Closed-Form Blossoms Evaluation

• The paper presents closed-form representations for blossoms that bypass recursion and directly compute subdivision control points for curves and surfaces.
• It details methods for evaluating polynomial and non-polynomial splines, unifying frameworks for Bézier, trigonometric, hyperbolic, and Müntz-type spaces.
• The work demonstrates that vectorized binomial and multinomial expansions enable efficient, parallelizable algorithms that outperform classical recursive schemes.

Closed-form formulae for blossoms evaluation provide explicit, non-recursive representations of the multiaffine, symmetric functional (the blossom) associated with curves and surfaces in computational geometry, as well as generalized non-polynomial function spaces. These formulae underpin efficient algorithms for subdivision, basis conversion, and evaluation of a wide class of geometric and approximation operators, especially within polynomial, trigonometric, hyperbolic, and Müntz-type spline spaces. Modern developments have unified polynomial and non-polynomial frameworks and have yielded efficient, parallelizable evaluation and subdivision schemes that complement and often outperform traditional recursive algorithms in practice (Zürnacı-Yetiş, 26 Dec 2025, Vlachkova, 11 Jan 2026).

1. Definition and Core Properties of Blossoms

The blossom of a degree-$n$ function $C(u)$ (for instance, a Bézier curve) is uniquely characterized as the symmetric, multi-affine function $B[u_1,\ldots,u_n]$ satisfying:

• Symmetry: $$B[u_{\sigma(1)},\ldots,u_{\sigma(n)}]=B[u_1,\ldots,u_n]$$ for any permutation $\sigma$;
• Multi-affinity: affine in each argument;
• Diagonal reduction: $B[t,\ldots,t]=C(t)$.

For non-polynomial spaces $$\pi_n(\gamma_1,\gamma_2) = \text{span}\{\gamma_1^{n-k}\gamma_2^k\}_{k=0}^n$$, the blossom generalizes to a function $g((x_1,w_1),\ldots,(x_n,w_n))$ on $([a,b]\times\mathbb{R})^n$ with analogous properties (Zürnacı-Yetiş, 26 Dec 2025).

2. Closed-form Blossom Formulae

Polynomial (Bézier/Bernstein) Case

The closed-form for the blossom of a Bézier curve $C(u)=\sum_{i=0}^n P_i u^i$ is: $$B[u_1, \ldots, u_n] = \sum_{i=0}^n P_i \cdot b_i(u_1, \ldots, u_n),$$ where

$$b_i(u_1, \ldots, u_n) = \frac{1}{\binom{n}{i}} \sum_{1 \leq \alpha_1<\cdots<\alpha_i \leq n} (u_{\alpha_1}\cdots u_{\alpha_i}).$$

At a tuple with $\nu$ entries $b$ and $n-\nu$ entries $a$: $$b_i(\underbrace{b,\ldots,b}_{\nu}, \underbrace{a,\ldots,a}_{n-\nu}) = \frac{1}{\binom{n}{i}} \sum_{k=\max(0,i+\nu-n)}^{\min(i,\nu)} \binom{\nu}{k} \binom{n-\nu}{i-k} b^k a^{i-k}.$$ This formula gives direct access to subdivision control points (see Section 4) (Vlachkova, 11 Jan 2026).

Extended Non-polynomial Homogeneous Blossoms

For $G\in\pi_{\ell}(\gamma_1,\gamma_2)$, the extended homogeneous blossom of order $k$ is: $$g_e\Bigl( (x_1,w_1),\ldots,(x_m,w_m) \big/ (u_1,v_1),\ldots,(u_n,v_n) \Bigr)$$ with $m=n+k$, uniquely determined by bisymmetry, multi-affinity, cancellation, and diagonal reproduction.

For positive order $k\ge 0$, the closed form is

$$g_e\Bigl((x_1,w_1),\ldots,(x_m,w_m) \big/ (u_1,v_1),\ldots,(u_n,v_n)\Bigr) = \sum_{\substack{ \{i_1,\ldots,i_\alpha\} \subset \{1,\ldots,m\}\ j_1,\ldots,j_\beta \in \{1,\ldots,n\}\ \alpha+\beta = k }} (-1)^{\beta} g\Bigl( (x_{i_1},w_{i_1}),\ldots,(x_{i_\alpha},w_{i_\alpha}), (u_{j_1},v_{j_1}),\ldots,(u_{j_\beta},v_{j_\beta}) \Bigr).$$

For negative order $k<0$, closed-form identities in terms of non-polynomial divided differences exist (see Section 3) (Zürnacı-Yetiş, 26 Dec 2025).

3. Connections with Divided Differences

The central connection between the non-polynomial divided difference operator $\Delta_{\gamma_1,\gamma_2}[\,\cdot\,]$ and blossoms is formalized as follows. For instance, for negative order, if $k=m-n<0$, $r=n-m-1>0$: $$h\Bigl( (x_1,w_1),\ldots,(x_m,w_m) \big/ (u_1,v_1),\ldots,(u_n,v_n) \Bigr) = \Delta_{\gamma_1,\gamma_2} \Bigl[ \varepsilon_1,\ldots,\varepsilon_n \Bigr] \Bigl\{ r!\prod_{i=1}^m (x_i\gamma_2(x)-w_i\gamma_1(x)) D_{\gamma_1,\gamma_2}^{-r}(H(x)) \Bigr\},$$ where $$(u_j, v_j) = (\gamma_1(\varepsilon_j), \gamma_2(\varepsilon_j))$$ and $D_{\gamma_1,\gamma_2}^{-r}$ denotes an appropriate differential operator.

