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arxiv: 2604.08697 · v1 · submitted 2026-04-09 · 🧮 math.NA · cs.NA

h-γ Blossoming, h-γ Bernstein Bases, and h-γ B\'{e}zier Curves for Translation Invariant left(γ₁,γ₂right) Spaces

Fatma Z\"urnac{\i}-Yeti\c{s} , Ron Goldman , Plamen Simeonov This is my paper

Pith reviewed 2026-05-10 16:46 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords h-gamma blossominggamma1 gamma2 spacesBernstein basesBezier curvestranslation invariant spacesrecursive evaluationdegree elevationsubdivision
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The pith

Merging γ-blossoming with h-blossoming produces h-γ Bernstein bases and Bézier curves for translation-invariant (γ1, γ2) spaces.

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

The paper constructs a novel h-γ blossom by combining γ-blossoming, which operates on spaces spanned by powers of two linearly independent functions γ1 and γ2, with h-blossoming, which incorporates a discrete step size h. This construction is possible only when the underlying space is translation invariant, meaning shifts by h can be rewritten as linear combinations of the original generators. From the h-γ blossom the authors define corresponding Bernstein bases and Bézier curves, then derive recursive evaluation, subdivision, Marsden identities, degree elevation, and interpolation procedures. A reader should care because the resulting schemes extend the familiar computational toolkit of polynomial Bézier curves to trigonometric, hyperbolic, and other non-polynomial spaces while preserving key algorithmic advantages.

Core claim

We merge γ-blossoming for (γ1, γ2) spaces with h-blossoming for h-Bernstein bases and h-Bézier curves to construct a novel h-γ blossom for translation invariant (γ1, γ2) spaces generated by two continuous, linearly independent functions γ1 and γ2. Based on this h-γ blossom, we define h-γ Bernstein bases and h-γ Bézier curves and study their properties. We derive recursive evaluation algorithms, subdivision procedures, Marsden identities, and formulas for degree elevation and interpolation for these h-γ Bernstein and h-γ Bézier schemes.

What carries the argument

The h-γ blossom, formed by merging γ-blossoming (for spaces generated by γ1 and γ2) with h-blossoming (for step size h) to produce bases and curves in translation-invariant (γ1, γ2) spaces.

If this is right

  • h-γ Bernstein bases admit recursive evaluation and subdivision algorithms analogous to the classical case.
  • Marsden identities hold for the h-γ bases, relating them to the underlying generators.
  • Degree elevation and interpolation formulas extend directly to the new h-γ Bézier curves.
  • The schemes apply uniformly to polynomial, trigonometric, hyperbolic, and discrete analogues of these spaces.

Where Pith is reading between the lines

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

  • The same merging strategy could be tested on spaces generated by three or more functions to see whether higher-order h-γ blossoms exist.
  • Implementation for specific γ pairs would allow direct numerical comparison of curve fairness and approximation quality against existing trigonometric Bézier methods.
  • The translation-invariance requirement suggests that the approach may extend naturally to other shift-invariant function spaces arising in approximation theory.

Load-bearing premise

The (γ1, γ2) space must be translation invariant, so that γ1(x-h) and γ2(x-h) remain linear combinations of γ1(x) and γ2(x).

What would settle it

For the trigonometric space with γ1(x)=cos x and γ2(x)=sin x, compute the explicit h-γ Bernstein basis functions and check whether they reproduce every function in the space under the blossoming axioms; failure to do so would refute the construction.

Figures

Figures reproduced from arXiv: 2604.08697 by Fatma Z\"urnac{\i}-Yeti\c{s}, Plamen Simeonov, Ron Goldman.

