3-D Earth Geometry
An interesting topic in 3-dimensional geometry is Earth geometry. The Earth is very close to a sphere (ball) shape, with an average radius of `6371\ "km"`. (It's actually a bit flat at the poles, but only by a small amount).
Earth geometry is a special case of spherical geometry. When we measure distances that a boat or aircraft travels between any 2 places on the Earth, we do not use straight line distances, since we need to go around the curve of the Earth from one place to another. (Think about the direct or straight-line distance between London and Sydney, through the Earth. That's going to be a lot less than the distance a plane flies around the surface of the Earth.)
Let's start with an example. What distance does a plane fly between Beijing, China and Perth, Western Australia?
To figure this out, we need to understand latitude and longitude first.
Latitude and Longitude
First, we represent the Earth by a sphere:
We slice the Earth through the equator and remove the top. We show the Earth's axis which (by convention) points to North at the top.
Parallels of Latitude
We want to be able to indicate how far North or South of the equator we are for any point on the Earth's surface.
For example, we wish to connect all the points that are 30° N of the equator. We do that by drawing a line from the centre of the Earth to the surface of the Earth, 30° up from the horizontal. We then draw a circle around the Earth parallel to the equator through the point where the 30° line meets the surface of the Earth. We can do the same thing for below the equator, in a southerly direction.
The angle from the horizontal through the equator determines the name given to each parallel of latitude. Five parallels of latitude are shown in the diagram; 30° N, 60° N, 0° N (the equator), 30° S and 60° S.
Each 1° of latitude on the Earth represents 60 nautical miles, or around 110.9 km.
Of course, 90° N represents the North pole and 90° S is the south pole.
Parallels of latitude give an indication of where a place (or a boat or a plane) are positioned in a northerly or southerly direction from 0° N (the equator).
Lines of Longitude
What about the East and West directions for a point on the Earth?
We choose some point on the equator and draw a line from the centre of the Earth (point C) to the chosen point on the equator. We then draw a circle around the whole Earth so that it goes through the point on the equator and the North and South poles. (Historically, the position for 0° was chosen so that the circle we have drawn goes through Greenwich in England).
The line we drew above is called a line of longitude. At the 'front' of our diagram, the line is called 0° E and at the back, the line is called 180° E, for reasons we'll see in a moment.
[The line of longitude that goes through 0° has a special name: the prime meridian.]
Lines of longitude have the same radius as the radius of the Earth. They are examples of great circles. (A great circle has the same radius and same centre as the earth itself).
Let's now define some more lines of longitude. We move around the Earth in an Easterly direction by 90° and draw a great circle through the poles as before. The front line of longitude is labeled 90° E and the one opposite to it at the back of our diagram is labeled 90° W.
We could then draw the lines of longitude for every angle between 0° E and 180° E (covering Europe through India, Asia and the Pacific Ocean) and then through 0° W and 180° W (through the Atlantic Ocean, the Americas and the Eastern Pacific Ocean).
Note: 0° E = 0° W = 0° and 180° E = 180° W = 180°.
Lines of longitude give an indication of where a place (or a boat or a plane) are positioned in an easterly or westerly direction from 0° E.
For lines of longitude, we cannot say that 1° represents any particular distance, because the physical distance changes between each line of longitude as we move from the equator to the poles. At the equator, 1° represents around 60 nautical miles or 111 km. but at the poles, it represents 0 km.
Finer Divisions of Longitude and Latitude
Since 1° of latitude represents 60 nautical miles on the Earth's surface, we need finer divisions of 1° to accurately determine places on the Earth's surface.
As we learned earlier in the Angles section, we can divide 1° into smaller divisions called minutes (where 1° = 60 minutes) or seconds (where 60 seconds = 1 minute).
For example, London is 51°32' North of the equator. This means 51 degrees plus 32/60 of a degree North.
Historically, it was the Babylonians who divided units of time and angles into 60 equal parts.
Examples of Latitudes and Longitudes for Cities
Any place on the Earth can be uniquely described by its latitude and longitude.
Some examples (moving from West to East):
Los Angeles: 34° 3' N, 118° 14' W
New York: 40° 45' N, 73° 59' W
London: 51°32' N, 0° 5' W. (London is just slightly west of Greenwich.)
New Delhi: 28°35' N, 77°12' E
Singapore: 1° 21' N, 103° 49' E (Singapore is almost on the equator).
Sydney: 33° 52' S, 151° 12' E
What is the Distance Between 2 Places on the Earth?
We need one more key bit of information to solve the problem posed before (distance from Beijing to Perth). What is the distance around the curve of the Earth along a great circle between any 2 points?
Let's call the latitude of the first place φ1 and the longitude of the first place as λ1. Similarly, the latitude of the second place is φ2 and the longitude of the second place is λ2.
We need to know the central angle (the angle from the centre of the Earth to the 2 places of interest). The formula for this is (angles in radians):
Central angle =
`2\ arcsin sqrt( sin^2 ((varphi_2-varphi_1)/2) +cos\ varphi_1\ cos\ varphi_2\ sin^2 ((lambda_2-lambda_1)/2) )`
(See Haversine Formula for a derivation of the above formula.)
In the case of Beijing to Perth, the central angle looks like the following, with O at the centre of the earth:
For Beijing, the latitude is φ1 = 39° 54' N and the longitude is λ1 = 116° 24' E.
For Perth, the latitude is φ2 = 31° 57' S and the longitude is λ2 = 115° 52' E.
We need to express these angles in radians (if you are rusty, see Radians):
Beijing: φ1 = 39° 54' N = 0.696386; λ1 = 116° 24' E = 2.031563.
Perth: φ2 = 31° 57' S = −0.557633; λ2 = 115° 52' E = 2.022255.
[Note that the latitude for Perth is negative, since it is South of the equator.]
Applying the formula gives us:
Central angle =
`2\ arcsin sqrt( sin^2 ((-1.254)/2) +cos\ (0.696)\ cos\ (-0.557)\ sin^2 ((-0.009)/2) )`
(radians, of course, and full calculator accuracy was used throughout, but not shown)
Now, to find the distance travelled, we need to use our formula for arc length that we learned before (see Arc Length). If r is the radius of the great circle and θ is the angle subtended at the centre (in radians), the arc length s is given by:
s = rθ
Now, the average radius of the Earth is `6371\ "km"`, and the angle we just found is `1.254049` radians, so the flying distance from Beijing to Perth is given by:
s = 6371 × 1.254049 = 7989 km.
Find the distance that an aircraft must fly from London to Los Angeles if it flies directly.