What is a great circle?

When I first learned about great circles, I found them to be one of the most eye-opening, head-scratching and strangely beautiful concepts in navigation. 

I had not come across them at all before my Instrument Rating aviation training and there is a good reason for this – this is the moment when navigation planning goes big.*

What Is a Great Circle?

A Great Circle – some simple definitions:

The line that divides a sphere in two. 

OR

Any circle that passes through two points that are opposite each other on a sphere. 

The largest circle that will fit around snuggly a sphere. Imagine a tight hoop that a beachball could only just fit perfectly through.

Great circles are longest circles you can draw on a sphere.

The earth is not a perfect sphere, but we can think of it as one. The Equator is a great circle, but the Tropics of Cancer and Capricorn are not (they are smaller than the Equator). And the Arctic and Antarctic Circles are even smaller than the Tropics and definitely not great circles.

What Is a Great Circle?On the left globe, all the lines running top to bottom (longitude) are great circles. But the Equator is the only one of the horizontal lines (latitude) one that is. 

The meridians – or lines of longitude – are also perfect circles. The Greenwich Meridian goes through Greenwich and also the North and South Poles and any line that goes through both the North and South Poles must be a great circle, because the Poles are opposite each other.

(Remember: any circle that passes through two points opposite each other on a sphere is a great circle.)

Why are great circles important in navigation? Because they show us the shortest routes between two points on a sphere. If we want to travel the shortest distance across any sphere, Earth being the obvious choice for most of us, you actually need to head towards the point on the opposite side of that sphere.

So far, reasonably logical and intuitive maybe. But things get a little bit more interesting when we bring cardinal directions and the compass into it. Why?

Because great circles mean you often have to head north to arrive somewhere a little south of you. And vice versa. Now maybe you can see why I found this concept a head-scratcher at first. But bear with it, it explains a lot of what you’ll see in long distance journeys. Let’s look at a real example. 

The other day I was on an Air Canada flight from Calgary, Alberta to London. These cities are on roughly the same latitude, approximately 51 N. So surely to fly from Calgary to London, which is almost exactly due east, we should fly due east?

Well we could, but it would be a much longer journey than if we found the great circle route and took that. And to do that we need to fly north of east for the first part of the journey, then east, then south of east. Because this is the straight line way to do it and it will shave hours off the journey. 

Below are the photos I took of the back of the seat route map on Air Canada.

What Is a Great Circle?The great circle track from Calgary to London
Heading away from Calgary, note the compass heading: Northeast.

What Is a Great Circle?Arriving into the UK, notice the compass heading: south of east.
Approaching London, heading well south of east.

 Time to see if you’ve got this nailed. Imagine you need to start a journey from London to Nagasaki, Japan and want to follow the shortest route. Japan is of course well to the east of London.

It is also more than a thousand miles south of London.

Have a guess of the heading you’d set off in to follow a great circle, then see how you did by reading the answer in this post: London to Nagasaki by Great Circle.

**(There are two types of flying – the type when you can see stuff and the type when you can’t. The Instrument Rating (‘IR’) is one of the biggest stepping stones in aviation, it is the ticket that qualifies a pilot to fly in what pilots call Instrument Meteorological Conditions (‘IMC’) – and what normal folk call CLOUDS.

See also:  How to write great dialogue

To get this rating you have to pass much more serious exams than are necessary to get a basic pilot’s licence. They include the sort of navigation and route planning necessary to fly a large jet thousands of miles. And this is where Great Circles come in.) 

Why Are Great Circles the Shortest Flight Path?

What Is a Great Circle?

Or why is it that when you see flight paths on a map they always take a curved route between 2 cities?

It’s because planes travel along the true shortest route in a 3-dimensional space.

This route is called a geodesic or great circle route. They are common in navigation, sailing and aviation.

But geodesics can be confusing when you’re looking at a 2-dimensional map as they follow quite the odd flight path. Let’s dig into this concept a bit deeper.

Great circle routes explained

In a flight path from New York to Madrid, if I asked you which line is shorter, you’d say the straight one, right?

What Is a Great Circle?

However, a straight line in a 2-dimensional map is not the same as a straight line on a 3-dimensional globe.

This is why flight paths take an arc route between an origin and a destination.

What is meaning of Great Circle Sailing ?

Great circle sailing involves the solution of courses, distances, and points along a great circle between two points.

