To see why, let L(θ) denote the line through the origin at an angle θ to the positive x-axis.

Thus the bound vector represented by (1,0,0) is a vector of unit length pointing from the origin along the positive x-axis.

Polar coordinates, for example, form a chart for the plane R minus the positive x-axis and the origin.

Also without loss of generality, we assume that the second sphere, with radius , is centered at a point on the positive x-axis, at distance from the origin.

The initial side is on the positive x-axis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns.

The positive x-axis in vehicles points always in the direction of movement.

We also assume that the sphere, with radius is centered at a point on the positive x-axis, at point .

The reference plane is assumed to be the xy-plane, and the origin of longitude is taken to be the positive x-axis.

The real number θ corresponds to the angle on the unit circle as measured from the positive x-axis.

To every point on the unit circle we can associate the angle of the positive x-axis with the ray connecting the point with the origin.