Manuscript VII-E@1 · VII-E.tex · proposition · b713cf3c

Proposition: A scalar speed does not determine a velocity set

Prescribed kinematics and schedule dependence › Directional dependence · lines 795–815 · proof included or linked

VII-E@1#prop:direction

In a square-lattice dependency model permitting at most one cardinal step of length $a$ per tick, or a wait, the asymptotic velocity set is

V = {(v_{x}, v_{y}) : ∣ v_{x} ∣ + ∣ v_{y} ∣ \leq a / τ} .

An isotropic physical speed $a / τ$ is not dominated in every direction by this set.

Proof. After $n$ ticks, $∣ Δ x ∣ + ∣ Δ y ∣ \leq na$ . Conversely, suitable proportions of cardinal steps and waits approximate every velocity in the displayed diamond. But

(\frac{a}{2 τ}, \frac{a}{2 τ})

has Euclidean norm $a / τ$ and lies outside that diamond.

Displayed formulas in this passage

unlabeled · line 799
$V = {(v_{x}, v_{y}) : ∣ v_{x} ∣ + ∣ v_{y} ∣ \leq a / τ} .$
unlabeled · line 811
$(\frac{a}{2 τ}, \frac{a}{2 τ})$

Nearby in this section

section: Directional dependence