Great Circle Sailing is used for long ocean passages. For this purpose, the earth is considered a perfect spherical shape; therefore, the shortest distance between two points on its surface is the arc of the great circle containing two points.

As the track is the circle, so the course is constantly changing, and the track must be broken down into a series of short rhumb lines at frequent intervals that can be used to sail on the Mercator chart. Doing this, the navigator would use the Gnomonic charts combined with the Mercator charts to draw the sailing track.

What Is a Great Circle?

In Other words we can understand great circle as:

A great circle is the shortest path between two points along the surface of a sphere. A great circle is the intersection of the surface with a plane passing through the center of the planet.

The equator and all meridians are great circles. All great circles other than these do not have a constant azimuth, the spherical analog of slope; they cross successive meridians at different angles. The Gnomonic Projection represents arbitrary great circles as straight lines.

Great circles are examples of geodesics. A geodesic is the shortest possible path constrained to lie on a curved surface, independent of the choice of a coordinate system.What Is a Great Circle?

Find Dist:

COS AB = (COS PA X COS PB) + (SIN PA X SIN PB X COS P)

Find Initial Course:

                ( COS PB – COS AB X COS PA)
COS A = ———————————–
                             (  SIN AB X SIN PA)

What is Great Circle?

A Great Circle is any circle that circumnavigates the Earth and
passes through the center of the Earth. A great circle always
divides the Earth in half, thus the Equator is a great circle (but
no other latitudes) and all lines
of longitude are great circles.
The shortest distance between any two points on the Earth lies along
a great circle.

Maptitude Mapping Software
gives you all of the tools, maps, and data you need to analyze and
understand how geography affects you and your business. Maptitude includes
tools for creating great circles.

Learn More  Buy Now  Free for Students/Teachers

What Is a Great Circle?

Great circles intersect the center of the Earth
and divide the Earth in half. The shortest distance between any two
locations on the Earth, such as New York City and New Delhi, lies
along a great circle.

Great circle

This article is about the mathematical notion. For its applications in geodesy, see great-circle distance. For other uses, see The Great Circle.

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (March 2018) (Learn how and when to remove this template message)
See also:  Punctuating questions

A great circle divides the sphere in two equal hemispheres

A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a small circle, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

For most pairs of distinct points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them.

In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry. The length of the minor arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry where such great circles are called Riemannian circles.

These great circles are the geodesics of the sphere.

The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center.
In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn + 1.

Derivation of shortest paths

See also: Great-circle distance

To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it.

Consider the class of all regular paths from a point

p

{displaystyle p}

to another point

q

{displaystyle q}

. Introduce spherical coordinates so that

p

{displaystyle p}

coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by

θ
=
θ
(
t
)
,

ϕ
=
ϕ
(
t
)
,

a

t

b

{displaystyle heta = heta (t),quad phi =phi (t),quad aleq tleq b}

provided we allow

ϕ

{displaystyle phi }

to take on arbitrary real values. The infinitesimal arc length in these coordinates is

d
s
=
r

θ

2

+

ϕ

2

sin

2


θ

d
t
.

{displaystyle ds=r{sqrt { heta '^{2}+phi '^{2}sin ^{2} heta }},dt.}

So the length of a curve

γ

{displaystyle gamma }

from

p

{displaystyle p}

to

q

{displaystyle q}

is a functional of the curve given by

S
[
γ
]
=
r

a

b

θ

2

+

ϕ

2

sin

2


θ

d
t
.

{displaystyle S[gamma ]=rint _{a}^{b}{sqrt { heta '^{2}+phi '^{2}sin ^{2} heta }},dt.}

According to the Euler–Lagrange equation,

S
[
γ
]

{displaystyle S[gamma ]}

is minimized if and only if

sin

2


θ

ϕ

θ

2

+

ϕ

2

sin

2


θ

=
C

{displaystyle {frac {sin ^{2} heta phi '}{sqrt { heta '^{2}+phi '^{2}sin ^{2} heta }}}=C}

,

where

C

{displaystyle C}

is a

t

{displaystyle t}

-independent constant, and

sin

θ
cos

θ

ϕ

2

θ

2

+

ϕ

2

sin

2


θ

=

d

d
t

θ

θ

2

+

ϕ

2

sin

2


θ

.

{displaystyle {frac {sin heta cos heta phi '^{2}}{sqrt { heta '^{2}+phi '^{2}sin ^{2} heta }}}={frac {d}{dt}}{frac { heta '}{sqrt { heta '^{2}+phi '^{2}sin ^{2} heta }}}.}

From the first equation of these two, it can be obtained that

ϕ

=

C

θ

sin

θ

sin

2


θ

C

2

{displaystyle phi '={frac {C heta '}{sin heta {sqrt {sin ^{2} heta -C^{2}}}}}}

.

Integrating both sides and considering the boundary condition, the real solution of

C

{displaystyle C}

is zero.

Thus,

ϕ

=
0

{displaystyle phi '=0}

and

θ

{displaystyle heta }

can be any value between 0 and

θ

0

{displaystyle heta _{0}}

, indicating that the curve must lie on a meridian of the sphere. In Cartesian coordinates, this is

x
sin

ϕ

0


y
cos

ϕ

0

=
0

{displaystyle xsin phi _{0}-ycos phi _{0}=0}

See also:  Possessives

which is a plane through the origin, i.e., the center of the sphere.

Applications

Some examples of great circles on the celestial sphere include the celestial horizon, the celestial equator, and the ecliptic. Great circles are also used as rather accurate approximations of geodesics on the Earth's surface for air or sea navigation (although it is not a perfect sphere), as well as on spheroidal celestial bodies.

The equator of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the land and water hemispheres. A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point.

The Funk transform integrates a function along all great circles of the sphere.

See also

  • Great-circle distance
  • Great-circle navigation
  • Rhumb line

External links

  • Great Circle – from MathWorld Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
  • Great Circles on Mercator's Chart by John Snyder with additional contributions by Jeff Bryant, Pratik Desai, and Carl Woll, Wolfram Demonstrations Project.
  • Navigational Algorithms Paper: The Sailings.
  • Chart Work – Navigational Algorithms Chart Work free software: Rhumb line, Great Circle, Composite sailing, Meridional parts. Lines of position Piloting – currents and coastal fix.

great circle | Примеры предложений

Словарь > Примеры для great circle

great circle isn’t in the Cambridge Dictionary yet. You can help!

For these samples the pole of the best-fit plane defined by the great circle path of the data was calculated. Из Cambridge English Corpus Each of these two-sphere generators have a great circle with the same name as a generator for the planar problem. Из Cambridge English Corpus Each of these two-spheres generators have a great circle with the same name as a generator for the planar problem. Из Cambridge English Corpus From what is left we remove all strips centered at great circle arcs that are intersections of twodimensional planes with the unit sphere while retaining the central arcs. Из Cambridge English Corpus This remains true on a hemisphere (within which the axis is a great semi-circle), but a full great circle could contain as many as 2t + 1 distinct points. Из Cambridge English Corpus Fuller was to construct or propose a series of transitional domes between 1949 and 1955 that corrected the mistaken structural assumptions of the first great circle dome. Из Cambridge English Corpus You must deal with this exceptional situation in such a way as not to violate the sense of moral justice in a great circle of these people. In order to reduce the distance to the 1,860 miles you have to fly a great circle, which takes you in the north up amongst the icebergs. This is not the shortest distance between the chosen endpoints on the parallel because a parallel is not a great circle. Из

Wikipedia

Этот пример из Wikipedia и может быть повторно использован лицензией CC BY-SA.

Topographic isolation is the minimum great circle distance to a point of higher elevation. Из

Wikipedia

Этот пример из Wikipedia и может быть повторно использован лицензией CC BY-SA.

The 109th meridian west forms a great circle with the 71st meridian east. Из

Wikipedia

Этот пример из Wikipedia и может быть повторно использован лицензией CC BY-SA.

The 70th meridian east forms a great circle with the 110th meridian west. Из

Wikipedia

Этот пример из Wikipedia и может быть повторно использован лицензией CC BY-SA.

The great circle separating hemispheres is called celestial horizon or rational horizon. Из

Wikipedia

Этот пример из Wikipedia и может быть повторно использован лицензией CC BY-SA.

The 115th meridian east forms a great circle with the 65th meridian west. Из

Wikipedia

Этот пример из Wikipedia и может быть повторно использован лицензией CC BY-SA.

Любые мнения в примерах не отражают мнение редакторов Cambridge Dictionary или издательства Cambridge University Press или ее лицензиаров.

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