Manuscript IX-E-3@1 · IX-E-3.tex · lemma · 8c41417a

Lemma: Sphere–ball intersection

The geometric bridge and the missed-ball branch · lines 870–887 · proof included or linked

IX-E-3@1#eq:sphere-ball

Let $t > 0$ , $s \geq 0$ , and $R = ∣ x - y ∣$ . Then

{∣ z - x ∣ = t} \cap {∣ z - y ∣ \leq s} \neq = \emptyset ⟺ t - s \leq R \leq t + s .

Proof. The smallest distance from $y$ to the sphere centered at $x$ is $∣ R - t ∣$ , including $R = 0$ . The claimed intersection is therefore equivalent to $∣ R - t ∣ \leq s$ .

Displayed formulas in this passage

IX-E-3@1#eq:sphere-ball · line 873
${∣ z - x ∣ = t} \cap {∣ z - y ∣ \leq s} \neq = \emptyset ⟺ t - s \leq R \leq t + s .$

Nearby in this section

section: The geometric bridge and the missed-ball branch
theorem: Theorem: Full exclusion from sphere and observation support
corollary: Corollary: Bridge for the flat realization
section: The geometric bridge and the missed-ball branch