iSoul In the beginning is reality

# Linear, radial, and scalar space and time

In 3D space the distance between points is the magnitude of the displacement vector, which is a scalar length. In 3D time the duration between instants (time points) is the magnitude of the distimement vector, which is a scalar 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 an instant is the magnitude of the radial distimement. 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 arc length (scalar space) between them. The duration between two instants along a path in time is the arc time (scalar time) between them. Each of these may be conceived of as one rectilinear dimension.

A measuring wheel can measure the arc length along the space path between points. An odologe can measure the arc time along the space path between events. An odologe is equivalent to a measuring wheel operating at a fixed rate. A stopwatch with a conversion speed is effectively an odologe.

The result of a measuring 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.

The result of an odologe operated throughout an event region or network is a temporal linear referencing system (TLRS), which produces a set of physical eventposts or conceptual eventpoints/eventicles. A visual or audible signal synchronized with an event would be an example of an eventpost. This might be a train conductor’s “all aboard!” or a signal at the stops of an automated transit system.