Different degree formats: Resolutions and conversions

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Latitude

The latitude of a point on Earth is the angle between the equatorial plane and the line passing through the point at right angle to the surface of the ellipsoid (see figure above). Earth is divided into 90 degrees of latitude going both northwards and southwards from the equator (0°) to each pole (90° N and 90° S). (Northern positions are considered to be positive (+) and southern positions to be negative (−).)

Longitude

The longitude of a point on Earth is the angle along the equator between the prime meridian passing through Greenwich and the meridian passing through the point (see figure above). Earth is divided into 180° degrees of longitude going both eastwards and westwards from the prime meridian (0°) until meeting approximately at the international date line (180° E = 180° W). (Eastern positions are considered to be positive (+) and western positions to be negative (−).) Divisions One degree ( ° ) is normally divided into 60 minutes ( ′ ) or decimals. One minute ( ′ ) is normally divided into 60 seconds ( ″ ) or decimals. One second ( ″ ) is normally divided into decimals (when divided).

Hence

   1° (degree)
  90° (degrees) 	
3600″ (seconds).
5400′ (minutes).

Widths of smallest units

The widths below are based on numbers found in the main table (21 KB).

Degree formats Smallest unit Width of smallest unit
At the poles At the equator
Fraction Decimal degree Decimal minute Decimal second Latitude Latitude Longitude
d (d)dd.ddddd° 1/100 000° 0.00001°     0.00060′ 0.036″   1.12 m   1.11 m   1.11 m
(d)dd.dddd° 1/10 000° 0.0001°       0.00600′ 0.360″ 11.17 m 11.06 m 11.13 m
D m <NOBR>(D)DD° mm.mmm′</NOBR> 1/1000 0.0000167° 0.001′     0.060″   1.86 m   1.84 m   1.86 m
(D)DD° mm.mm′ 1/100 0.0001667° 0.01′       0.600″ 18.62 m 18.43 m 18.55 m
<NOBR>D M s</NOBR> (D)DD° MM′ ss.s″ 1/10 0.0000278° 0.00167′ 0.1″       3.10 m   3.07 m   3.09 m
(D)DD° MM′ ss″ 1/1 0.0002778° 0.01667′ 1″        31.03 m 30.72 m 30.92 m

Note!

Be aware that the resolution indicated by most recreational GPS receivers is much better than their position accuracy. The position accuracy of most non-augmented 12-channel parallel recreational GPS receivers is specified as better than 15 m (95 % of the time) when reception conditions are good (and SA (Selective Availability) is switched off).

Why is a minute of latitude longer at the poles than at the equator? As the equator is farther from the centre of Earth than the poles are, how can a minute of latitude be longer at the poles than at the equator? It is correct that a minute of arc is longer on the surfaces of large spheres than on the surfaces of small ones. On the surfaces of ellipsoids (and spheroids) the length of a minute of arc at a certain point is dependent on the curvature at that point and not the distance from some common centre. Imagine yourself placed at the North Pole, being able to travel the surrounding area. Further imagine yourself being able to make very accurate observations there. Based on these observations you can calculate that you are placed on a sphere with a radius of 6399.6 km. Then imagine yourself moved to a point on the equator, being able to travel north and south of this point. Here you do the same type of observations. Based on these observations you can now calculate that you are placed on a sphere with a radius of 6335.4 km (64.2 km shorter!). Now the crucial question: which sphere of the two has the longest minute of arc on its surface?

Degree conversion formula

Most GPS receivers can display the different degree formats (default format is usually D m). Hence they can be used to convert easily from one format to the other two.

Information Small letters (d, m, s) mean decimal numbers (e.g. 58.65375, 39.225, 13.5). Capital letters (D, M) mean integers (part of number in front of decimal point) (e.g. 58, 39). The TRUNC (truncate) function converts a decimal number to an integer by keeping the part of the number in front of the decimal point and discarding the rest (e.g. TRUNC(58.65375) = 58, TRUNC(−58.65375) = −58). Note: When converting a negative coordinate (e.g. southern or western positions) to another degree format (d, D m or D M s), let all starting values (degrees, minutes, seconds) be negative.

From decimal-degrees (d) to Degrees decimal-minutes (D m)

D = TRUNC(d)
m = (d − D) × 60

Example: d = 58.65375°

D = TRUNC(58.65375°) = 58°
m = (58.65375 − 58) × 60′ = 0.65375 × 60′ = 39.225′
So 58.65375° corresponds to 58° 39.225′.

