Cylindrical and Spherical Coordinates
Cylindrical and Spherical Coordinates
In , the following are the three most commonly-used coordinate systems:
- Rectangular (Cartesian) coordinates: A point is specified by the coordinates , where and are real numbers indicating the projection of the point onto three axes.
- Cylindrical coordinates: A point is specified by the coordinates , where is the -coordinate in rectangular coordinates and are the polar coordinates of the projection of the point onto xy plane. Note that here we assume and .
- Spherical coordinates: A point is specified by the coordinates , where is the -coordinate in cylindrical coordinates, is the distance between the point and the origin, and is the angle measured from -axis to the line segment joining the point to the origin. Note that , and
The following are the conversions between these three coordinate systems:
( Rectangular, Cylindrical, Spherical)
:
:
, where
:
:
, where
, where
:
, where
:
Exercise: Let in rectangular coordinates. Find the cylindrical and spherical coordinates of .
Surfaces in cylindrical and spherical coordinates
Here are some examples of equations of various geometric objects in different coordinate systems:
- A unit sphere centered at the origin
- Rectangular coordinates:
- Cylindrical coordinates:
- Spherical coordinates:
- A cylinder whose cross-section on xy plane is the unit circle centered at the origin
- Rectangular coordinates:
- Cylindrical coordinates:
- Spherical coordinates:
- A inverted cone whose vertex is the origin and its vertical angle is a right angle
- Rectangular coordinates:
- Cylindrical coordinates:
- Spherical coordinates: