Perhaps we have different starting points because I cannot derive either the Lorentz or Galilean transformation from the equations without additional information. The equations, at least for the Lorentz transformation, can be either for a circle where r^2 = x^2 + y^2 or for a sphere where r^2 = x^2 + y^2 + z^2. Assuming a constant speed of light c where

*x’* = α<sub>1</sub>

*x* + α<sub>2</sub> *t*, y’ = y, z’ = z

*t’* = β<sub>1</sub>

*x* + β<sub>2</sub> *t*,

I arrive at the equation

(α<sub>1</sub><sup>2</sup>

– β<sub>1</sub><sup>2</sup> c<sup>2</sup> – 1) *x*<sup>2</sup> – 2 (β<sub>1</sub> β<sub>2</sub> c<sup>2</sup> + α<sub>1</sub><sup>2</sup>

v) *xt* + (c<sup>2</sup> – β<sub>2</sub><sup>2</sup>

c<sup>2</sup> + α<sub>1</sub><sup>2</sup> v<sup>2</sup>) *t*<sup>2</sup> = 0

where v is the speed between reference frames. This equation is of the form A x^2 + B xt + C t^2 = 0 and cannot be solved without more information.

On the other hand, the values for the components from the Lorentz transformation are known as

α<sub style=”background-color: transparent; font-family: inherit;”>1</sub>

= β<sub style=”background-color: transparent; font-family: inherit;”>2</sub> = γ and β<sub style=”background-color: transparent; font-family: inherit;”>1</sub>

= – γ v/c<sup style=”background-color: transparent; font-family: inherit;”>2</sup>

where γ is the Lorentz factor. Substituting these values for the components produces

(0) *x*<sup>2</sup> – 2 (0) *xt* + (0) *t*<sup>2</sup> = 0

which is degenerate.