Linear, radial, and scalar space and time

In 3D space the distance between points is the magnitude of the displacement vector, which is a path length. In 3D time the duration between instants (time points) is the magnitude of the dischronment vector, which is a path time.

Radial space is space with radial coordinates, 2D polar or 3D spherical, in which the distance from or to a point is the magnitude of the radial displacement. Radial time is time with radial coordinates, 2D polar or 3D spherical, in which the duration to a timepoint is the magnitude of the radial dischronment. The radial space and time are “as the crow flies,” with the imaginary crow flying at a constant rate.

The distance between two points along a path (or curve) is the path length between them. The duration between two instants along a path in time is the path time between them. Each of these may be conceived of as one rectilinear dimension.

The result of a measuring wheel (or surveyor’s wheel) operated throughout a region or network is a (spatial) linear referencing system (LRS or SLRS), which produces a set of physical wayposts or conceptual waypoints. These are often used by highway departments with geographic information systems (GIS).

For more on linear referencing systems, see here and here.