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Theorem of Pappus

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The Theorem of Pappus, also known as the Pappus–Guldinus Theorem, provides a geometric method for determining the volume and surface area of solids generated by the revolution of a plane region or a plane curve about an external axis. The theorem consists of two related statements. The first addresses the volume of solids formed by rotating plane areas, while the second addresses the surface area generated by rotating plane curves. Both results depend on the location of the centroid, which...

Spherical Coordinates

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Surfaces of revolution are formed when a two-dimensional curve is rotated around an axis, producing a three-dimensional shape. This concept is used in engineering tasks like determining the surface area of a rocket nozzle, where precise calculations are critical for applying uniform heat-resistant coatings. When a curve is revolved about the x-axis, it sweeps out a continuous surface whose area must be calculated accurately to estimate material requirements.Approximating with Conical BandsTo...

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The two theorems developed by Pappus and Guldinus are widely used in mathematics, engineering, and physics to find the surface area and volume of any body of revolution. This is done by revolving a plane curve around an axis that does not intersect the curve to find its surface area or revolving a plane area around a non-intersecting axis to calculate its volume.
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Related Experiment Video

Updated: May 8, 2026

Quantitative Assessment Protocol for Facial Soft Tissue Volumetric Changes with Stereophotogrammetry

Quantitative Assessment Protocol for Facial Soft Tissue Volumetric Changes with Stereophotogrammetry

Published on: December 9, 2025

Ricci Flow-based Spherical Parameterization and Surface Registration.

X Chen¹, H He, G Zou

¹State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing, China, 100090.

Computer Vision and Image Understanding : CVIU

|September 11, 2013

Summary

This summary is machine-generated.

This study introduces an improved Euclidean Ricci flow for spherical parameterization and feature extraction, enhancing surface registration accuracy. The novel method offers superior robustness and efficiency compared to existing techniques.

Keywords:

Discrete Ricci flow surface surface parameterization

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Area of Science:

Computational geometry
Computer vision
Medical image analysis

Background:

Surface registration is crucial for comparing and analyzing 3D shapes.
Existing methods often struggle with robustness and accuracy, especially for complex surfaces.
Ricci flow offers intrinsic geometric properties beneficial for surface mapping.

Purpose of the Study:

To present an improved Euclidean Ricci flow method for spherical parameterization.
To develop a scale-space processing technique using Ricci energy for robust feature extraction.
To achieve more accurate surface registration compared to current methods.

Main Methods:

Developed an improved Euclidean Ricci flow algorithm for spherical parameterization.
Introduced a novel scale-space processing method based on Ricci energy.
Applied the method to extract robust surface features for registration.

Main Results:

The proposed Euclidean Ricci flow method enhances conformality, robustness, and intrinsicalness for surface mapping.
The Ricci energy-based scale-space processing effectively extracts robust surface features.
Demonstrated significant improvements in surface registration accuracy compared to curvature and sulci pattern methods.

Conclusions:

The improved Euclidean Ricci flow and Ricci energy-based feature extraction offer a powerful new approach for surface registration.
The method's intrinsic properties ensure efficient and effective surface mapping.
Ricci energy effectively captures local surface differences, enabling advanced surface analysis.