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Mathematics
Volume 14
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10.3390/math14010012
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

On Some Novel Uses of Strain Tensors Beyond Visualization in Modern Shape Analysis

by
Paolo Piras
1,*,
Antonio Profico
2,
Franco Milicchio
3,
Luciano Teresi
4,
Stefano Gabriele
5 and
Valerio Varano
5,*
1
CPIA1, Via Carlo Alberto Cortina 70, 00159 Roma, Italy
2
Dipartimento di Biologia, Università di Pisa, Via Luca Ghini 13, 56126 Pisa, Italy
3
Dipartimento di Ingegneria Civile, Informatica e delle Tecnologie Aeronautiche, Università Roma Tre, Via della Vasca Navale 79/81, 00146 Roma, Italy
4
Dipartimento di Ingegneria Industriale, Elettronica e Meccanica, Università Roma Tre, Via Vito Volterra 62, 00146 Roma, Italy
5
Dipartimento di Architettura, Università Roma Tre, Largo Giovanni Battista Marzi 10, 00153 Roma, Italy
*
Authors to whom correspondence should be addressed.
Mathematics 2026, 14(1), 12; https://doi.org/10.3390/math14010012 (registering DOI)
Submission received: 24 October 2025 / Revised: 9 December 2025 / Accepted: 16 December 2025 / Published: 20 December 2025
Versions Notes

Abstract

Modern Shape Analysis integrates mathematical and statistical methods to study morphology using geometric data from 2D and 3D digitization. Beyond visualization, local deformation analysis offers deeper insight into longitudinal changes like ontogenetic growth. This study explores strain tensors as analytical tools for shape transformations, evaluating Thin Plate Spline, Quadratic Trend, and Cubic Trend interpolants in estimating deformation tensors at landmarks. Simulated 2D data and 3D hominid skulls confirm that tensor-based multivariate analyses, such as Principal Component Analysis, yield results comparable to Parallel Transport methods. Thin Plate Spline more accurately recovers assigned tensors than least square interpolants, making it preferable for local deformation analysis. This method integrates traditional shape comparisons, proving particularly useful for studying ontogenetic trajectories and biomechanical deformations.
Keywords: strain tensor; local deformation; Parallel Transport; interpolant; first derivative; Principal Component Analysis; deformation trajectories strain tensor; local deformation; Parallel Transport; interpolant; first derivative; Principal Component Analysis; deformation trajectories

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MDPI and ACS Style

Piras, P.; Profico, A.; Milicchio, F.; Teresi, L.; Gabriele, S.; Varano, V. On Some Novel Uses of Strain Tensors Beyond Visualization in Modern Shape Analysis. Mathematics 2026, 14, 12. https://doi.org/10.3390/math14010012

AMA Style

Piras P, Profico A, Milicchio F, Teresi L, Gabriele S, Varano V. On Some Novel Uses of Strain Tensors Beyond Visualization in Modern Shape Analysis. Mathematics. 2026; 14(1):12. https://doi.org/10.3390/math14010012

Chicago/Turabian Style

Piras, Paolo, Antonio Profico, Franco Milicchio, Luciano Teresi, Stefano Gabriele, and Valerio Varano. 2026. "On Some Novel Uses of Strain Tensors Beyond Visualization in Modern Shape Analysis" Mathematics 14, no. 1: 12. https://doi.org/10.3390/math14010012

APA Style

Piras, P., Profico, A., Milicchio, F., Teresi, L., Gabriele, S., & Varano, V. (2026). On Some Novel Uses of Strain Tensors Beyond Visualization in Modern Shape Analysis. Mathematics, 14(1), 12. https://doi.org/10.3390/math14010012

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