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Related Concept Videos

Finding Volume Using Cross-Sectional Area01:24

Finding Volume Using Cross-Sectional Area

For solids whose cross-sectional areas vary in a predictable way, volume can be determined by integrating these areas along an axis perpendicular to the slices. This approach is particularly useful for polyhedral solids, where classical geometric formulas may not be immediately applicable. A tetrahedron provides a clear example of how cross-sectional integration can be applied to a three-dimensional object with continuously changing geometry.Consider a tetrahedron with height h and a base that...
Work Done During Volume Change01:17

Work Done During Volume Change

In mechanics, work is done on an object when the force acting on it displaces the object. In thermodynamics, work done on a system can be estimated when the system's volume changes during any thermodynamic process.
Consider a gas confined to a cylinder fitted with a movable piston at one end. If the gas expands from volume V1 to volume V2, it exerts a force on the piston, such that the piston moves by a distance dr.
The work done by the gas on the piston can be expressed as
Unsoundness of Aggregate due to Volume Change01:26

Unsoundness of Aggregate due to Volume Change

Unsoundness in aggregates due to volume changes is primarily caused by the physical alterations aggregates undergo, such as freezing and thawing, thermal changes, and wetting and drying. Unsound aggregates, when subjected to these changes, result in volume change upon disintegration. This, in turn, contributes to the deterioration of concrete, including scaling, pop-outs, and cracking. Particular types of aggregates, such as porous flints, cherts, and those containing clay minerals, are...
Volumes of Solids of Revolution01:29

Volumes of Solids of Revolution

Volumes of irregularly shaped objects can be systematically determined using the concept of solids of revolution. This approach begins with a region defined by a curve in a two-dimensional plane. When this region is rotated about a fixed line, known as the axis of revolution, it generates a three-dimensional object with rotational symmetry. Such objects frequently arise in mathematical modeling, physics, and engineering applications.When the region being rotated lies directly against the axis...
Calculation of Volume of Solids by Integration01:27

Calculation of Volume of Solids by Integration

Volume calculation often begins with simple geometric solids. For example, the volume of a rectangular box is obtained by multiplying the area of its base by its height. This straightforward approach relies on the fact that the cross-sectional area of the box remains constant throughout its length. Many real-world objects, however, do not have uniform cross-sections, and their volumes cannot be determined using elementary geometric formulas.To address this limitation, the Slicing Method...
Area Computation by the Alternative Coordinate Method01:24

Area Computation by the Alternative Coordinate Method

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...

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Quantitative Assessment Protocol for Facial Soft Tissue Volumetric Changes with Stereophotogrammetry
06:26

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Published on: December 9, 2025

Projection-based volume alignment.

Lingbo Yu1, Robert R Snapp, Teresa Ruiz

  • 1University of Vermont, Department of Computer Science, Burlington, VT 05405, USA.

Journal of Structural Biology
|February 16, 2013
PubMed
Summary
This summary is machine-generated.

A new projection-based volume alignment (PBVA) algorithm accurately aligns 3D electron microscopy (3DEM) reconstructions, outperforming traditional methods for improved macromolecular assembly analysis.

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Area of Science:

  • Structural Biology
  • Biophysics
  • Microscopy

Background:

  • Heterogeneous samples in 3D electron microscopy (3DEM) yield multiple reconstructions.
  • Volume alignment is crucial for 3D classification and averaging to achieve higher resolution.
  • Existing alignment methods can be computationally intensive.

Purpose of the Study:

  • To introduce a novel projection-based volume alignment (PBVA) algorithm.
  • To enhance the accuracy and efficiency of volume alignment in 3DEM.
  • To enable more robust 3D classification and averaging of macromolecular assemblies.

Main Methods:

  • Developed a projection-based volume alignment (PBVA) algorithm.
  • Utilized projection alignment by maximizing cross-correlation functions for rotation and translation.
  • Implemented normalization for missing data and averaging/interpolation for accurate alignment.
  • Compared PBVA computation time against traditional 3D cross-correlation methods.

Main Results:

  • PBVA significantly outperforms traditional 3D cross-correlation methods in computation time.
  • The algorithm demonstrates robustness and high accuracy across various signal-to-noise ratios and missing data percentages.
  • PBVA was successfully applied to align reconstructions of a yeast Complex I subcomplex.

Conclusions:

  • PBVA is an accurate, robust, and computationally efficient algorithm for volume alignment in 3DEM.
  • The method facilitates improved 3D classification and averaging of macromolecular assemblies.
  • This advancement aids in the structural determination of complex biological molecules.