Spherical Bézier Fusion

• Spherical Bézier fusion is a technique that constructs smooth curves and patches on the unit sphere by fusing multiple Bézier segments using spherical geometry.
• It employs optimal parameter selection, radial error minimization, and SLERP-based subdivision to achieve G0 and G1 continuity in spherical patch assemblies.
• The method is crucial for mesh modeling and computational geometry, though challenges remain in attaining higher-order smoothness and handling non-uniqueness in optimal patch approximations.

Spherical Bézier Fusion refers to the construction, analysis, and optimal approximation of continuous or smooth curves and patches on the 2-sphere $S^2$ by fusing together multiple Bézier segments or patches, specifically exploiting the manifold structure of $S^2$. This domain intersecting geometric modeling and computational mathematics is characterized by the use of radial and geodesic error metrics, special interpolation algorithms adapted to the sphere, and rigorous continuity constraints to assemble global spherical splines from local Bézier elements (Vavpetič et al., 2023, Mancinelli et al., 2021).

1. Mathematical Foundations and Error Metrics

Spherical Bézier fusion fundamentally relies on extending Bézier curve and patch definitions from Euclidean spaces to the manifold structure of $S^2$. The tensor-product quadratic Bézier patch is parameterized as

$$\mathbf p(u,v) = \sum_{i,j=0}^2 B_i^2(u)\,B_j^2(v)\,\mathbf b_{ij},\quad u,v\in[-1,1]$$

where $\{\mathbf b_{ij}\}$ are control points on or near the sphere, and $B_i^2$ are the usual quadratic Bernstein polynomials.

The core metric for optimality is the (simplified) radial error: $$E_{\text{simp}}(\alpha, \beta) = \max_{(u,v)\in[-1,1]^2}|\|\mathbf p(u,v)\|^2-1|$$ This measures deviation in squared-radius between the approximating patch and the unit sphere; positive values indicate lying outside the sphere, negatives inside. For implementation, the error is minimized by optimizing over two scalar parameters $\alpha, \beta > 0$ that determine the non-corner control points, under the symmetry and boundary constraints dictated by the spherical geometry (Vavpetič et al., 2023).

2. Algorithms for Patch Optimization and Assembly

The construction of an optimal spherical-squared Bézier patch proceeds as follows:

1. Parameter Selection: Choose $a$ (half-side of projected square to the tangent plane). Corner points are fixed as $\pm(a,a,\sqrt{1-2a^2})$.
2. Determination of Control Points: The only degrees of freedom for the interior control points are $(\alpha, \beta)$:

$$\begin{aligned} &\mathbf b_{00}=(-a,-a,\sqrt{1-2a^2}),~ \mathbf b_{20}=(+a,-a,\sqrt{1-2a^2}), \ &\mathbf b_{02}=(-a,+a,\sqrt{1-2a^2}),~ \mathbf b_{22}=(+a,+a,\sqrt{1-2a^2}), \ &\mathbf b_{10}=\alpha(\mathbf b_{00}+\mathbf b_{20}),~ \mathbf b_{01}=\alpha(\mathbf b_{00}+\mathbf b_{02}),\ &\mathbf b_{21}=\alpha(\mathbf b_{20}+\mathbf b_{22}),~ \mathbf b_{12}=\alpha(\mathbf b_{02}+\mathbf b_{22}), \ &\mathbf b_{11}=\beta \cdot (0,0,1) \end{aligned}$$

1. Optimality by Equioscillation: Minimize the maximum error by enforcing the “equioscillation” condition,

$f_s(0,\alpha) = f_d(0, \alpha, \beta)$

where $f_s$ and $f_d$ are the restrictions of $f$ to side and diagonal, respectively.

2. Numerical Solution: This leads to a closed-form for $\beta$ in terms of $\alpha$, and ultimately to a nonlinear $2 \times 2$ system for the double extremum condition, which is solved numerically (e.g., by Newton-Raphson iteration with strict convergence thresholds).

For practical implementation, this process is repeated for each face-configuration needed, with symmetry exploited for sphere assemblies (Vavpetič et al., 2023).

