## Description of Arc Length on Spherical Surfaces

Many problems require computing the distance between two points. When we are interested in
the distance between points along a straight line, we apply the
well-known distance formula derived from the Pythagorean theorem.
However, if we are interested in the distance between points along a
curved surface, the problem becomes more difficult. Fortunately,
computing the minimum distance, or *arc length*,
between two points on a spherical surface is a special case that is
relatively simple. To begin, let’s look at two different coordinate
systems, *rectilinear coordinates *and
*spherical coordinates*.

### Rectilinear and Spherical Coordinates

The rectilinear coordinate system is the coordinate system that is most familiar to us.
In rectilinear coordinates, a point’s location is specified using
three values, *x*, *y*,
*z*, which are its positions along the
*x*-axis, *y*-axis, and
*z*-axis. Referring to Figure 17.5, the
*z*-axis is positive going upward. Standing at
the arrow looking forward, the *x*-axis is
positive to the right, and the *y*-axis is
positive straight ahead. From this vantage point, the positive
directions for *x* and *y*
look the same as in two dimensions. Thus, to locate (3, 4, 5), for
example, we move three units to the right along the
*x*-axis, four units ahead parallel to the
*y*-axis, and five units up parallel to the
*z*-axis (see Figure
17.5).

Get *Mastering Algorithms with C* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.