A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force

Zamora Cisneros, David Uriel; Harrington, Matthew J.; Dorval Courchesne, Noémie-Manuelle; Rey, Alejandro D.

doi:10.3390/sym18061022

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A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force

by

David Uriel Zamora Cisneros

¹

,

Matthew J. Harrington

²

,

Noémie-Manuelle Dorval Courchesne

¹

and

Alejandro D. Rey

^1,*

¹

Department of Chemical Engineering, McGill University, 3610 University Street, Montreal, QC H3A 0C5, Canada

²

Department of Chemistry, McGill University, 801 Sherbrooke Street West, Montreal, QC H3A 0B8, Canada

^*

Author to whom correspondence should be addressed.

Symmetry 2026, 18(6), 1022; https://doi.org/10.3390/sym18061022 (registering DOI)

Submission received: 26 April 2026 / Revised: 9 June 2026 / Accepted: 9 June 2026 / Published: 13 June 2026

(This article belongs to the Special Issue Mathematics: Feature Papers 2026)

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Abstract

This paper develops, implements, and uses a novel methodology that integrates global and local geometric modeling of ellipsoids and transforms curvatures and shape descriptors into energy metrics that provide curvature-driven mechanistic insight into self-assembly driving force pathways associated with fiber and film formation. Global parameterization, based on eccentricities, and local parameterization, based on mean and Gaussian curvatures, are mapped into shape and curvedness parameterizations that clearly distinguish critical shape effects from curvedness effects in contrast to classical mean and Gaussian curvature methodologies. Finally, the mechanistic insights are illustrated using a liquid membranology model, transforming the geometric descriptors into bending energy densities that point the way to fiber and film assembly pathways.

Keywords: biaxiality; triaxiality; spheroids; curvature; shape coefficient; umbilic points; lines of curvature

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MDPI and ACS Style

Zamora Cisneros, D.U.; Harrington, M.J.; Dorval Courchesne, N.-M.; Rey, A.D. A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force. Symmetry 2026, 18, 1022. https://doi.org/10.3390/sym18061022

AMA Style

Zamora Cisneros DU, Harrington MJ, Dorval Courchesne N-M, Rey AD. A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force. Symmetry. 2026; 18(6):1022. https://doi.org/10.3390/sym18061022

Chicago/Turabian Style

Zamora Cisneros, David Uriel, Matthew J. Harrington, Noémie-Manuelle Dorval Courchesne, and Alejandro D. Rey. 2026. "A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force" Symmetry 18, no. 6: 1022. https://doi.org/10.3390/sym18061022

APA Style

Zamora Cisneros, D. U., Harrington, M. J., Dorval Courchesne, N.-M., & Rey, A. D. (2026). A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force. Symmetry, 18(6), 1022. https://doi.org/10.3390/sym18061022

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A Novel Differential Geometry Methodology for Curvature and Shape Characterization of Ellipsoids: Shape Transition Symmetry Breaking and Mechanistic Insights into Self-Assembly Curvature Driving Force

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