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Alternative 3D Coordinates Help

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

Alternative 3D Coordinates - Latitude and Longitude

Here are some coordinate systems that are used in mathematics and science when working in 3D space.

Latitude And Longitude

Latitude and longitude angles 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. 10-15A. The polar axis connects two specified points at antipodes 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 90° angle to the polar axis. It is assigned longitude φ = 0°.

 

Alternative Coordinates Alternative 3D Coordinates Latitude And Longitude

Alternative Coordinates Alternative 3D Coordinates Latitude And Longitude

Fig. 10-15 . (A) Latitude and longitude coordinates for locating points on the earth’s surface. (B) Declination and right ascension coordinates for locating points in the sky.

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

− 90° ≤ θ ≤ 90°

− 180° < φ ≥ 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.

Space and Time - Celestial Coordinates and Hours, Minutes, and Seconds

Celestial Coordinates - Celestial Latitude and Longitude

Celestial latitude and celestial longitude are extensions of the earth’s latitude and longitude 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,φ) appears at the zenith in the sky (directly overhead) from the point on the earth’s surface whose latitude and longitude coordinates are (θ,φ) .

Alternative Coordinates Alternative 3D Coordinates Latitude And Longitude

Fig. 10-15 . (A) Latitude and longitude coordinates for locating points on the earth’s surface. (B) Declination and right ascension coordinates for locating points in the sky.

Celestial Coordinates - Declination and Right Ascension

Declination and right ascension define the positions of objects in the sky relative to the stars. Figure 10-15B applies to this system. Declination (θ) is identical to celestial latitude. Right ascension (φ) is measured eastward from the vernal equinox (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° ≤ φ < 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°. As if that isn’t confusing enough, the minutes and seconds of right ascension are not the same as the fractional degree units by the same names more often encountered. One minute of right ascension is 1/60 of an hour or ¼ of a degree, and one second of right ascension is 1/60 of a minute or 1/240 of a degree.

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