An *observer* is a body capable of use as a measurement apparatus. An *inertial observer* is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. An observer here shall mean an inertial observer.

An observer makes measurements relative to a frame of reference. A *frame of reference* is a physical system relative to which motion and rest may be measured. An *inertial frame* is a frame in which Newton’s first law holds (a body either remains at rest or moves in uniform motion, unless acted upon by a force). A frame of reference here shall mean an inertial frame.

A *rest frame* of observer P is a frame at rest relative to P. A *motion frame* of observer P is a frame in uniform motion relative to P. Each observer has at least one rest frame and at least one motion frame associated with it. An observer’s rest frame is three-dimensional, but their motion frame is effectively one-dimensional, that is, only one dimension is needed.

*Space* is the geometry of places and lengths in R^{3}. A *place point* (or *placepoint*) is a point in space. The space origin is a reference place point in space. The *location* of a place point is the space vector to it from the space origin. *Trime* (3D time) is the geometry of times and durations in R^{3}. A *time point* (or *timepoint*) is a point in trime. The time origin is a reference time point in trime. The *chronation* of a time point is the trime vector to it from the time origin.

A frame of reference is *unmarked* if there are no units specified for its coordinates. A frame of reference is *marked* by specifying (1) units of either length or duration for its coordinates and (2) an origin point. A *space frame* of observer P is a rest frame of P that is marked with units of length. A *time frame* of observer P is a motion frame of P that is marked with units of duration.

Speed, velocity, and acceleration require an independent motion frame. Pace, lenticity, and retardation require an independent rest frame. These independent frames are standardized as clocks or odologes so they are the same for all observers.

Let there be a frame K_{1} with axes *a*_{1}, *a*_{2}, and *a*_{3}, that is a rest frame of observer P_{1}, and let there be a motion frame K_{2} with axes *a´*_{1}, *a´*_{2}, and *a´*_{3}, that is a motion frame of P_{1} along the coincident *a*_{1}*-a´*_{1 } axis. See Figure 1.