The value of an independent variable may be selected first, and so is arbitrary, even subjective. One may select anything or everything within its range. A graph normally covers a whole range of the independent variable.

Given an independent time interval, different travel rates result in different travel distances or, the other way around, different travel distances have different average rates. Similarly, given an independent space interval, different travel rates result in different travel times or different travel times have different average rates. These are shown on an event map, which is either events projected on a geographic map or shown graphically with a consistent time-scale.

Boston T Map with Time-Scale

Time is measured by a clock, which moves at a standard rate: the hour hand at one revolution per hour, the minute hand at one revolution per minute. A monthly calendar is updated at a rate of once per month, with the day updated once per day.

In space-time, time is measured by rotating or oscillating motion, which is independent of the surrounding space; space is measured by linear motion. In time-space, space is measured by rotating or oscillating motion, which is independent of the surrounding time; time is measured by linear motion.

If the travel rate is the speed of light, then distances and durations are proportional. Distance can be defined in terms of duration or *vice versa*. The difference between time and space then is only how they are measured. Light is the standard mode for modern physics.

For transportation the expected rate of travel in each mode is the standard, the modal rate. This is either determined by management, as with scheduled transport services, or empirically, as with measurement or experience. For physics, the modal rate is measured or determined from theory.

The modal rate is a standard for the mode; it reflects the mode rather than any particular travel in the mode (although a set of travel data may be used to estimate it). It is used to understand the past or to set expectations for the future. In transportation, trip planning and system management are the main applications. There are many applications in physics.

*Related*