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Alternative 3D Coordinates Help (page 3)

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

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.

 

 

Alternative Coordinates Alternative 3D Coordinates Spherical Coordinates

Alternative Coordinates Alternative 3D Coordinates Spherical Coordinates

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°.

 

Alternative Coordinates Alternative 3D Coordinates Spherical Coordinates

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:

Alternative Coordinates Alternative 3D Coordinates Spherical Coordinates

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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