Chapter 4: Geometry

and the positive polar axis, and the angle (azimuth) between the initial ray and the
projection of ÇÈ to the equatorial plane. See Figure 4.32. As in the case of polar
and cylindrical coordinates, is only defined up to multiples of ¿¼ Æ , and likewise .
Usually is assigned a value between ¼ and ½¼ Æ , but values of between ½¼ Æ and
¿¼ Æ can also be used; the triples ´ µ and ´ ¿¼ Æ ½¼ Æ · µ represent the
same point. Similarly, one can extend to negative values; the triples ´ µ and
´ ½¼ Æ ½¼ Æ · µ represent the same point.
FIGURE 4.32
A set of spherical coordinates for È is ´ µ ´½¼ ¼ Æ ¿¼ Æ µ.
È
FIGURE 4.33
Standard relations between Cartesian, cylindrical, and spherical coordinate systems. The
origin is the same for all three. The positive Þ-axes of the Cartesian and cylindrical systems
coincide with the positive polar axis of the spherical system. The initial rays of the cylindrical
and spherical systems coincide with the positive Ü-axis of the Cartesian system, and the rays
¼ Æ coincide with the positive Ý-axis.
Þ
È
Ü
Ý
Ö
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