LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Polyline interpolation and equal-length redistribution approximate a smooth arc-length path

Statement

Let $L > 0$ , let $M \geq 1$ be an integer, and let $R : [0, L] \to R^{d}$ be a unit-speed $C^{3}$ curve with $M$ equal arc-length intervals of length $h = L / M$ . Suppose $∥ R^{''} ∥_{2} \leq K_{2}$ and the coordinate third derivatives satisfy the bound in Second-order centered and unequal-step tangent estimates with constant $M_{3}$ . For sufficiently small $h$ , each chord length $ℓ_{i} = ∥ R (i h) - R ((i - 1) h) ∥_{2}$ satisfies $∣ ℓ_{i} - h ∣ \leq C_{1} h^{3}$ , where $C_{1}$ depends only on $K_{2}, M_{3}$ . The piecewise-linear interpolant through the samples differs from $R$ in supremum norm by at most $C_{2} h^{2}$ . If this polygon is redistributed by linear interpolation at equally spaced cumulative-length targets, the $j$ th redistributed image is within $C_{3} h^{2}$ of $R (j h)$ , for every $j = 0, \dots, M$ .

Here $C^{3}$ on $[0, L]$ means that $R$ is the restriction of a $C^{3}$ curve on an open interval containing $[0, L]$ ; derivatives at the endpoints are the continuous one-sided values, and the unit-speed and derivative bounds are required on $[0, L]$ .

Facts & Assumptions

Given: The unit-speed $C^{3}$ curve, equal arc-length mesh, derivative bounds, and small-mesh condition in the Statement.

[F1]

The unit-speed hypothesis [A1] means $∥ R^{'} (s) ∥_{2} = 1$ on $[0, L]$ . Thus the normalized tangent convention of Regular minimum-energy paths on a declared collision-free tube gives $t (s) = R^{'} (s)$ on this interval.

[F2]

Second-order centered and unequal-step tangent estimates: the coordinate third-derivative bound used here is $∣ R_{j}^{'''} ∣ \leq M_{3} / \sqrt{d}$ on the path interval.

[F3]

The Lagrange and Cauchy forms of Taylor's remainder: its one-dimensional Lagrange formula applies to a scalar $C^{3}$ function on a closed interval when derivatives through order three are continuous to the endpoints; with degree two it bounds the remainder by $\sup ∣ g^{'''} ∣ ∣ u ∣^{3} / 6$ , and with degree one it bounds the remainder by $\sup ∣ g^{''} ∣ ∣ u ∣^{2} / 2$ .

[F4]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : differentiating a composition gives the derivative of the outer function applied to the derivative of the inner function.

[A1]

The curve has the endpoint $C^{3}$ convention in the Statement, has unit speed on $[0, L]$ , and has the stated bounds $∥ R^{''} ∥_{2} \leq K_{2}$ and coordinate third-derivative bound $M_{3} / \sqrt{d}$ on the whole interval; the interior $C^{3}$ convention is $C^{k}$ maps and multi-index derivative notation in Euclidean space.

Given: The data in the Statement.

Proof

1.1A1F1F2F3F4givenalgebra

On each mesh interval $[s, s + h]$ , apply the one-dimensional degree-two formula [F3] to each coordinate of the extended curve at $s$ ; the endpoint convention supplies the required derivative continuity, and [A1, F2] gives a remainder vector $r$ with $∥ r ∥_{2} \leq M_{3} h^{3} / 6$ . Thus $R (s + h) - R (s) = h T + \frac{h^{2}}{2} K + r$ with $T = R^{'} (s)$ , $K = R^{''} (s)$ , and $∥ K ∥_{2} \leq K_{2}$ . Differentiating $∥ R^{'} (s) ∥_{2}^{2} = 1$ gives $T^{T} K = 0$ by the chain rule [F4]. Hence $∣ ∥ R (s + h) - R (s) ∥_{2}^{2} - h^{2} ∣ \leq C h^{4}$ for a constant depending only on $K_{2}, M_{3}$ ; for sufficiently small $h$ the chord is at least $h / 2$ , so division by its sum with $h$ yields $∣ ℓ_{i} - h ∣ \leq C_{1} h^{3}$ .

1.2A1F1F3givenalgebra

For $s = i h + θ h$ with $0 \leq θ \leq 1$ , apply the one-dimensional degree-one formula [F3] on $[i h, s]$ and $[i h, (i + 1) h]$ . The remainder bound in [F3] and $∥ R^{''} ∥_{2} \leq K_{2}$ give $R (s) = R (i h) + θ h R^{'} (i h) + O (K_{2} h^{2})$ and $R ((i + 1) h) = R (i h) + h R^{'} (i h) + O (K_{2} h^{2})$ . Subtracting the linear interpolant cancels the first-order term, so the pointwise interpolation error is bounded by $C_{2} h^{2}$ , uniformly over all intervals.

2.1givenstep 1.1algebra

Let $p_{i} = \sum_{m = 1}^{i} ℓ_{m}$ and $L_{h} = p_{M}$ . Step 1.1 gives $∣ p_{i} - i h ∣ \leq i C_{1} h^{3} \leq C_{1} L h^{2}$ and $∣ L_{h} - L ∣ \leq C_{1} L h^{2}$ . Thus the redistributed target $q_{j} = j L_{h} / M$ satisfies $∣ q_{j} - j h ∣ \leq C_{1} L h^{2}$ . Each segment has length at least $h / 2$ for small $h$ . If target $q_{j}$ lies on segment $i$ at fraction $θ$ , then $p_{i - 1} + θ ℓ_{i} = q_{j}$ ; combining $∣ p_{i - 1} - (i - 1) h ∣ \leq C_{1} L h^{2}$ , $∣ q_{j} - j h ∣ \leq C_{1} L h^{2}$ , and $∣ ℓ_{i} - h ∣ \leq C_{1} h^{3}$ gives $∣ (i - 1) h + θ h - j h ∣ \leq C h^{2}$ .

3.1F1step 1.2step 2.1algebra∎

The point on the true curve at parameter $(i - 1) h + θ h$ is within $C_{2} h^{2}$ of the redistributed polygon point by step 1.2, and it is within $C h^{2}$ of $R (j h)$ by unit speed and step 2.1. The triangle inequality gives the claimed $C_{3} h^{2}$ bound, including endpoints where redistribution fixes the endpoints.

Depends on

Used by

Dependency tree · two levels

19 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)
Weinan E, Weiqing Ren, and Eric Vanden-Eijnden, Simplified and improved string method for computing the minimum energy paths in barrier-crossing events (standard reference, not scraped)