Figure 1
Figure 1. Figure 1: Computing g(Γ(u1), . . . , Γ(un); h) from the initial h– γ blossom values Q0 i , i = 0, . . . , n. Here we illustrate the case n = 3 and we use the notation u, v, w to represent the blossom value g(Γ(u), Γ(v), Γ(w); h). for i = 0, . . . , n − k − 1 and k = 0, . . . , n − 1. Then P k i (x) = g(Γ(a − (k + i)h), . . . , Γ(a − (n − 1)h), Γ(b), Γ(b − h), . . . , Γ(b − (i − 1)h), Γ(x − (σ(1) − 1)h), . . . , Γ(x … view at source ↗
Figure 2
Figure 2. Figure 2: Recursive evaluation algorithm for G(x) for σ(i) = i. Here we illustrate the case n = 3 and we use the notation u, v, w to represent the blossom value g(Γ(u), Γ(v), Γ(w); h). 5 Bernstein Basis Functions In this section, the notation [a, b] denotes the interval determined by the endpoints a and b, regardless of their order. Thus, x ∈ [a, b] means that x lies between a and b, i.e., min{a, b} ≤ x ≤ max{a, b}.… view at source ↗
Figure 4
Figure 4. Figure 4: Recursive evaluation algorithm for G(x) with σ(i) = n + 1 − i. Here we illustrate the case n = 2 and we use the notation u, v to represent the blossom value g(Γ(u), Γ(v); h). B 2 0 (x, [a, b]; γ, h) = d(x, b)d(x − h, b) d(a, b)d(a − h, b) , B 2 1 (x, [a, b]; γ, h) = d(a, x)d(x − h, b) d(a, b)d(a − h, b) + d(x, b − h)d(a − h, x − h) d(a − h, b − h)d(a − h, b) , B 2 2 (x, [a, b]; γ, h) = d(a − h, x)d(a − h, … view at source ↗
Figure 5
Figure 5. Figure 5: Placing 1 at the apex and propagating down yields the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original abstract

A $\left(\gamma_{1}, \gamma_{2}\right)$ space of order $n$ is a space of univariate functions spanned by $\left\{\gamma_{1}^{n-k}(x), \gamma_{2}^{k}(x)\right\}_{k=0}^{n}$. A $\left(\gamma_{1}, \gamma_{2}\right)$ space is said to be translation invariant if $\gamma_{1}(x-h)$ and $\gamma_{2}(x-h)$ can be expressed as nonsingular linear combinations of $\gamma_{1}(x)$ and $\gamma_{2}(x)$. Translation invariant $\left(\gamma_{1}, \gamma_{2}\right)$ spaces include polynomials $\left(\gamma_{1}(x)=1, \gamma_{2}(x)=x\right)$, trigonometric functions $\left(\gamma_{1}(x)=\cos x, \gamma_{2}(x)=\sin x\right)$, hyperbolic functions $\left(\gamma_{1}(x)=\cosh x, \gamma_{2}(x)=\sinh x\right)$, and their discrete analogues. We merge $\gamma$-blossoming for $\left(\gamma_{1}, \gamma_{2}\right)$ spaces with $h$-blossoming for $h$-Bernstein bases and $h$-B\'{e}zier curves to construct a novel $h$-$\gamma$ blossom for translation invariant $\left(\gamma_{1}, \gamma_{2}\right)$ spaces generated by two continuous, linearly independent functions $\gamma_{1}$ and $\gamma_{2}$. Based on this $h$-$\gamma$ blossom, we define $h$-$\gamma$ Bernstein bases and $h$-$\gamma$ B\'{e}zier curves and study their properties. We derive recursive evaluation algorithms, subdivision procedures, Marsden identities, and formulas for degree elevation and interpolation for these $h$-$\gamma$ Bernstein and $h$-$\gamma$ B\'{e}zier schemes.

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

0 major / 2 minor

Summary. The paper merges existing γ-blossoming techniques for translation-invariant (γ1, γ2) spaces with h-blossoming to define a new h-γ blossom. From this it constructs h-γ Bernstein bases and h-γ Bézier curves for spaces spanned by {γ1^{n-k}(x), γ2^k(x)} and derives their standard algebraic properties: recursive evaluation, subdivision, Marsden identities, degree elevation, and interpolation formulas. The constructions rely on the translation-invariance assumption that γ1(x-h) and γ2(x-h) are nonsingular linear combinations of γ1(x) and γ2(x), together with continuity and linear independence of γ1 and γ2.

