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Spatial Coordinates Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Latitude And Longitude

Latitude and longitude are directional angles that uniquely define the positions of points on the surface of a sphere or in the sky. The scheme for geographic locations on the earth is illustrated in Fig. 6-1 A. The polar axis connects two specified points at antipodes, or points directly opposite each other, on the sphere. These points are assigned latitude θ = 90° (north pole) and θ = –90° (south pole). The equatorial axis runs outward from the center of the sphere at a right angle to the polar axis. It is assigned longitude ø = 0°.

Three-Space and Vectors Spatial Coordinates Latitude And Longitude

Three-Space and Vectors Spatial Coordinates Latitude And Longitude

Fig. 6-1. (A) Latitude and longitude angles for locating points on the earth’s surface. (B) Declination and right ascension angles for locating points in the sky.

Latitude θ is measured positively (north) and negatively (south) relative to the plane of the equator. Longitude ø is measured counterclockwise (positively) and clockwise (negatively) relative to the equatorial axis. The angles are restricted as follows:

– 90° ≤ θ ≤ 90°

– 180° < ø ≤ 180°

On the earth’s surface, the half-circle connecting the 0° longitude line with the poles passes through Greenwich, England (not Greenwich Village in New York City!) and is known as the Greenwich meridian or the prime meridian. Longitude angles are defined with respect to this meridian.

Latitude and longitude angles translate into points on the surface of a sphere (such as the earth’s surface) centered at the point where the polar axis intersects the equatorial plane. But latitude and longitude angles can also translate into positions in the sky. These positions are not really points, but are rays pointing out from the observer’s eyes indefinitely into space.

Celestial Coordinates

Celestial latitude and celestial longitude are extensions of the earth’s latitude and longitude angles into the heavens. The same set of coordinates used for geographic latitude and longitude applies to this system. An object whose celestial latitude and longitude coordinates are (θ, ø) appears at the zenith in the sky, that is, directly overhead, from the point on the earth’s surface whose latitude and longitude coordinates are (θ, ø).

Declination and right ascension define the positions of objects in the sky relative to the stars, rather than the earth. Figure 6-1B applies to this system.

Three-Space and Vectors Spatial Coordinates Latitude And Longitude

Fig. 6-1. (B) Declination and right ascension angles for locating points in the sky.

Declination ( θ ) is identical to celestial latitude. Right ascension (ø) is measured eastward from the vernal equinox, which is the position of the sun in the heavens at the moment spring begins in the northern hemisphere. The angles are restricted as follows:

– 90° ≤ θ ≤ 90°

0° ≤ ø < 360°

Hours, Minutes, And Seconds

Astronomers use a peculiar scheme for right ascension. Instead of expressing the angles of right ascension in degrees or radians, they use hours, minutes , and seconds based on 24 hours in a complete circle, corresponding to the 24 hours in a day. That means each hour of right ascension is equivalent to 15°.

If that isn’t confusing enough for you, the minutes and seconds of right ascension are not the same as the fractional degree units by the same names that are used by mathematicians and engineers. One minute of right ascension is 1/60 of an hour or Three-Space and Vectors Spatial Coordinates Hours, Minutes, And Seconds of an angular degree, and one second of right ascension is 1/60 of a minute or 1/240 of an angular degree.

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