Spherical Coordinates

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Spherical coordinates are similar to polar coordinates, except for the fact that these are used in 3D space.

Spherical coordinates consist of three components:

The azimuth angle indicates in which direction on the horizontal plane the point lies. This angle varies from 0° and 360° (or -180° and +180°, which is exactly the same, since -90° equals 270° etc.) The angle is measured counterclockwise. On the above figure, ROQ is the azimuth angle and should be approximately -60°.
The elevation angle corresponds to the angle between OP and the vertical plane. It varies between -90° and +90°. On the figure, QOP is the elevation angle and is about +60°.
The radius tells you how far away the point lies in the direction specified by the azimuth and elevation angles. On the figure, the radius corresponds to the distance |OP|.

Example

Say you are located at the origin (0,0,0). To the left of you is a 5 km height mountain. It is 10 km away.

The Cartesian coordinates of the top are (-10, 5, 0).
The spherical coordinates are
- Azimuth angle: 90° because the mountain top is to your left.
- Elevation angle: 27° because you have to look up to see the top.
- Radius: 11.18 which is the distance between you and the mountain top.

More information can be found on Wikipedia. Note that there are many ways to define spherical coordinates. For example, it is possible to replace the elevation angle by the zenith angle, which corresponds to the angle between the \(Y\) axis and \(OP\). In this course, we choose to work with the elevation angle.