Significance. If the derivations hold, the work supplies a parameter-free unification of generalized Bézier schemes across polynomial, trigonometric, hyperbolic, and discrete spaces that satisfy translation invariance. Explicit algorithms for subdivision, degree elevation, and interpolation increase the immediate applicability in CAGD. The manuscript credits the two source blossoming frameworks and states the translation-invariance hypothesis explicitly, which is a strength.

minor comments (2)
  1. The abstract and introduction should include a short explicit statement of the precise regularity assumed on γ1 and γ2 beyond continuity (e.g., whether C^1 or higher is needed for the blossoming operator to be well-defined).
  2. Notation for the h-γ blossom operator itself could be introduced with a displayed definition before the subsequent basis and curve constructions to improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the h-γ blossom and its applications to Bernstein bases and Bézier curves in translation-invariant (γ1, γ2) spaces. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs the h-γ blossom explicitly by merging the pre-existing γ-blossoming operator (defined on translation-invariant (γ1, γ2) spaces) with the h-blossoming operator, using the translation-invariance hypothesis to ensure the two frameworks commute via linear combinations. All subsequent objects—h-γ Bernstein bases, h-γ Bézier curves—and their algebraic properties (recursion, subdivision, Marsden identity, degree elevation, interpolation) are then derived directly from this merged definition and the stated continuity/linear-independence assumptions. No step reduces a claimed result to a fitted parameter, renames an internal quantity as a prediction, or relies on a load-bearing self-citation whose content is itself unverified; the derivation remains a self-contained definitional extension of prior independent frameworks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of translation invariance for the (γ1, γ2) spaces and on the ability to combine two existing blossoming frameworks; no free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption A (γ1, γ2) space is translation invariant if γ1(x-h) and γ2(x-h) can be expressed as nonsingular linear combinations of γ1(x) and γ2(x).
    This property is explicitly required in the definition of the spaces considered and is the key enabler for merging h-blossoming with γ-blossoming.

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [1]

    Di¸ sib¨ uy¨ ukC ¸., Goldman R. (2015). A unifying structure for polar forms and for Bernstein B´ ezier curves, J. Approx. Theory, 192, 234–249

    work page 2015

  2. [2]

    Di¸ sib¨ uy¨ uk C ¸ ., Goldman R. (2016). A unified approach to non-polynomial B-spline curves based on a novel variant of the polar form. Calcolo, 53(4), 751-781

    work page 2016

  3. [3]

    Lyche, T. (1999). Trigonometric splines; a survey with new results. Shaping Preserving Representations in Computer-Aided Geometric Design, 201-227

    work page 1999

  4. [4]

    Goldman, R. (1985). P´ olya’s urn model and computer aided geometric design. SIAM J. Alg. Disc. Meth. 6, 1–28

    work page 1985

  5. [5]

    Goldman, R., Barry, P. (1991). Shape parameter deletion for P´ olya curves. Numer. Algorithms 1, 121–137

    work page 1991

  6. [6]

    Goldman, R., Simeonov, P. (2015). Two essential properties of ( q, h)-Bernstein-B´ ezier curves. Applied Numerical Mathematics, 96, 82-93

    work page 2015

  7. [7]

    Gonsor, D., Neamtu, M. (1994). Non-polynomial polar forms. Curves and Surfaces in Geometric Design (Chamonix-Mont-Blanc, 1993), 193-200

    work page 1994

  8. [8]

    Simeonov, P., Zafiris, V., Goldman, R. (2011). h-Blossoming: A new approach to algorithms and identities forh-Bernstein bases andh-B´ ezier curves. Computer Aided Geometric Design, 28(9), 549-565

    work page 2011

  9. [9]

    Stancu, D. (1968). Approximation of functions by a new class of linear polynomial operators. Rev. Roumaine Math. Pures Appl. 13, 1173–1194

    work page 1968

  10. [10]

    Stancu, D. (1984). Generalized Bernstein approximating operators. In: Itinerant Seminar on Functional Equations, Approximation and Convexity. Cluj-Napoca, 1984. Univ. ”Babes-Bolyai”, Cluj-Napoca, pp. 185–192. Preprint 84-6. 17

    work page 1984