A pivotal identity relates divided differences of products to (extended) blossoms, establishing a duality central to dual basis and spline theory (Zürnacı-Yetiş, 26 Dec 2025).

4. Subdivision and Practical Evaluation

Closed-form blossom formulae yield immediate expressions for the subdivided control points of Bézier curves and surfaces over arbitrary intervals, bypassing recursion. For degree-$n$ Bézier curves with control points $P_i$, subdivision over $[a,b]$ gives new control points $W_{\nu}$ as: $$W_{\nu} = \sum_{i=0}^n P_i \frac{1}{\binom{n}{i}} \sum_{k=\max(0,i+\nu-n)}^{\min(i,\nu)} \binom{\nu}{k}\binom{n-\nu}{i-k}b^k a^{i-k},$$ for $0 \leq \nu \leq n$.

For tensor-product and triangular Bézier surfaces, analogous binomial/multinomial closed-form expansions apply. These formulae are:

• Tensor-product (bidegree $(n,m)$):

$$Q_{\nu\mu} = \sum_{i=0}^{n}\sum_{j=0}^{m} P_{ij} \left[ \frac{1}{\binom{n}{i}} \sum_{k} \binom{\nu}{k} \binom{n-\nu}{i-k} b^k a^{i-k} \right] \left[ \frac{1}{\binom{m}{j}} \sum_{r} \binom{\mu}{r} \binom{m-\mu}{j-r} d^r c^{j-r} \right].$$

• Triangular (total degree $N$) with barycentric vertices $A,B,C$:

$$R_{\nu\mu} = \sum_{i+j \leq N} P_{ij} \frac{1}{\binom{N}{i,j}} \sum_{\substack{ i_\alpha+i_\beta+i_\gamma=i\ j_\alpha+j_\beta+j_\gamma=j }} \prod_{*} \binom{\nu}{i_\alpha} \binom{\mu}{i_\beta} \binom{\lambda}{i_\gamma} a_1^{i_\alpha}b_1^{i_\beta}c_1^{i_\gamma} a_2^{j_\alpha}b_2^{j_\beta}c_2^{j_\gamma}$$

with the indicated multinomial indices (Vlachkova, 11 Jan 2026).

Pointwise evaluation using these formulas can be efficiently vectorized and implemented without recursion, enabling rapid subdivision in CAD/CAM and real-time systems.

5. Special Function Spaces and Determinantal Forms

Closed-form blossom and divided difference formulae extend to:

• Trigonometric splines: With $\gamma_1(t) = \cos t$, $\gamma_2(t) = \sin t$, explicit ratios of sine-determinants and Vandermonde-type terms yield trigonometric B-spline evaluations, e.g.,

$$B[f;u_1,\ldots,u_n](t) = \sum_{j=0}^n f(t_j) \prod_{i\neq j} \frac{\sin(t - t_i)}{\sin(t_j - t_i)}.$$

• Hyperbolic splines: Analogous determinant and ratio formulas with $\sinh$ replace $\sin$ in the above.
• Müntz-type spaces: With $\gamma_1(t)=1$, $\gamma_2(t)=t^\alpha$, products $(x_i \gamma_2 - w_i \gamma_1)$ yield classical Müntz-Vandermonde ratios, and the extended blossom recovers dual functionals for the Müntz–Bernstein basis (Zürnacı-Yetiş, 26 Dec 2025).

These explicit forms generalize classical determinant/ratio-based constructions and provide practical tools for basis conversion and dual functional computation in these spaces.

6. Algorithmic Implications and Efficiency

Closed-form blossom evaluation enables direct computation of control nets for subdivision and refinement, with the following practical consequences:

• Pointwise binomial or multinomial sum–based evaluation, suitable for SIMD, GPU, and memory-efficient platforms.
• For curves, each output point requires $O(n^2)$ operations, total $O(n^3)$—comparable to or outperforming recursive de Casteljau algorithms when multiple subintervals or arbitrary intervals are required.
• Surface schemes generalize with similar complexity but benefit more substantially from the memory and parallelization advantages.

Empirical results in highly optimized settings demonstrate that these closed forms outperform classical recursive methods, especially for repeated or block subdivision tasks in applications such as CAD/CAM, animation, and adaptive rendering (Vlachkova, 11 Jan 2026).

7. Illustrative Examples

Polynomial Case (Cubic Bézier Segment):

For control points $P_0=(0,0,0)$, $P_1=(1,0,0)$, $P_2=(1,1,0)$, $P_3=(0,1,0)$ and parameter interval $[0.2,0.8]$, the closed-form yields, for $\nu=0$ to $3$,

$$W_\nu = \sum_{i=0}^3 P_i\left( \text{binomial sum coefficients for } [a,b] \right),$$

recovering the control points for the subdivided segment (Vlachkova, 11 Jan 2026).

Trigonometric and Hyperbolic Blossoms

For $\sin t$ in $\pi_1(\cos,\sin)$: $$g((x_1,w_1),(x_2,w_2)) = \frac{1}{2}(w_1 x_2 + x_1 w_2).$$ For $\sinh t$ in $\pi_1(\cosh,\sinh)$: $$h((x_1,w_1),(x_2,w_2)) = \frac{1}{2}(w_1 x_2 + x_1 w_2).$$ The extended blossom of order $+1$ in both cases reduces to alternated sums of this basic form (Zürnacı-Yetiş, 26 Dec 2025).

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These developments establish closed-form formulae for blossom evaluation as a central tool for analytic, symbolic, and numerical studies of geometric modeling spaces, dual bases, and subdivision, supporting both theoretical investigation and practical implementation in scientific computing and design.