Every depiction of motion requires a reference that is not in motion. We may identify that which is not in motion as a reference frame or even a point of orientation. Every depiction or description of motion, however verbal or however mathematical, may be rendered as a geometrical sketch. If nothing else, the sketchpad itself, of whatever medium, is a frame of reference.
The Ultimate or Absolute Reference Frame
Of course, there is as an ultimate or absolute reference frame. It is the self-orientation of the individual observer of motion or the self-orientation of the individual device recording motion. One cannot get out of one’s own observational skin, but one can get out of one’s own skin analytically through an analytic depiction of motion from a point of orientation of one’s choice.
Funny Thing about Communicating a Depiction of Motion
Obviously the most valid communication of an observed motion to someone else would be a record of the motion as experienced. Funny thing is that in our own minds as well as in our communication of an observed motion to others, we do not typically present the motion as it was observed by us. We typically express it to others (and even to ourselves) analytically for the sake of simplicity and for the sake of effective communication (and for the sake of effective self-memorization).
If I were to direct you to the grocery store from my house, the most realistic presentation would be to show you a record of the actual motion as observed by a camcorder in the passenger seat of my car and aimed out the windshield. However, for the clarity of communication, I wouldn’t do that. Instead, I would give you analytical directions on how to drive to the grocery store. I would say, “Make a left turn out of my driveway. Turn left at the second traffic signal. Turn left at the third traffic signal and you will be in the parking lot of the grocery store.” That analysis is readily communicated and followed. However, the route is not the one I would take. The route I would take would pass through only one traffic signal and pass through one 4-way stop. However, an analysis of that route is not as easily communicated or as clearly landmarked as the depiction above involving five traffic signals.
The key to depicting and communicating motion is not in the details of its visual experience. The key to depiction is analytical simplicity. This is particularly evident in the choice of a point of orientation (or the choice of a reference frame) for the analysis. The choice of a point of analytical orientation is not right or wrong. The choice is usually a question of simplicity.
The analytical directions above from my house through five traffic signals to the grocery store are based on an observational orientation point that is the driver of the car. This is the same orientation point of my experience as a driver, but it skips most of the visual details of a camcorder. If the directions were based on a map, marked to show the route from my house to the grocery store, the observational orientation point of the analytical depiction would be that of a virtual observer above the route. An analyst of motion has the option of considering himself as a virtual observer outside the analytical depiction or as a virtual observer within the depiction. An actual observer of motion has no choice but to observe from his actual vantage point.
Comparison of Two Analytical Depictions
As an example, both of the following two depictions or sketches are of only three objects, namely three Spheres, A, B, and C. Both depictions are on a two dimensional sketchpad.
In Sketch 1, Sphere A is represented as a circle at the top center of the pad, but it is in the background, i.e. beyond the surface of the plane of the sketchpad.
Sphere B is in a circular orbit around point B’ . Sphere B itself is represented by six circles at different locations along its orbital path. Point B’ is in the plane of the sketchpad.
Sphere C is in a vertical line with Sphere A and point B’. This alignment appears at the center of the sketchpad. However, Sphere C is in the foreground, i.e. above the plane of the sketchpad.
Thus, the centers of Spheres A and C and point B’ form a plane perpendicular to the plane of the sketchpad which plane intersects the plane of the sketchpad as a vertical line.
The virtual observer is outside of the range of the three spheres and is elevated above the plane of Sphere B’s orbit. (If the virtual observer were in the plane of Sphere B’s orbit, the orbit would be a horizontal line rather than an oval as depicted.) The virtual observer would describe Sphere B as in a planar orbit, whose center point B’ was stationary with respect to his observation point. Spheres A and C are also stationary with respect to the virtual observation point.
In Sketch 2, Sphere A has been moved forward, but kept in the plane defined by Spheres A and C and point B’. It is then on a line perpendicular to the plane of Sphere B’s orbit and intersects the plane of the orbit at its center point, B’, i.e. Sphere A is on the axis of Sphere B’s orbit. Also, in Sketch 2, Sphere A has been moved to a point along the axis just above the orbital plane of Sphere B.
