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°.
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) .
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.
Cylindrical Coordinates
Figures 10-16A and 10-16B show two systems of cylindrical coordinates for specifying the positions of points in three-space.
Schematic for Mathematicians, Engineers, and Scientists
In the system shown in Fig. 10-16A, we start with Cartesian xyz -space. Then an angle θ is defined in the xy -plane, measured in degrees or radians (but usually radians) counterclockwise from the positive x axis, which is called the reference axis . Given a point P in space, consider its projection P′ onto the xy -plane. The position of P is defined by the ordered triple (θ, r,h ) . In this ordered triple, θ represents the angle measured counterclockwise between P′ and the positive x axis in the xy -plane, r represents the distance or radius from P′ to the origin, and h represents the distance, called the altitude or height, of P above the xy -plane. (If h is negative, then P is below the xy -plane.) This scheme for cylindrical coordinates is preferred by mathematicians, and also by some engineers and scientists.
Schematic for Navigators and Aviators
In the system shown in Fig. 10-16B, we again start with Cartesian xyz -space. The xy -plane corresponds to the surface of the earth in the vicinity of the origin, and the z axis runs straight up (positive z values) and down (negative z values). The angle θ is defined in the xy -plane in degrees (but never radians) clockwise from the positive y axis, which corresponds to geographic north. Given a point P in space, consider its projection P′ onto the xy -plane. The position of P is defined by the ordered triple (θ,r,h) , where θ represents the angle measured clockwise between P′ and geographic north, r represents the distance or radius from P′ to the origin, and h represents the altitude or height of P above the xy -plane. (If h is negative, then P is below the xy -plane.) This scheme is preferred by navigators and aviators.
Fig. 10-16 . (A) Mathematician’s form of cylindrical coordinates for defining points in three-space. (B) Astronomer’s and navigator’s form of cylindrical coordinates for defining points in three-space.
Spherical Coordinates
Figures 10-17A to 10-17C show three systems of spherical coordinates for defining points in space. The first two are used by astronomers and aerospace scientists, while the third one is of use to navigators and surveyors.
Angles Represent Declination and Right Ascension - Astronomers and Aerospace Scientists
In the scheme shown in Fig. 10-17A, the location of a point P is defined by the ordered triple (θ,φ, r ) such that θ represents the declination of P, φ represents the right ascension of P , and r represents the distance or radius from P to the origin. In this example, angles are specified in degrees (except in the case of the astronomer’s version of right ascension, which is expressed in hours, minutes, and seconds as defined earlier in this chapter). Alternatively, the angles can be expressed in radians. This system is fixed relative to the stars.
Angles Represent Celestial Latitude and Longitude - Astronomers and Aerospace Scientists
Instead of declination and right ascension, the variables θ and φ can represent celestial latitude and celestial longitude respectively, as shown in Fig. 10-17B. This system is fixed relative to the earth, rather than relative to the stars.
Fig. 10-17 . (A) Spherical coordinates for defining points in three-space, where the angles represent declination and right ascension. (B) Spherical coordinates for defining points in three-space, where the angles represent celestial latitude and longitude.
Angles Represent Elevation and Azimuth - Navigators and Surveyors
There’s yet another alternative: θ can represent elevation (the angle above the horizon) and φ can represent the azimuth (bearing or heading), measured clockwise from geographic north. In this case, the reference plane corresponds to the horizon, not the equator, and the elevation can range between, and including, −90° (the nadir, or the point directly underfoot) and +90° (the zenith, or the point directly overhead). This is shown in Fig. 10-17C. In a variant of this system, the angle θ is measured with respect to the zenith, rather than the horizon. Then the range for this angle is 0° ≤ θ ≤ 180°.
Fig. 10-17 . (C) Spherical coordinates for defining points in three-space, where the angles represent elevation (angle above the horizon) and azimuth (also called bearing or heading).
Alternative 3D Coordinates Practice Problems
PROBLEM 1
What are the celestial latitude and longitude of the sun on the first day of spring, when the sun lies in the plane of the earth’s equator?
SOLUTION 1
The celestial latitude of the sun on the first day of spring (March 21, the vernal equinox) is 0°, which is the same as the latitude of the earth’s equator. The celestial longitude depends on the time of day. It is 0° (the Greenwich meridian) at high noon in Greenwich, England or any other location at 0° longitude. From there, the celestial longitude of the sun proceeds west at the rate of 15° per hour (360° per 24 hours).
PROBLEM 2
Suppose you stand in a huge, perfectly flat field and fly a kite on a string 500 meters long. The wind blows directly from the east. The point on the ground directly below the kite is r meters away from you, and the kite is 400 meters above the ground. If your body represents the origin and the units of a coordinate system are one meter in size, what is the position of the kite in the cylindrical coordinate scheme preferred by navigators and aviators?
SOLUTION 2
The position of the kite is defined by the ordered triple (θ,r,h) , where θ represents the angle measured clockwise from geographic north to a point directly under the kite, r represents the radius from a point on the ground directly under the kite to the origin, and h represents the distance (height or altitude) of the kite above the ground. Because the wind blows from the east, you know that the kite must be directly west of the origin (represented by your body), so θ = 270°. The kite is 400 meters off the ground, so h = 400. The value of r can be found by the Pythagorean theorem:
Therefore, (θ,r,h) = (270°,300,400) in the system of cylindrical coordinates preferred by navigators and aviators.
Practice problems for these concepts can be found at: Polar Coordinates Practice Test.
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