Register-Efficient Linear-Time Evaluation in the Bernstein Basis

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dc.contributor.authorValasek, Gábor
dc.contributor.authorHorváth, Anna Lili
dc.contributor.editorMasia, Belen
dc.contributor.editorThies, Justus
dc.date.accessioned2026-04-17T14:03:02Z
dc.date.available2026-04-17T14:03:02Z
dc.date.issued2026
dc.description.abstractWe investigate the evaluation of points and derivatives of Bézier curves and surfaces on modern architectures, focusing on performance and guided by numerical error bounds. While the de Casteljau algorithm remains the reference for numerical robustness, its linear working-set size imposes substantial register pressure on GPUs. We introduce a linear-time, constant-storage evaluation framework derived from the ladder algorithm that attains de Casteljau-level robustness and demonstrate that it outperforms other methods both on the GPU and CPU. Our analysis provides backward-error bounds for points and derivatives and it is also supported by empirical tests across degrees commonly used in rendering of curves and surfaces. Moreover, we show that fused multiply-add (FMA) instructions, now ubiquitous in hardware, can improve robustness even for linear interpolation. We advocate a nested FMA formulation that reconstructs endpoints exactly, in contrast to the subtraction-and-FMA pattern prevalent in shader compilers. Together, these results yield reduced memory bandwidth and register pressure, and improved performance.
dc.description.number2
dc.description.sectionheadersParametric and Structured Geometry
dc.description.seriesinformationComputer Graphics Forum
dc.description.volume45
dc.identifier.doi10.1111/cgf.70403
dc.identifier.issn1467-8659
dc.identifier.pages14 pages
dc.identifier.urihttps://diglib.eg.org/handle/10.1111/cgf70403
dc.identifier.urihttps://doi.org/10.1111/cgf.70403
dc.publisherThe Eurographics Association and John Wiley & Sons Ltd.
dc.rightsCC-BY-4.0
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.subjectMathematics of computing → Computations on polynomials
dc.subjectComputing methodologies → Parametric curve and surface models
dc.titleRegister-Efficient Linear-Time Evaluation in the Bernstein Basis
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