3. Fusion, Spline Construction, and Regularity

Assemblies of Bézier patches form spherical splines of varying regularity:

• $G^0$ Continuity: Direct assemblies of optimal patches (e.g., two for a hemisphere, six for a cubic cube-like tessellation) ensure $G^0$ continuity by matching edge control points via symmetry. For instance, six optimal spherical-square patches with $a = \sqrt{3}/3$ cover the inscribed faces of a cube, providing a global $G^0$ approximation to $S^2$.
• $G^1$ Continuity: Achieving $G^1$ continuity along patch boundaries requires matching tangent planes, i.e., enforcing for all $t\in[-1,1]$:

$$\partial_v \mathbf p^{(1)}(t, -1) \parallel R\,\partial_u \mathbf p^{(2)}(-1, t)$$

with $R$ the corresponding rotational operator. This translates into linear constraints for edge control points. For six-patch spherical splines, a unique $G^1$ solution exists, given by $\alpha_G=\tfrac{3}{4}$, $\beta_G=\tfrac{7\sqrt{3}}{6}$; however, the resulting radial error ($\approx 0.0825$) is significantly worse than $G^0$-optimal patch fusion and the surface lies entirely outside the sphere (Vavpetič et al., 2023).

4. Spherical Bézier Curves: Evaluation, Subdivision, and Fusion

Adapting Bézier curve algorithms to $S^2$ requires respecting its Riemannian structure, replacing affine combinations by the geodesic "manifold average." On $S^2$, the geodesic interpolation is given closed-form as spherical linear interpolation (SLERP): $$A_{S^2}(p, q; t) = \frac{\sin((1-t) \theta) p + \sin(t \theta) q}{\sin \theta}$$ with $\theta = \arccos(\langle p, q\rangle)$ (Mancinelli et al., 2021).

Two robust subdivision schemes are employed on $S^2$ for evaluation and fusion:

• Recursive De Casteljau Bisection (RDC): Adaptively subdivides control polygons using SLERP until the polygon segments are sufficiently small, guaranteeing $C^1$ smoothness for $k\geq 2$ (Proposition 4.1).
• Open-Uniform Lane–Riesenfeld Subdivision (OLR): Produces $C^{k-1}$ curves for degree-$k$ Bézier splines by repeated mid-point SLERP averaging, matching B-spline subdivision but on the sphere (Proposition 4.2).

Fusing multiple segments into spherical composite splines, $C^1$ continuity is enforced by aligning tangent vectors at anchor points using parallel transport in the tangent plane. $C^2$ smoothness requires further matching of second derivatives, which is nontrivial and generally unsolved for general configurations (Mancinelli et al., 2021).

5. Spherical Rectangles and Non-Uniqueness

The extension from squares to spherical rectangles introduces additional degrees of freedom ($\alpha_1, \alpha_2, \beta$) in the ansatz for the patch control points. Empirical results show that, for certain aspect ratios, the minimax radial error problem admits a continuum of solutions forming a convex polygonal region in parameter space. Consequently, there can exist multiple equally-optimal Bézier patch approximants to the same spherical rectangle, which complicates the problem of fusing such patches with regularity constraints (e.g., $G^1$ or higher). Practical assembly in such cases may require auxiliary selection principles such as energy or fairness functionals (Vavpetič et al., 2023).

6. Practical Guidelines, Stability, and Implementation

Practical procedures for spherical Bézier fusion include precomputing optimal parameters for each desired face or rectangle half-angle, exploiting patch symmetry in assembly, and choosing $G^0$ or $G^1$ continuity according to application error tolerance. The adapted RDC and OLR subdivision methods on $S^2$ leverage closed-form SLERP, resulting in robust, efficient, and interactive algorithms—demonstrated on mesh models with millions of faces. Numerical stability requires avoiding nearly-antipodal control point pairs (where $\sin\theta \to 0$), for which deterministic branch selection or rejection is used.

Complexity analysis reveals that each geodesic SLERP is $O(1)$, De Casteljau evaluation is $O(k^2)$, and geometric subdivision and fusion scale favorably with the number of control points and curve resolution (Mancinelli et al., 2021).

7. Current Limitations and Open Problems

While $C^1$ and $G^1$ fusion is well-understood for both curves and patches on the sphere, achieving higher-order regularity (e.g., $C^2$ or $G^2$) at joints remains unresolved in the general setting—even for $S^2$. The non-uniqueness of optimal patches for certain configurations introduces additional ambiguity for automated fusion methods. $G^1$ fusion of patches can yield approximations with notably worse radial error compared to $G^0$, presenting a trade-off between regularity and approximation fidelity (Vavpetič et al., 2023). The theoretical and algorithmic exploration of fairness, energy-based criteria, and well-conditioned higher-order smooth fusion remains an open and active topic in the literature.

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References:

• "Optimal approximation of spherical squares by tensor product quadratic Bézier patches" (Vavpetič et al., 2023)
• "B/Surf: Interactive Bézier Splines on Surfaces" (Mancinelli et al., 2021)