ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Squared distance to a circle is Morse for centers off the medial axis and degenerate at the center

Example

On the unit circle $S^{1} \subseteq R^{2}$ , the squared-distance function from a center $p = (a, b)$ is Morse for every $p \neq 0$ , while at the center it is constant and therefore maximally degenerate. For this example, the medial axis is just the single point $0$ .

Facts & Assumptions

Given: The unit circle $S^{1} \subseteq R^{2}$ and a center $p = (a, b) \in R^{2}$ .

[L1]

Generic centers give Morse squared-distance functions on a compact embedded manifold (For a compact manifold embedded in Euclidean space, the squared-distance function from a generic center is Morse).

Verification

technique · direct computation

1.1givenalgebra

Parametrize the circle by $x (θ) = (\cos θ, \sin θ)$ . Then $d_{p} (θ) = ∥ x (θ) - p ∥^{2} = 1 + a^{2} + b^{2} - 2 (a \cos θ + b \sin θ) .$ Therefore $d_{p}^{'} (θ) = 2 (a \sin θ - b \cos θ) and d_{p}^{''} (θ) = 2 (a \cos θ + b \sin θ) .$

2.1L1step 1.1algebra

If $p \neq 0$ , the equation $a \sin θ = b \cos θ$ has exactly two solutions modulo $2 π$ , corresponding to the two points where the radius through $p$ meets the circle. At those two points one has $d_{p}^{''} (θ) = \pm 2 \sqrt{a^{2} + b^{2}} \neq 0$ , so both critical points are nondegenerate and $d_{p}$ is Morse. This matches the generic-center theorem [L1].

2.2step 1.1algebra

If $p = 0$ , then $d_{0} (θ) \equiv 1$ is constant. Every point of the circle is critical, so the function is degenerate and not Morse.

3.1step 2.1step 2.2∎

Thus centers off the medial axis yield Morse squared-distance functions, while the center itself is the exceptional degenerate case.

Depends on

For a compact manifold embedded in Euclidean space, the squared-distance function from a generic center is Morse

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)
Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)