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Related Concept Videos

Deformations in a Transverse Cross Section01:21

Deformations in a Transverse Cross Section

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When a material is subjected to uniaxial stress, it elongates or contracts in the direction of the applied force, and also undergoes changes in the perpendicular directions. This behavior is crucial for understanding how materials behave under stress and is governed by mechanical properties such as Poisson's ratio v, which measures the ratio of transverse strain to axial strain.
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Deformations in a Symmetric Member in Bending01:18

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When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
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Three-Dimensional Analysis of Strain01:29

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Three-dimensional strain analysis is crucial for understanding how materials deform under stress, particularly in elastic, homogeneous materials. This method employs principal stress axes to simplify complex stress states into more understandable forms. Subjected to stress, a small cubic element within a material either expands or contracts along these axes, transforming into a rectangular parallelepiped. This transformation effectively illustrates the material's deformation. The principal...
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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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Curvilinear motion characterizes the movement of a particle or object along a curved path, notably evident when envisioning a car navigating a winding road. If the car starts at point A, its position vector is established within a fixed frame of reference, where the ratio of the position vector to its magnitude signifies the unit vector pointing in the position vector's direction.
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Updated: May 1, 2026

Three-Dimensional Shape Modeling and Analysis of Brain Structures
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A hierarchical geodesic model for diffeomorphic longitudinal shape analysis.

Nikhil Singh, Jacob Hinkle, Sarang Joshi

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    We introduce a novel hierarchical geodesic model (HGM) to analyze longitudinal shape changes on manifolds. This method extends hierarchical linear models (HLMs) for non-Euclidean data, like brain scans.

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    Area of Science:

    • Computational geometry
    • Statistical modeling
    • Medical imaging analysis

    Background:

    • Hierarchical linear models (HLMs) are standard for longitudinal data analysis.
    • HLMs are limited to Euclidean data, restricting their use in complex shape analysis.

    Purpose of the Study:

    • To generalize HLMs to the manifold setting for analyzing longitudinal shape data.
    • To introduce a novel hierarchical geodesic model (HGM) for shape analysis on the group of diffeomorphisms.

    Main Methods:

    • Developed a hierarchical geodesic model (HGM) applicable to manifold-valued data.
    • Derived solutions for HGMs to estimate individual and group geodesics, and residual geodesics.
    • Applied the model to synthetic shapes and 3D MRI brain scans.

    Main Results:

    • The proposed HGM effectively explains longitudinal trends in shapes represented as diffeomorphisms.
    • Individual and population-level shape trajectories were successfully estimated.
    • Demonstrated the model's efficacy on both synthetic and real-world neuroimaging data.

    Conclusions:

    • HGMs provide a powerful framework for longitudinal analysis of non-Euclidean shape data.
    • This approach extends statistical modeling capabilities to complex manifold spaces.
    • HGMs show promise for applications in medical imaging and other fields involving shape dynamics.