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°.
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.
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 
of an angular degree, and one second of right ascension is 1/60 of a minute or 1/240 of an angular degree.
Cartesian Three-space
An extension of rectangular coordinates into three dimensions is Cartesian three-space (Fig. 6-2), also called xyz-space . The independent variables are usually plotted along the x and y axes; the dependent variable is plotted along the z axis. Each axis is perpendicular to the other two. They all intersect at the origin , which is usually the point where x = 0, y = 0, and z = 0.
Fig. 6-2. Cartesian three-space, also called xyz -space.
The scales in Cartesian three-space are all linear. This means that, along any given individual axis, equal distances represent equal changes in value. But the divisions (that is, the spaces between hash marks) on different axes do not necessarily have to represent the same increments. For example, the x axis might be designated as having 1 unit per division, the y axis 10 units per division, and the z axis 5 units per division.
Points in Cartesian three-space are represented by ordered triples ( x , y , z ). As with ordered pairs, there are no spaces between the variables and the commas when denoting an ordered triple; they’re all scrunched up together.
Cylindrical Coordinates
Figure 6-3 shows two systems of cylindrical coordinates for specifying the positions of points in three-space.
In the system shown in Fig. 6-3A, we start with Cartesian xyz -space. Then an angle θ is defined in the xy -plane, measured in degrees or radians (usually radians) counterclockwise from the positive x axis or reference axis. Given a point P in space, consider its projection P′ onto the xy -plane, such that line segment PP′ is parallel to the z axis. 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 (altitude or height) of P above the xy -plane. This scheme for cylindrical coordinates is preferred by mathematicians, and also by some engineers and scientists.
Fig. 6-3. (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.
In the system shown in Fig. 6-3B, 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, such that line segment PP′ is parallel to the z axis. 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 rom P′ to the origin, and h represents the distance (altitude or height) of P above the xy -plane. This scheme is preferred by navigators and astronomers.
Spherical Coordinates
Figure 6-4 shows 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 preferred by navigators and surveyors.
In the scheme shown in Fig. 6-4A, the location of a point P is defined by the ordered triple ( θ,ø,r ) such that θ represents the declination of P, ø represents the right ascension of P, and r represents the radius from P to the origin, also called the range. 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.
Instead of declination and right ascension, the variables θ and ø can represent celestial latitude and celestial longitude respectively, as shown in Fig. 6-4B. This system is fixed relative to the earth, rather than relative to the stars.
Fig. 6-4. (A) Spherical coordinates for defining points in three-space, where the angles θ and ø represent declination and right ascension, and r represents radius or range. (B) Spherical coordinates for defining points in three-space, where the angles θ and ø represent latitude and longitude, and r represents radius or range.
There’s yet another alternative: θ can represent elevation (the angle above the horizon) and ø 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 cover the span of values between, and including, –90° (the nadir, or the point directly underfoot) and +90° (the zenith). This is shown in Fig. 6-4C. In a variant of this system used by mathematicians, the angle θ is measured with respect to the zenith (or the positive z axis), rather than the plane of the horizon. Then the range for this angle is 0° ≤ θ ≤ 180°.
Fig. 6-4. (C) Spherical coordinates for defining points in three-space, where the angles θ and ø represent elevation (angle above the horizon) and azimuth (also called bearing or heading), and r represents radius or range.
Spatial Coordinates Practice Problems
Practice 1
Suppose you fly a kite above a perfectly flat, level field. The wind is out of the east-southeast, or azimuth 120°. Thus, the kite flies in a west-northwesterly direction, at azimuth 300°. Suppose the kite flies at an elevation angle of 50° above the horizon, and the kite line is 100 meters long. Imagine that it is a sunny day, and the sun is exactly overhead, so the kite’s shadow falls directly underneath it. How far from you is the shadow of the kite? How high is the kite above the ground? Express your answers to the nearest meter.
Solution 1
Let’s work in navigator’s cylindrical coordinates. The important factors are the length of the kite line (100 meters) and the angle at which the kite flies (50°). Figure 6-5 shows the scenario. Let r be the distance of the shadow from you, as expressed in meters. Let h be the height of the kite above the ground, also in meters.
First, let’s find the ratio of h to the length of the kite line, that is, h /100. The line segment whose length is h, the line segment whose length is r, and the kite line form a right triangle with the hypotenuse corresponding to the kite line. From basic circular trigonometry, we can surmise the following:
sin 50° = h/100
Fig. 6-5. Illustration for Solution 1.
- sing a calculator, we derive h as follows:
We also know, from basic circular trigonometry, this:
cos 50° = r /100
- sing a calculator, we derive r as follows:
In this situation, the wind direction is irrelevant. But if the sun were not directly overhead, the wind direction would make a difference. It would also make the problem a lot more complicated. If you like difficult problems, try this one again, but imagine that the sun is shining from the southern sky (azimuth 180°) and is at an angle of 35° above the horizon.
Practice problems for these concepts can be found at: Three-Space and Vectors Practice Test
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From Trigonometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.
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