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Other Coordinate Systems for Physics Help (page 2)

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Cylindrical Coordinates

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

Fig. 3-16 . Cylindrical coordinates for defining points in three-space.

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.

Spherical Coordinates

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 )

Fig. 3-17 . Spherical coordinates for defining points in three-space.

such that

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:

Fig. 3-18 . Semilog xy plane with linear x axis and logarithmic y axis.

−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:

Fig. 3-19 . Semilog xy plane with logarithmic x axis and linear y axis.

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.

Log-log Coordinates

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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