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(someone show me how to write eqs.)

If for example we take theta = arcsin v/c, then the rotational transformation equations would be (assuming the moving frame moves parallel to the x-axis)

x' = x * cos theta + ct sin theta = x * cos theta + vt

y' = y

z' = z

ct' = ct * cos theta - x sin theta = ct * cos theta - vx/c

Note that in this framework the Lorentz factor becomes gamma = sec theta.

The distance between any two points would still be an invariant among any frames. Taking an example, v = 0.5c, x = 10, and ct = 2, gives sqrt (104) as the distance in the original frame. The y and z coordinates will be ignored here for simplicity (they are zero) but you should be able to see that a transform will work with them as well. theta = 30 deg, so the Lorentz factor is sec 30 deg = 1.155. This agrees with 1/sqrt(1-v^2/c^2). x' is then 10 * 0.866 + 2 * 0.5 = 9.66 and ct' is -3.27. You can see from this that the distance is the same as the original frame, sqrt(104). So this preserves intervals across frame rotations due to relative velocity as well.

So why is this approach not used?