Let’s start with one-dimensional, i.e., scalar, functions, f, g, h, and k. Say there is the following functional relation:
s = f(t) = f(h(s)) ≡ g(s) = t,
t = g(s) = g(k(t)) ≡ f(t) = s,
in which s and t are parameters with different units. By implication the functions are either f or its inverse:
s = f(t) = f(f-1(s)) = t,
t = f-1(s) = f-1(f(t)) = s.
Function f takes t-units into s-units, and function f-1 takes s-units into t-units. The vector versions are as follows:
s = f(t) = f(f-1(s)) = t,
t = f-1(s) = f-1(f(t)) = s.
Motion space is an ordered pair of vectors s and t: (s, t), resulting in their direct sum vector space. Addition is conducted by components: (r, w) + (s, t) = (r + s, w + t). Scalar multiplication is also by component: (a, b) (s, t) = (as, bt). To multiply a scalar and only one component requires the other component to be unity. Thus additive unity is (0, 0) and multiplicative unity is (1, 1).
There are two ways to mask an ordered pair of vectors: left mask (s, t)⌊ = (s, t) and right mask (s, t)⌋ = (s, t), where s = |s| and t = |t|. What was described here as expansion and contraction may now be shown more clearly as masking and unmasking. A parametric length vector function is converted to a parametric duration vector function as follows:
r(t) = masked r(t) ↑ unmasked r(t) ↔ (inverted) = unmasked t(r) ↓ masked t(r) = t(r).
r(t) = [r(t), θ(t), φ(t)] ↑ [r´(t´, χ´, ψ´), θ´(t´, χ´, ψ´), φ´(t´, χ´, ψ´)] ↔ (t´(r´, θ´, φ´), χ´(r´, θ´, φ´), ψ´(r´, θ´, φ´)) ↓ [t(r), χ(r), ψ(r)] = t(r).