## 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).

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