1 Orthogonal curvilinear coordinates
The results shown in Section 28.2 have been given in terms of the familiar Cartesian
coordinate system. However, other coordinate systems can be used to better describe some physical situations. A set of coordinates
,
and
where the directions at any point indicated by
,
and
are orthogonal (perpendicular) to each other is referred to as a set of
orthogonal curvilinear coordinates
. With each coordinate is associated a
scale factor
,
or
respectively where
(with similar expressions for
and
). The scale factor gives a measure of how a change in the coordinate changes the position of a point.
Two commonly-used sets of orthogonal curvilinear coordinates are
cylindrical polar coordinates
and
spherical polar coordinates
. These are similar to the plane polar coordinates introduced in
HELM booklet
17.2 but represent extensions to three dimensions.
1.1 Cylindrical polar coordinates
This corresponds to plane polar
coordinates with an added
-coordinate directed out of the
plane. Normally the variables
and
are used instead of
and
to give the three coordinates
,
and
. A cylinder has equation
constant.
The relationship between the coordinate systems is given by
(i.e. the same is used by the two coordinate systems). See Figure 20(a).
Figure 20:
The scale factors , and are given as follows
1.2 Spherical polar coordinates
In this system a point is referred to by its distance from the origin and two angles and . The angle is the angle between the positive -axis and the line from the origin to the point. The angle is the angle from the -axis to the projection of the point in the plane.
A useful analogy is of latitude, longitude and height on Earth.
- The variable plays the role of height (but height measured above the centre of Earth rather than from the surface).
- The variable plays the role of latitude but is modified so that represents the North Pole, represents the equator and represents the South Pole.
- The variable plays the role of longitude.
A sphere has equation
constant.
The relationship between the coordinate systems is given by
. See Figure 21.
Figure 21:
The scale factors , and are given by