Basic Transformation of Coordinates and Measures, Corresponding Axes Parallel, Co-Linear x-Axes Parallel to β, and t = 0 When Origins Coincide.

Table 1: Transformation of Coordinates of a Space-Time Point

rest coordinates to moving coordinates	moving coordinates to rest coordinates
tβ = t0, Tβ = T0 [subscripts superfluous]	t0 = tβ, T0 = Tβ [subscripts superfluous]

In the moving frame we find the angle φ transformed, ; if then . In the rest frame, the absolute span s at any instant of a moving solid rod of length l, (in its proper frame with velocity β,) with the angle φ₀ (in the rest frame) between β and the axis of the rod, is given by . In the moving system, a point at absolute rest (at rest in the rest frame) will have velocity . Time in all frames of reference is the same as in the rest frame.

Table 2: Transformation of Measure

tan(φ)
cos(φ)	,
sin(φ)	,
absolute span of moving length	, Derivation (with x-axis parallel to β)
velocity	,
ψ dihedral angle between two angles φ1 and φ2	ψ0 = ψβ
θ angle between the vectors β1 and β2 of two inertial frames	, ,  , ,

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