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A corner has no unique tangent and a cusp gives a zero centered chord

Example

Two elementary paths show why regularity and nondegenerate centered chords are explicit hypotheses. The corner path is $R_{c} (s) = (s, 0)$ for $s \leq 0$ and $R_{c} (s) = (0, s)$ for $s \geq 0$ . The cusp path is $R_{u} (s) = (s^{2}, 0)$ . For equal samples at $- h$ and $h$ , the centered chord is $R (h) - R (- h)$ .

Facts & Assumptions

Given: The two displayed paths and any $h > 0$ small enough that their samples lie in the stated parameter intervals.

[F1]

Second-order centered and unequal-step tangent estimates assumes a regular $C^{3}$ curve and a nonzero centered chord before normalizing it.

[F2]

Finite-image projected-gradient string with arc-length redistribution returns geometry-failure if an interior centered chord is zero.

Proof

Given: The paths and sample spacing in the Example.

1.1givenalgebra

For the corner, the one-sided derivatives at $0$ are $R_{c}^{'} (0^{-}) = (1, 0)$ and $R_{c}^{'} (0^{+}) = (0, 1)$ . They are distinct, so there is no single tangent at the corner. The centered chord is $R_{c} (h) - R_{c} (- h) = (h, h)$ , whose unit direction is $(1, 1) / \sqrt{2}$ ; it agrees with neither one-sided tangent.

1.2givenalgebra

For the cusp, $R_{u} (h) = R_{u} (- h) = (h^{2}, 0)$ , so its centered chord is exactly zero. Also $R_{u}^{'} (0) = (0, 0)$ , so the path is not regular there.

2.1F1F2step 1.1step 1.2∎

The corner violates the differentiability/unique-tangent hypotheses of [F1], while the cusp violates regularity and makes the algorithmic centered direction undefined. By [F2], a finite string update encountering this zero chord must return geometry-failure; neither example can be covered by the second-order normalized-tangent estimate of [F1].

Depends on

Used by

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Dependency tree · two levels

5 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

Weinan E, Weiqing Ren, and Eric Vanden-Eijnden, String method for the study of rare events (standard reference, not scraped)