Quantities (called magnitudes) combined with direction are called vectors. Quantities not combined with direction are called scalars. A space is a geometry or topology that contains vectors (which may or may not equal a vector space or Euclidean space as defined in mathematics).
The kind of a space depends on the units of the magnitude. If direction is combined with distance, the result is a distance space, which is 3D space. If direction is combined with duration, the result is a duration space, which is 3D time. Direction may be combined with other quantities, such as speed in a velocity space or pace in a lenticity space.
Position vectors are directed from an origin or destination point to a point position. A metric may be defined between positions: distance for distance space and distime for distime space.
Some spaces are associated or combined with a particular scalar. Distance space associated with scalar distime associated with distance space is 3+1 space. Distance space combined with time is spacetime, which is 4D. Distime space associated with scalar distance is 1+3 space. Distime space combined with stance is timespace, which is a different 4D space.
A 2D map of a space associates the direction and metric on the map with the direction and metric in the space. The association may not be linear, as in the case of a map of the Earth’s surface. In the context of a particular map, the values of a scalar associated with a point may be represented by concentric circles around the point, like contour lines. These represent constant values of the metric. Such a scalar is an associated scalar. For example, time may be represented by the distime from a point to any point at a particular duration away.
A curve within a space is determined by its directional and metric properties. The arc length of the curve is the metric length of the curve if it were made linear without stretching or compressing it. This ‘length’ may not be a physical length; it is a quantity with the units of the space’s metric. If the equation of the curve is known, then (in principle) the arc length of the curve may be calculated. However, if the curve is represented within a space in which the associated scalar is the metric of the original space, the arc length may be read from the concentric circles.
Let’s look at arclength and distance. Let the figure below represent a map with a superimposed route in 2D length space with three legs starting in the center with three circular isodistance lines:
Since the circles are evenly spaced, the distances of each leg are equal, like waves from a pebble dropped in water.
The red line represents the displacement from beginning to end. The distance covered could be read from these circles since the concentric circles represent isodistances from the center.
The length of the route, the arclength, is the sum of the lengths of the three line segments. The average speed is the distance, the magnitude of the displacement, divided by the arctime, the time taken to traverse the route.
Now look at this same figure as representing a zig-zag route in 2D time space so that each line segment represents a duration. Since the circles are evenly spaced, the distimes of each leg are equal.
In this case, the red line represents the dischronment from beginning to end. The distime covered could be read from these circles since the concentric circles represent isochrons from the center.
The length of the route, the arctime, is the sum of the lengths of the three line segments. The average pace is the distime, the magnitude of the dischronment, divided by the arclength, the length of the route.