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arxiv: 2605.08087 · v1 · submitted 2026-01-26 · 💻 cs.GR

Explicit Inversion of Planar NURBS Curves

Michelangelo Marsala , Pablo Maz\'on This is my paper

Pith reviewed 2026-05-16 10:45 UTC · model grok-4.3

classification 💻 cs.GR
keywords NURBScurve inversionrational splinesplanar curvesparametric curvesinverse mappingcomputer-aided design
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The pith

A planar NURBS curve parametrization has an explicit inverse given by rational splines.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that any planar NURBS curve comes with a built-in inverse function from points on the curve back to its parameter values. This inverse is constructed as a linear combination of a specific family of rational spline functions defined directly on the curve. Explicit formulas are given for these functions, making the inversion computable in closed form. If this holds, it replaces numerical search methods with direct evaluation in computer graphics and design applications.

Core claim

A general planar NURBS curve parametrization φ from an interval to the curve C admits an inverse φ inverse from C to the interval that is defined by rational splines. A family of rational spline functions is constructed on C with explicit computation formulas, and the inverse parametrization is proved to be a linear combination of these functions.

What carries the argument

Family of rational spline functions on the curve C that serve as a basis for expressing the inverse parametrization as their linear combination.

If this is right

  • The inverse can be evaluated directly by combining the rational splines without solving equations iteratively.
  • Applications in CAD can use this for exact point-to-parameter mapping on planar NURBS.
  • Several examples confirm the formulas produce correct inverses for sample curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the construction generalizes, similar explicit inverses might exist for non-planar or higher-order splines.
  • This explicit form could improve stability in algorithms that repeatedly invert curves, such as in rendering or animation.
  • Testing on curves with near-singular points would reveal practical limits of the method.

Load-bearing premise

The curve is planar and its parametrization is sufficiently regular that an inverse exists at every point without self-intersections or singularities.

What would settle it

Finding a planar NURBS curve for which the constructed rational spline basis fails to reproduce the true inverse parametrization would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.08087 by Michelangelo Marsala, Pablo Maz\'on.

Figure 1
Figure 1. Figure 1: NURBS curve with associated control net (a) and its decomposition in rational polynomial pieces (b). [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The physical B-spline N0,1 (a), N1,1 (b), N2,1 (c) and the graph of the inverse NURBS in (12). In dashed line is the input NURBS curve ϕ. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: NURBS curve with associated control net (a) and the graph of its inverse (b). In dashed line is the input NURBS [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: From (a) to (f): the quadratic physical rational functions [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: NURBS curve of degree d = 4 with associated control net (a) and the graph of its inverse (b). In dashed line is the input curve ϕ. Note the C0 continuity in correspondence of P5. Note that, since µ(2/3) = 4, it results ϕ(2/3) = P5 as [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: From (a) to (i): the set of cubic physical rational splines [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Quintic NURBS curve with a self insersection point and its associated control net (a) and the graph of its inverse [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: From (a) to (f): the quartic physical rational splines [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
read the original abstract

We prove that a general planar NURBS curve parametrization $\phi: [u_0,u_m] \xrightarrow{} C \subset \mathbb{R}^2$ admits an inverse map $\phi^{-1}: C \xrightarrow{} [u_0,u_m]$ defined by rational splines. More specifically, we construct a family of rational spline functions on the curve $C$, present explicit formulas for their computation, and prove that the inverse parametrization admits a representation as a linear combination of these functions. Several examples are provided to illustrate the effectiveness of the proposed approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proves that a general planar NURBS curve parametrization φ: [u0, um] → C ⊂ R² admits an inverse map φ^{-1}: C → [u0, um] defined by rational splines. It constructs a family of rational spline functions on C, gives explicit formulas for their computation, and proves that the inverse admits a representation as a linear combination of these functions, with several examples provided.

Significance. If the central claim holds, the result supplies an explicit constructive inversion procedure for planar NURBS curves via rational splines. This would be useful in CAD, geometric modeling, and graphics applications that require reliable curve inversion. The manuscript supplies explicit formulas, a proof, and examples, which are positive features for reproducibility and verification.

major comments (1)
  1. [Abstract] Abstract and §1: the statement is made for a 'general' planar NURBS parametrization without listing injectivity or regularity assumptions. For φ^{-1} to be a single-valued function on the full image C, the parametrization must be injective (no self-intersections) and regular (φ' ≠ 0, no cusps). The construction appears to proceed segment-wise via local rational inversion; without global handling of these cases the linear-combination representation cannot hold on the entire C. This assumption is load-bearing for the existence claim.
minor comments (1)
  1. [§3] Notation for the rational spline basis functions on C should be introduced with a clear definition of their support and degree before the linear-combination statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The point raised about missing assumptions is valid and will be addressed by explicit clarification in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and §1: the statement is made for a 'general' planar NURBS parametrization without listing injectivity or regularity assumptions. For φ^{-1} to be a single-valued function on the full image C, the parametrization must be injective (no self-intersections) and regular (φ' ≠ 0, no cusps). The construction appears to proceed segment-wise via local rational inversion; without global handling of these cases the linear-combination representation cannot hold on the entire C. This assumption is load-bearing for the existence claim.

    Authors: We agree that injectivity and regularity are necessary for φ^{-1} to be a well-defined single-valued function on all of C. Our construction defines local rational inverses on each knot interval and assembles them into a global linear combination; this assembly is valid precisely when the parametrization has no self-intersections and no vanishing derivatives. In the revision we will replace the word 'general' in the abstract and §1 with an explicit statement of these two assumptions, add a short paragraph in §2 explaining why they are required for the global representation, and note that the method can be applied piecewise on sub-curves when the assumptions fail globally. No change to the core theorems or formulas is needed. revision: yes

Circularity Check

0 steps flagged

No circularity: constructive proof of inverse via explicit rational spline basis

full rationale

The paper's central claim is a constructive existence proof: it defines a family of rational spline functions on C, gives explicit formulas for them, and shows the inverse is their linear combination. No equation reduces to a fitted parameter renamed as prediction, no self-citation chain carries the load-bearing step, and no ansatz is smuggled in. The derivation is self-contained against the stated assumptions on the NURBS parametrization; the result does not presuppose the inverse representation it derives.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard properties of NURBS basis functions and rational splines; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math NURBS curves are piecewise rational functions formed from B-spline basis functions multiplied by weights.
    Standard definition invoked to construct the inverse functions.

pith-pipeline@v0.9.0 · 5381 in / 1131 out tokens · 42317 ms · 2026-05-16T10:45:51.108170+00:00 · methodology

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