Angle matrices are essentially identical to distance matrices, except they contain measures of the angular distances between two locations (in fact, in PASSaGE 1, angles were only for two-dimensional data and were simply stored in normal distance matrices). Angle matrices can represent the relationship between two- or three-dimensional coordinates. Angles are stored internally as radians, but are converted to degrees in most input/output. For two-dimensional coordinates, the angle measures the counter-clockwise arc between the positive x-axis (due East) and the vector connecting the two points. Because the order of the two points is arbitrary, the angular measure is symmetric around the circle and is generally measured from 0 to π (0 to 180°).
Diagram of definition of angles in PASSaGE: Two-dimensional coordinate system.
For three-dimensional coordinates, two angles describe the relative orientation of the points to each other. The first, θ, measures the angle between the two points projected orthogonally into the x,y plane (sometimes called the angle of ascension). This measure is exactly equivalent to the two-dimensional angle described above, except that it is measured around the full circle, from 0 to 2π (0 to 360°). If we imagine one of the points is at the center of the earth and the other is on the surface, θ is equivalent to longitude. The second angle, φ, represents the angle between the vector connecting the two points and the z-axis (sometimes called the angle of declination), and is generally measured from 0 (the vector connecting the points is parallel to the z-axis, i.e., the points have identical x- and y-coordinates) to π/2 or 90° (the vector connecting the points is perpendicular to the z-axis, i.e., both points have identical z-coordinates). If we again imagine the points as being at the center and surface of the globe, φ is similar to latitude, except with 0° at the North Pole, 90° at the equator, and 180° at the South Pole.
Diagram of definition of angles in PASSaGE: Three-dimensional coordinate system.
While useful for describing the three-dimensional relationship among points, one problem with this conventional definition of three-dimensional angles is that the meaning of θ and φ are not precisely independent. One degree of arc along θ covers a different physical distance across a sphere when φ is small than when φ is large (just as one degree of longitude is a large distance near the equator but a small distance near the pole). When φ is zero (the points have identical x and y coordinates), θ is undefined. Thus, when performing analyses which require three-dimensional angles, we will use a form of spherical tessellation to divide space into equal size angular sectors.