**Cylindrical Coordinates**

Figures 10-16A and 10-16B show two systems of *cylindrical coordinates* for specifying the positions of points in three-space.

**Schematic for Mathematicians, Engineers, and Scientists**

In the system shown in Fig. 10-16A, we start with Cartesian *xyz* -space. Then an angle *θ* is defined in the *xy* -plane, measured in degrees or radians (but usually radians) counterclockwise from the positive *x* axis, which is called the *reference 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 *(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, called the altitude or height, of

*P*above the

*xy*-plane. (If

*h*is negative, then

*P*is below the

*xy*-plane.) This scheme for cylindrical coordinates is preferred by mathematicians, and also by some engineers and scientists.

**Schematic for Navigators and Aviators**

In the system shown in Fig. 10-16B, 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. 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 from *P′* to the origin, and *h* represents the altitude or height of *P* above the *xy* -plane. (If *h* is negative, then *P* is below the *xy* -plane.) This scheme is preferred by navigators and aviators.

**Fig. 10-16** . (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.

**Spherical Coordinates**

Figures 10-17A to 10-17C show 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 of use to navigators and surveyors.

**Angles Represent Declination and Right Ascension - Astronomers and Aerospace Scientists**

In the scheme shown in Fig. 10-17A, the location of a point *P* is defined by the ordered triple *(θ,φ, r )* such that

*θ*represents the declination of

*P, φ*represents the right ascension of

*P*, and

*r*represents the distance or radius from

*P*to the origin. 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.

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