Cylindrical and Spherical Coordinates - Analytic Geometry and Vectors - University Mathematics Handbook

University Mathematics Handbook (2015)

III. Analytic Geometry and Vectors

Chapter 9. Cylindrical and Spherical Coordinates

9.1  Cylindrical Coordinates

a.  A cylindrical coordinate system combines the polar coordinates in a plane (see II 1.2), with vertical coordinate in a space. The cylindrical coordinates of point in space are ordered set of three numbers , when are the polar coordinates of projection of point on plane (see illustration). Remember that is distance, and therefore can be either positive or zero, and angle varies from zero to .

Formulas of conversion from cylindrical coordinates to Cartesian coordinates are

To reverse the conversion, from to

b.  In cylindrical coordinates, the cylindrical surface equation has a simpler form.

Example: The equation of cylinder is

, ,

That is, it is the equation of the locus for all sets of points at distance from -axis.

9.2  Spherical Coordinates

a.  Point in space is defined by the set of three numbers , when is the length of vector (when is the origin), is the angle of projection with the positive direction of -axis on plane , and is the angle formed by vector with the positive direction of -axis.

are called the spherical coordinates of
point .

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b.  Relation between spherical and Cartesian coordinates:

1.  If is a point of spherical coordinates , then

.

2.  If are Cartesian coordinates of point , then

when only has positive values, values vary from zero to and angle varies from zero to .

c.  Examples:

1.  The spherical coordinates of point (2,1,2) are:

2.  The equation of sphere in spherical coordinates is

d.  Note: Sometimes angle is referred to as the angle between vector and plane . In the illustration above, it is .

In such a case, is substituted with , so the equation of sphere in spherical coordinates is

.