From decimal-degrees (d) to Degrees Minutes decimal-seconds (D M s)

D = TRUNC(d)
M = TRUNC((d − D) × 60)
s = (d − D − M/60) × 3600 = (d − D) × 3600 − M × 60

Example: d = 58.65375°

D = TRUNC(58.65375°) = 58°
M = TRUNC((58.65375 − 58) × 60′) = TRUNC(39.225′) = 39′
s = (58.65375 − 58 − 39/60) × 3600″ = 0.0375 × 3600″ = 13.5″
s = (58.65375 − 58) × 3600″ − 39 × 60″ = 2353.5″ − 2340″ = 13.5″

So 58.65375° corresponds to 58° 39′ 13.5″.

From Degrees decimal-minutes (D m) to decimal-degrees (d)

d = D + m/60

Example: D = 58°, m = 39.225′

d = 58° + 39.225°/60 = 58° + 0.65375° = 58.65375°
So 58° 39.225′ corresponds to 58.65375°.

From Degrees decimal-minutes (D m) to Degrees Minutes decimal-seconds (D M s)

D = D
M = TRUNC(m)
s = (m − M) × 60

Example: D = 58°, m = 39.225′

D = 58°
M = TRUNC(39.225′) = 39′
s = (39.225 − 39) × 60″ = 0.225 × 60″ = 13.5″

So 58° 39.225′ corresponds to 58° 39′ 13.5″.

From Degrees Minutes decimal-seconds (D M s) to decimal-degrees (d)

d = D + M/60 + s/3600

Example: D = 58°, M = 39′, s = 13.5″

d = 58° + 39°/60 + 13.5°/3600 = 58° + 0.65° + 0.00375° = 58.65375°

So 58° 39′ 13.5″ corresponds to 58.65375°.

From Degrees Minutes decimal-seconds (D M s) to Degrees decimal-minutes (D m)

D = D
m = M + s/60

Example: D = 58°, M = 39′, s = 13.5″

D = 58°
m = 39′ + 13.5′/60 = 39′ + 0.225′ = 39.225′

So 58° 39′ 13.5″ corresponds to 58° 39.225′.

From an NMEA 0183 sentence (Dm,H) to Degrees decimal-minutes Hemisphere (D m H)

NMEA = National Marine Electronics Association (in the US).

Geographical coordinates in NMEA 0183 sentences look like this: …,Dm,H,…

where H = hemisphere (N = north, S = south, E = east, W = west).

Hence the above means: D° m′ H.

D = TRUNC(Dm/100)
m = Dm − 100 × D
H = H

Example: Part of an NMEA 0183 sentence containing a position: …,5839.225,N,00910.660,E,…

D = TRUNC(5839.225/100) = 58°
m = 5839.225 − 100 × 58 = 5839.225 − 5800 = 39.225′
H = N 	     	D = TRUNC(00910.660/100) = 009°
m = 00910.660 − 100 × 009 = 910.660 − 900 = 10.660′
H = E

So the position 5839.225,N,00910.660,E in an NMEA 0183 sentence equals the position 58° 39.225′ N 009° 10.660′ E.

From Degrees decimal-minutes Hemisphere (D m H) to NMEA 0183 (Dm,H) (see above too)

Dm,H = 100 × D + m, H
Example: D = 58°, m = 39.225′, H = N
Dm,H = 100 × 58 + 39.225, N = 5800 + 39.225, N = 5839.225,N

So 58° 39.225′ N equals 5839.225,N in an NMEA 0183 sentence.

Position example

       The coordinate formats  	5839.225,N,00910.660,E
	58.65375° N  009.17767° E
	58° 39.225′ N  009° 10.660′ E
	58° 39′ 13.5″ N  009° 10′ 39.6″ E  all represent the same position.


More decimal-degree (circle = 360 degrees) conversion formulae

From decimal-degrees (d) to radians (r) (circle = 2π radians)

r = d × π/180

Example: d = 58.65375°

r = 58.65375 radians × π/180 = 1.023701 radians

From decimal-degrees (d) to gons or grad(e)s (g) (circle = 400 gons or grad(e)s)

g = d × 10/9

Example: d = 58.65375°

g = 58.65375 gons × 10/9 = 65.17083 gons or grad(e)s

From decimal-degrees (d) to mils (ml) (circle = 6400 mils)

ml = d × 160/9

Example: d = 58.65375°

ml = 58.65375 mils × 160/9 = 1042.733 mils

From radians (r) (circle = 2π radians) to decimal-degrees (d)

d = r × 180/π

Example: r = 1.023701 radians

d = 1.023701° × 180/π = 58.65375°

From gons or grad(e)s (g) (circle = 400 gons or grad(e)s) to decimal-degrees (d)

d = g × 0.9

Example: g = 65.17083 gons or grad(e)s

d = 65.17083° × 0.9 = 58.65375°

From mils (ml) (circle = 6400 mils) to decimal-degrees (d)

d = ml × 9/160

Example: ml = 1042.733 mils

d = 1042.733° × 9/160 = 58.65375°