, eastbound

Is this route a great circle, a rhumb line, or both?

  1. S1The plane of the 50° N parallel is parallel to the Equator but well above Earth's centre.
  2. S2A plane that misses the centre produces a small circle, not a great circle.
  3. S3The parallel does cross every meridian at 90°, so the track stays constant at 090°.
Mercator chart with a curved great circle and a straight rhumb lineOn a Mercator graticule the rhumb line between two mid-latitude positions is a straight dashed purple line, while the great circle is a red curve bowing towards the North Pole.Equator

Example 4

A long mid-latitude route curving polewards on a Mercator chart

A long route between two mid-latitude points appears curved toward the pole on a Mercator chart. What is it?

  1. S1Mercator is conformal, so a constant-track rhumb line must plot as a straight line.
  2. S2The plotted route is curved, so it cannot be a rhumb line.
  3. S3A great circle plots as a curve concave to the Equator — bending toward the nearer pole.
A route crossing every meridian at the same angleA spherical Earth with a purple rhumb line crossing several meridians, with the identical crossing angle marked at each meridian.AB

Example 5

A route crossing every meridian at the same angle

What is this route called, and is it the shortest path?

  1. S1A constant angle with every meridian is precisely the definition of a loxodrome.
  2. S2Only the Equator and the meridians combine that property with a great circle.
  3. S3For any other oblique case the great circle between the same points is shorter.
13Section 13

Common Mistakes

Each claim below is one of the classic traps. Open the panel for the correction.

“A straight line on every chart is the shortest route.”Incorrect — open for the correction

It depends on the projection. On a Mercator chart a straight line is a rhumb line; on a Lambert or polar stereographic chart a straight line is very nearly a great circle.

“A great-circle track is constant.”Incorrect — open for the correction

The track normally changes continuously, because local north changes direction along the route. The Equator and the meridians are the only exceptions.

“Every parallel is a great circle.”Incorrect — open for the correction

Only the Equator is. Every other parallel is a small circle, because its plane does not pass through Earth's centre.

“A rhumb line is always much longer.”Incorrect — open for the correction

The difference may be very small on short routes, and it is exactly zero along the Equator or along a meridian. The penalty grows with distance, with east-west orientation and with latitude.

“The route midpoint is always the vertex.”Incorrect — open for the correction

The vertex is where the great circle reaches its maximum latitude. It coincides with the midpoint only when departure and destination are at the same latitude — and it may lie entirely outside the segment flown.

“A Mercator chart shows great circles as straight lines.”Generally incorrect — open for the correction

Great circles plot as curves concave to the Equator. Only the Equator itself and the meridians plot as straight lines and are also great circles.

“A rhumb line reaches the pole.”Needs care — open for the correction

An oblique rhumb line approaches and spirals toward the pole mathematically, circling it an infinite number of times without ever arriving. A meridian is the exception: it does reach the pole.

14Section 14

Knowledge Check

Fourteen questions in ATPL style. One retry is allowed per question, 75% is a pass, and you can restart as often as you like.

Question 1 of 14 · Multiple choice

Great-circle definition

0/14

Which route is normally the shortest path between two points on Earth?
15Section 15

Chapter Summary

Everything from this chapter on one revision sheet.

Great circle

  • Plane

    Passes through Earth's centre

  • Division

    Divides Earth into equal hemispheres

  • Distance

    Shortest spherical route

  • Track

    True track normally changes

  • On Mercator

    Curved, concave to the Equator

  • Vertices

    Has a northern and a southern vertex

Rhumb line

  • Angle

    Constant angle with all meridians

  • Track

    Constant true track

  • Distance

    Normally longer

  • On Mercator

    Straight line

  • Extended

    Spirals toward the poles

  • Practical

    Historically easy to steer

Special cases

  • Equator

    Great circle and rhumb line

  • Meridian

    Great circle and rhumb line

  • Other parallels

    Rhumb line only

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