Geometric Modeling of Bézier and Spline Curves

In geometric modelling, Bézier and spline curves serve as fundamental tools for the representation and manipulation of smooth shapes. Bézier curves, defined by a set of control points and Bernstein polynomial basis functions, provide intuitive shape control and are widely used in computer graphics and industrial design. Spline curves—including B-splines and non-uniform rational B-splines (NURBS)—extend this framework by piecing together polynomial segments with prescribed continuity, enabling accurate modelling of free-form geometries. Recent innovations introduce additional shape parameters and alternative basis functions—trigonometric, hybrid, fractional or blended—to enhance local adjustability, maintain higher-order smoothness across segments and improve computational efficiency. The careful management of parametric continuity (Cⁿ) and geometric continuity (Gⁿ) ensures visual and curvature smoothness, critical for applications from automotive and aerospace design to animation and digital fabrication.

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Geometric Modeling of Bézier and Spline Curves publication trend

The graph below shows the total number of articles in geometric modeling of bézier and spline curves across all publications each year (not limited to Nature Index journals).

Technical terms

Bezier curve: A parametric curve defined by control points and Bernstein basis functions, offering global and local shape control.

B-spline curve: A piecewise polynomial curve constructed from control points and basis functions, allowing local modification via knot placement.

Control point: A coordinate that influences the form and trajectory of a parametric curve or surface.

Parametric continuity (Cn): Smoothness measured by equality of derivatives up to order n at segment boundaries.

Geometric continuity (Gn): Smoothness criterion based on alignment of tangents or curvature, independent of parameterisation speed.

Shape parameter: An extra variable within basis functions that adjusts curve or surface shape without altering control points.

Basis function: A mathematical function used to blend control points into the final curve or surface geometry.

References

1. Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications. Mathematics (2020).
2. Geometric modeling and applications of generalized blended trigonometric Bézier curves with shape parameters. Advances in Continuous and Discrete Models (2020).
3. Generalized Fractional Bézier Curve with Shape Parameters. Mathematics (2021).
4. Geometric Modeling Using New Cubic Trigonometric B-Spline Functions with Shape Parameter. Mathematics (2020).