Let it be that the optics of the virtual observer were such that he couldn’t distinguish the location of Sphere A as that of Sketch 1 or of Sketch 2. Depth perception and vertical perception in the case of Sphere A is lacking due to deficiency in the optics employed by the virtual observer.
Notice that the illumination of Sketches 1 and 2 implicitly is in the same direction from which the Spheres are viewed by the virtual observer so that all three Spheres are fully lighted in the locations in which they are depicted.
A Different Source of Illumination
Let the illumination implicit in Sketches 1 and 2 be turned off. Also, let Sphere A be the source of light. The light emitted by Sphere A, would be reflected off of the surface of Sphere B. Let the virtual observer still lack depth and vertical perception with respect to the location of Sphere A relative to the planar orbit of Sphere B. However, let the new optics employed be such that the virtual observer could discern the pattern of light reflected by Sphere B in the six locations of Sphere B depicted. The location of Sphere A relative to the planar orbit of Sphere B could then be determined based on the pattern of light reflected.
The Two Depictions Modified when Sphere A is the Source of Light
Given the relative location of Sphere A in Sketch 1 with A as a source of light, the observed pattern of reflection from Sphere B, in the six depicted locations in its orbit, would be from the top of Sphere B as depicted.
Given the relative location of Sphere A in Sketch 2 with A as a source of light, the observed pattern of reflection from Sphere B, in the six depicted locations in its orbit, would vary as depicted.
Let it be that the pattern of light from Sphere A, reflected by Sphere B and detected by the virtual observer is that of Sketch 2 with A as a source of light. The virtual observer would be prompted to state that Sphere B was in orbit around Sphere A.
Implicitly in all four Sketches, the views of Spheres A and B would not be significantly altered, if the location of the virtual observer were Sphere C.
All four Sketches would not be significantly altered, if the implicit location of the virtual observer were explicitly identified as that of Sphere C. Consequently, all four depictions are essentially Sphere-C-ocentric depictions in which Spheres A and C are not in relative motion to one another.
For a set of two analytical depictions comparable to the set of two Sketches with A as a light source, see the University of Arizona archives. These archives present two locations of Sphere A, the Sun, with respect to the orbit of Venus (Sphere B). The two different patterns of Sunlight reflected from Venus are viewed by an observer on Sphere C, the Earth, which is depicted as not in relative motion to the Sun. These two analytical depictions are thus comparable to Sketches 1 and 2 with Sphere A as a source of light.
In the referenced University of Arizona archives, simplicity of analytical depiction is achieved by presenting the Sun and Earth as stationary with respect to one another. No one accepts that as true, yet this does not invalidate these two analytical depictions for their purpose, which is simplicity in illustrating two different patterns in the phases of Venus dependent upon the location of the Sun with respect to the orbit of Venus.
Another example: Simplicity of analytical expression is achieved by depicting the Earth as not rotating on its axis when the Sun’s diurnal motion relative to the Earth is accurately identified in common parlance by Sunrise and Sunset.
It is simplicity of expression and communication which determines how we analytically depict motions. An analytical depiction of motion in its choice of orientation is validated by simplicity and utility, not by its being an exclusionary singularity of what is true. A different choice of orientation may be rejected for being more complex and less useful, but not for being inaccurate.
In the case of the two depictions of motion yielding different patterns in the phases of Venus in the University of Arizona archives, the orientation is the same for both depictions. It is geocentric. The depictions are equal in simplicity and utility. The two depictions differ in the location of the Sun relative to a static orbit of Venus and a static Earth. With the Earth as the orientation point for both depictions, the static location of the Sun could be determined by the pattern of the phases of Venus actually observed, which is that of the second depiction. It would be wrong to discredit these two depictions because they represent the Sun as static with respect to the Earth. That is not their purpose. In fact to represent the Sun’s motion with respect to the Earth during the cycle of the orbit of Venus would obscure the purpose of these analytical depictions.