Rational-Quadratic Splines

• Rational-quadratic splines are C1-smooth, piecewise quadratic parametric functions constructed using rational B-spline segments to exactly represent conic sections.
• They ensure smooth transitions in geometric models via continuity-enforcing extraction matrices and control point constraints in CAD and isogeometric analysis.
• They support tensor-product extensions for constructing bivariate surfaces and offer adjustable shape control through weights and the parameter alpha.

Rational-quadratic splines are $C^1$ smooth, piecewise quadratic parametric functions constructed from rational quadratic B-spline or NURBS segments and assembled using continuity-enforcing extraction matrices. These splines generalize classical quadratic B-splines by allowing rational (weight-modified) Bernstein or B-spline segments, facilitating the exact representation of conic sections such as ellipses and ellipsoids. Rational-quadratic splines are foundational in geometric modeling, CAD, CAE, and isogeometric analysis, providing local support, partition of unity, and highly flexible approximation spaces (Speleers et al., 2020, Allaoui et al., 2018).

1. Definition and Construction of Rational-Quadratic Spline Bases

The construction of rational-quadratic splines is rooted in the theory of quadratic NURBS and Bernstein polynomials. On each knot span $[u_i, u_{i+3}]$ with open knot vector $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$, the three quadratic basis functions are given as $N_{i,2}(u)$, $N_{i+1,2}(u)$, and $N_{i+2,2}(u)$. Given positive weights $w_i, w_{i+1}, w_{i+2}$, the rational basis on the segment is defined by

$$R_{i+k}(u) = \frac{w_{i+k} N_{i+k,2}(u)}{\sum_{j=0}^2 w_{i+j} N_{i+j,2}(u)}, \quad k=0,1,2.$$

Mapping the segment to $[0,1]$ reduces the $N_{i,2}(u)$ to standard quadratic Bernstein polynomials $[u_i, u_{i+3}]$0 with $[u_i, u_{i+3}]$1. Setting $[u_i, u_{i+3}]$2 recovers exact rational quadratic representation of conic sections on each piece (Speleers et al., 2020).

A distinct family of rational B-spline bases was introduced by Allalouï & Goudjo, parameterized by a shape index $[u_i, u_{i+3}]$3. They define the homographic “weights” and employ a modified Cox–de Boor recurrence for the rational-quadratic basis construction, with explicit formulas and piecewise rational-quadratic segments (Allaoui et al., 2018).

2. Continuity and Extraction Matrices

For practical usage, multiple rational-quadratic NURBS patches must be joined with $[u_i, u_{i+3}]$4 continuity across segment boundaries. The $[u_i, u_{i+3}]$5 conditions at the join parameter $[u_i, u_{i+3}]$6 between two quadratic patches comprise positional continuity—control points shared at the join—and derivative continuity, enforced by linear relations on the control points and weights:

$[u_i, u_{i+3}]$7

where $[u_i, u_{i+3}]$8 and $[u_i, u_{i+3}]$9 depend on the knot spans and weights around the join.

Global spline basis functions $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$0 are expressed as linear combinations of local NURBS bases via a sparse extraction matrix $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$1:

$\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$2

where $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$3 collects all local bases. The matrix $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$4 is constructed to ensure partition of unity, convexity, and to embed the $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$5 conditions into its structure. Such matrices are termed DTA-compatible (Design-Through-Analysis), enabling seamless integration between geometric modeling and numerical analysis (Speleers et al., 2020).

3. Exact Representation of Conics and Quadrics

By selecting appropriate weights and control point configurations, rational-quadratic splines can exactly represent ellipses, ellipsoids, and other conic sections. For instance, four quadratic NURBS segments, each with weights $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$6 and local knot vectors, together with four control points positioned at $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$7, $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$8, $\{u_i, u_i, u_i, u_{i+3}, u_{i+3}, u_{i+3}\}$9, $N_{i,2}(u)$0, yield an exact ellipse via the parametric curve:

$N_{i,2}(u)$1

on $N_{i,2}(u)$2 with periodic identification. This construction ensures that

$N_{i,2}(u)$3

Thus, piecewise rational-quadratic splines reproduce conic sections exactly with minimal polynomial degree (Speleers et al., 2020).

4. Tensor-Product Extensions and Surface Construction

The tensor-product approach extends rational-quadratic spline spaces to construct $N_{i,2}(u)$4 smooth bivariate and multivariate surfaces. The product of two univariate rational-quadratic spaces yields bivariate splines, enabling smooth surfaces with four boundary curves and compatibility with standard NURBS-based design workflows.

For representing surfaces with polar singularities (e.g., hemispherical caps, full ellipsoids), specialized extraction blocks collapse boundary B-splines into basis functions carrying Hermite data at the pole. The methodology supports constructing globally $N_{i,2}(u)$5 surfaces with control over tangent planes at singular points, exact ellipsoid reproduction with only six control points, and compatibility with arbitrary polynomial degrees on each parameter direction (Speleers et al., 2020).

5. Properties: Locality, Partition of Unity, and Shape Control

Rational-quadratic spline bases inherit locality: each basis function is nonzero only on a limited number of knot spans or subintervals. The non-negativity of basis functions and the partition of unity property are retained via normalization and matrix construction, ensuring the convex hull property of the associated curves and surfaces.

The rational B-spline family of Allalouï & Goudjo introduces an additional free parameter $N_{i,2}(u)$6 for shape control. For $N_{i,2}(u)$7 ($N_{i,2}(u)$8 concave), the basis peaks move inward; for $N_{i,2}(u)$9 ($N_{i+1,2}(u)$0 convex), the peaks shift outward. As $N_{i+1,2}(u)$1, the rational family recovers standard polynomial quadratic B-splines, ensuring classical smoothness (piecewise $N_{i+1,2}(u)$2 at simple knots, $N_{i+1,2}(u)$3 at double knots). For finite $N_{i+1,2}(u)$4, smoothness at knot points is generally only $N_{i+1,2}(u)$5, with one-sided derivatives existing except at high multiplicities (Allaoui et al., 2018).

6. Approximation and Refinement

Rational-quadratic splines constitute a basis of $N_{i+1,2}(u)$6 and support standard spline approximation properties: local quasi-interpolation and stable local projection. As the parameter $N_{i+1,2}(u)$7 is increased, approximation constants converge to the standard B-spline case, facilitating asymptotic error estimation by analogy to the polynomial theory.

Knot insertion and local degree elevation are performed analogously to standard NURBS operations, using matrix multiplication of the extraction and refinement matrices. This compatibility ensures that existing CAD and CAE systems supporting NURBS can implement rational-quadratic splines with minimal additional infrastructure (Speleers et al., 2020, Allaoui et al., 2018).

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Summary Table: Rational-Quadratic Spline Key Features

Feature	Speleers & Toshniwal (Speleers et al., 2020)	Allalouï & Goudjo (Allaoui et al., 2018)
Basis type	Rational NURBS, quadratic segments	Rational B-spline, $N_{i+1,2}(u)$8-param. family
$N_{i+1,2}(u)$9 continuity	Extraction matrix ($N_{i+2,2}(u)$0), control pt. constraints	Piecewise $N_{i+2,2}(u)$1, $N_{i+2,2}(u)$2 as $N_{i+2,2}(u)$3
Conic/ellipse/ellipsoid	Exact with $N_{i+2,2}(u)$4 weights and control geometry	Not explicitly constructed for conics
Partition of unity	Ensured by matrix normalization	Inductive proof, construction
Shape control	Local via weights and $N_{i+2,2}(u)$5 constraints	Via parameter $N_{i+2,2}(u)$6