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Generalized Hooke's Law

Generalized Hooke's Law

The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...

Gauss's Law: Cylindrical Symmetry

Gauss's Law: Cylindrical Symmetry

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,...

Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

Gauss's Law: Spherical Symmetry

Gauss's Law: Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a uniform...

Deformations in a Symmetric Member in Bending

Deformations in a Symmetric Member in Bending

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.
When the member is segmented into tiny cubic elements, it is observed that the primary stress...

Rotation of Asymmetric Top

Rotation of Asymmetric Top

By definition, a spherically symmetric body has the same moment of inertia about any axis passing through its center of mass. This situation changes if there is no spherical symmetry. Since most rigid bodies are not spherically symmetric, these require special treatment.
The relationship between the angular momentum of any rigid body and its angular velocity, both of which are vectors, involves the moment of inertia. The moment of inertia is a scalar quantity only for spherically symmetric...

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An invariant shape representation using the anisotropic Helmholtz equation.

A A Joshi¹, S Ashrafulla, D W Shattuck

¹Signal & Image Processing Institute, Univ. of Southern California, Los Angeles, CA, USA.

Medical Image Computing and Computer-Assisted Intervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention

|January 5, 2013

Summary

This summary is machine-generated.

We developed a new method to analyze brain sulcal curves, creating a stable and unique shape representation. This technique accurately measures symmetry in cortical folding patterns between brain hemispheres.

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Last Updated: May 15, 2026

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

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Published on: July 19, 2016

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Published on: August 30, 2013

Area of Science:

Neuroscience
Computational Geometry
Medical Imaging

Background:

Analyzing the geometry of human cortical sulcal curves is crucial for understanding brain structure and function.
Existing methods often lack invariance to Euclidean motion, limiting their application in comparative studies.

Purpose of the Study:

To introduce a novel, motion-invariant shape representation for sulcal curves.
To establish a method for quantifying shape similarity and establishing correspondences between curves.
To apply this representation to analyze left-right sulcal shape symmetry in the human brain.

Main Methods:

Developed a shape representation based on the eigensystem of the anisotropic Helmholtz equation.
Implemented a finite element formulation for accurate eigensystem computation with irregularly sampled data.
Generated 26 sulcal curves from 24 subjects' brain data for analysis.

Main Results:

The novel representation is stable, unique, and invariant to scaling and isometric transformations.
A point-wise shape distance and bijective correspondence between curves were established.
Demonstrated utility in measuring left-right sulcal shape symmetry, highlighting the pattern of cortical folding.

Conclusions:

The proposed invariant shape representation is effective for analyzing the geometry of sulcal curves.
This method provides a robust tool for quantifying cortical folding patterns and inter-hemispheric symmetries.
The findings advance the field of neuroimaging and computational anatomy.