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Volumes of Solids of Revolution

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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...

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Calculation of Volume of Solids by Integration

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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 of a Surface of Revolution

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Curvilinear Motion: Rectangular Components

Curvilinear Motion: Rectangular Components

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Related Experiment Video

Updated: May 10, 2026

Serial Block-Face Scanning Electron Microscopy (SBF-SEM) of Biological Tissue Samples

Serial Block-Face Scanning Electron Microscopy (SBF-SEM) of Biological Tissue Samples

Published on: March 26, 2021

Virtual volume resection using multi-resolution triangular representation of B-spline surfaces.

László Ruskó¹, Ilona Mátéka, András Kriston

¹GE Hungary Healthcare Division, Petőfi Sándor sgt. 10, Szeged 6722, Hungary. laszlo.rusko@ge.com

Computer Methods and Programs in Biomedicine

|June 4, 2013

Summary

A new tool enables physicians to easily partition segmented organs, like the liver, into lobes or segments using 3D medical image processing. This efficient method aids in clinical diagnosis and surgical planning.

Keywords:

B-spline surface Liver lobe and segment separation Multi-resolution triangulation Virtual volume resection

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Area of Science:

Medical Image Processing
Computational Anatomy
Surgical Planning

Background:

Accurate organ segmentation and manipulation are crucial for clinical diagnosis and therapy planning.
Existing medical image processing tools require efficient methods for partitioning complex 3D anatomical structures.

Purpose of the Study:

To develop an efficient and user-friendly tool for partitioning segmented organs into their constituent segments or lobes.
To enable physicians to precisely divide 3D anatomical objects based on user-defined cuts.

Main Methods:

Developed an algorithm interpolating user-defined 2D traces with B-spline surfaces to create cutting surfaces.
Computed binary cutting volumes using multi-resolution triangulation of the B-spline surface.
Integrated the algorithm into an open-source medical image processing framework for interactive partitioning.

Main Results:

The tool successfully performed free-form cuts for liver segment separation and virtual liver tumor resection.
Evaluated on 14 liver segment separation cases and 1 virtual tumor resection case.
Volume quantification of segments showed good correlation with existing methods, demonstrating clinical usability.

Conclusions:

The developed tool is efficient and user-friendly for complex organ partitioning tasks.
The method supports anatomical segment separation and virtual surgical planning.
The approach shows significant clinical potential for improving diagnostic and therapeutic procedures.