By common experience, we know there are three dimensions of motion. That is, space, which is the space of motion, is three dimensional. To measure the extent of motion requires comparing one motion with another, of which there are two ways: distance and duration. The distance of a motion is measured by comparing it with synstancial but not necessarily synchronous motion. The duration of a motion is measured by comparing it with synchronous but not necessarily synstancial motion.
Length of motion considered by itself forms a length space, which is space with a metric of length. Duration of motion considered by itself forms a duration space, which is space with a metric of duration. Since there are three dimensions of motion, length space and duration space are both three dimensional metric spaces. By convention, both are Euclidean. The length metric is called distance. The duration metric may be called distime.
Each placepoint has a length position (LP) vector that begins with the space origin. Each timepoint has a duration position (DP) vector that begins with the duration origin. The magnitude of a length position vector is called the stance. Every placepoint that is equidistant from the origin has the same stance. The magnitude of a duration position vector is called the time. Every timepoint in duration space that is an equal distime from the origin has the same time.
Stance and time are vector magnitudes, with their direction ignored. Stance is a radius from the origin of length space. A unit of length is the absolute value difference between two stances, that is, between the radii of two length vectors with unit difference. Time is a radius from the origin of duration space. A unit of duration is the absolute value difference between two times, that is, between the radii of two duration vectors with unit difference.
The rate of motion measured by the length of motion per unit of duration is called speed. The rate of motion measured by the duration of motion per unit of length is called pace. Note that a faster speed is a larger ratio, whereas a faster pace is a smaller ratio. Also, the ratio of a slower speed to a faster speed is less than one but the ratio of a faster pace to a slower pace is less than one.
The vector rate of change in the length vector per unit of duration is called velocity. The vector rate of change in the duration vector per unit of length is called lenticity. The vector rate of change in velocity per unit of duration is called acceleration. The vector rate of change in lenticity per unit of length is called relentation.
The length position vector of a trajectory evolves as a function of the time. The duration position vector of a trajectory evolves as a function of the stance. These functions are inverses of one another.