Other Coordinate Systems for Physics Help (page 2)
Here are some other coordinate systems that you are likely to encounter in your journeys through the world of physics. Keep in mind that the technical details are simplified for this presentation. As you gain experience using these systems, you will be introduced to more details, but they would confuse you if we dealt with them now.
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. 3-14 a . The polar axis connects two specified points at antipodes on the sphere. These points are assigned latitude +90 (north pole) and −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 0. Latitude is measured positively (north) and negatively (south) relative to the plane of the equator. Longitude is measured counterclockwise (east) and clockwise (west) relative to the equatorial axis. The angles are restricted as follows:
−90° ≤ latitude ≤ +90°
−180° ≤ longitude ≤ +180°
On the earth’s surface, the half-circle connecting the zero-longitude line with the poles passes through Greenwich, England, and is known as the Greenwich meridian or the prime meridian . Longitude angles are defined with respect to this meridian.
Celestial latitude and celestial longitude are extensions of the earth’s latitude and longitude into the heavens. An object whose celestial latitude and longitude coordinates are ( x, y ) appears at the zenith (straight overhead) in the sky from the point on the earth’s surface whose latitude and longitude coordinates are ( x, y ).
Declination and right ascension define the positions of objects in the sky relative to the stars. Figure 3-14 b applies to this system. Declination (abbreviated dec ) is identical to celestial latitude. Right ascension (abbreviated RA ) is measured eastward from the vernal equinox (the position of the sun in the heavens at the moment spring begins in the northern hemisphere). Right ascension is measured in hours (symbolized h ) rather than degrees, where there are 24h in a 360° circle. The angles are restricted as follows:
−90° ≤ dec ≤ +90°
0h ≤ RA < 24h
An extension of rectangular coordinates into three dimensions is cartesian three-space (Fig. 3-15), also called rectangular three-space or xyz space . Independent variables are usually plotted along the x and y axes; the dependent variable is plotted along the z axis. “Graphs” of this sort show up as snakelike curves winding and twisting through space or as surfaces such as spheres, ellipsoids, or those mountain-range-like displays you have seen in the scientific magazines. Usually, the scales are linear; that is, the increments are the same size throughout each scale. However, variations of these schemes can employ nonlinear graduations for one, two, or all three scales.
Computers are invaluable in graphing functions in rectangular three-space. Computers can show perspective, and they let you see the true shape of a surface plot. A good three-dimensional (3D) graphics program lets you look at a graph from all possible angles, even rotating it or flipping it over in real time.
Figure 3-16 shows a system of cylindrical coordinates for specifying the positions of points in three-space. Given a set of cartesian coordinates or xyz space, an angle q is defined in the xy plane, measured in radians counterclockwise from the x 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 ( q, r, z ) such that
q = angle between P ′ and the x axis in the xy plane
r = distance (radius) from P to the origin
z = distance (altitude) of P above the xy plane
You can think of cylindrical coordinates as a polar plane with the addition of an altitude coordinate to define the third dimension.
Figure 3-17 shows a system of spherical coordinates for defining points in space. This scheme is similar to the system for longitude and latitude with the addition of a radius r representing the distance of point P from the origin. The location of a point P is defined by the ordered triple (long, lat, r )
long = longitude of P
lat = latitude of P
r = distance (radius) from P to the origin
In this example, angles are specified in degrees; alternatively, they can be expressed in radians. There are several variations of this system, all of which are commonly called spherical coordinates .
Semilog ( X -linear) Coordinates
Figure 3-18 shows semilogarithmic (semilog) coordinates for defining points in a portion of the xy plane. The independent-variable axis is linear, and the dependent-variable axis is logarithmic. The numerical values that can be depicted on the y axis are restricted to one sign or the other (positive or negative). In this example, functions can be plotted with domains and ranges as follows:
−1 ≤ x ≤ 1
0.1 ≤ y ≤ 10
The y axis in Fig. 3-18 spans two orders of magnitude (powers of 10). The span could be larger or smaller than this, but in any case the y values cannot extend to zero. In the example shown here, only portions of the first and second quadrants of the xy plane can be depicted. If the y axis were inverted (its values made negative), the resulting plane would cover corresponding parts of the third and fourth quadrants.
Semilog ( Y -linear) Coordinates
Figure 3-19 shows semilog coordinates for defining points in a portion of the xy plane. The independent-variable axis is logarithmic, and the dependent-variable axis is linear. The numerical values that can be depicted on the x axis are restricted to one sign or the other (positive or negative). In this example, functions can be plotted with domains and ranges as follows:
0.1 ≤ x ≤ 10
−1 ≤ y ≤ 1
The x axis in Fig. 3-19 spans two orders of magnitude (powers of 10). The span could be larger or smaller, but in any case the x values cannot extend to zero. In the example shown here, only portions of the first and fourth quadrants of the xy -plane can be depicted. If the x axis were inverted (its values made negative), the resulting plane would cover corresponding parts of the second and third quadrants.
Figure 3-20 shows log-log coordinates for defining points in a portion of the xy plane. Both axes are logarithmic. The numerical values that can be depicted on either axis are restricted to one sign or the other (positive or negative). In this example, functions can be plotted with domains and ranges as follows:
0.1 ≤ x ≤ 10
0.1 ≤ y ≤ 10
Fig. 3-20 . Log-log xy plane.
The axes in Fig. 3-20 span two orders of magnitude (powers of 10). The span of either axis could be larger or smaller, but in any case the values cannot extend to zero. In the example shown here, only a portion of the first quadrant of the xy plane can be depicted. By inverting the signs of one or both axes, corresponding portions of any of the other three quadrants can be covered.
Practice problems for these concepts can be found at: Graphing Schemes for Physics Practice Test
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