**Purpose**

Given the introduction of Complete and Incomplete Coordinate Systems, we use the example of a bird flying in the cage on the back of a moving truck to create our equations for an Incomplete Coordinate System. In this case we use three birds; one flying from the back to the front and returning to the back, traveling along the X axis, one flying from the left side of the cage to the right and back to the left, traveling along the Y axis, and one flying from the bottom of the cage to the top and back to the bottom, traveling along the Z axis. Each bird is a surrogate for a wave.

Notice that the mathematics governing the bird’s cyclical behavior – traveling rear to front to rear (or left to right, or top to bottom), repeatedly – is dependent on the position of the cage that is moving forward at velocity

v. However, in an Incomplete Coordinate System, the bird’s travel speed is governed by the progress he can make through the air, regardless of how fast the truck carrying the cage is going. (As long as it is going slower than the bird can fly). In the case of an Incomplete Coordinate System, the air is not moving forward with velocityv.

**Equations**

We can compute total time and distance for one cycle (or oscillation) or we can compute half the distance for time and distance. To align our equations with Einstein’s, we will present the equations in the form of One-Half an Oscillation.

The equations for an Incomplete Coordinate System are derived in the manuscripts provided in the Papers section and are summarized here.

Generalizations required to accommodate time normalization and adjust for a Complete Coordinate System are not shown.

**Equations for the Coordinate System**

We have developed the equations for the wave fronts and one-half oscillations when the coordinate system is moving a velocity

v. We still need to determine the position of the coordinate system. In this case, we use the Newtonian transformations.For example, if the back of the trailer is located at the origin and the length of the trailer is

x’, then the front of the trailer is positioned as locationx’. We use this length,x’, in the one-half the oscillations equations for the X axis. If the trailer is moving forward at velocityvfortseconds, then the front of the trailer is located at positionx, wherex=x’+vt.This equation enables us to determine the length of the trailer if we are instead given

xandt, such thatx’=x-vt. We can use this equation to first calculatex’before using the one-half oscillation equations for the X-axis or we can simply replacex’withx-vtin the numerator.