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Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates01:21

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Understanding the motion of particles is a fundamental aspect of classical mechanics, and the choice of the coordinate system plays a pivotal role in unraveling the complexities of their dynamics.
When a particle moves relative to an inertial frame, the equations of motion can be expressed using rectangular components. If the motion is confined to the x-y plane, the equations having the x and y coordinates only can be used to simplify the mathematical representation.
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Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to another reference frame. The concept of relative velocity can be used to describe motion in two dimensions. Consider a particle P and two reference frames S and S′. The position of the origin of S′ as measured in S is , the position of P as measured in S′ is , and the position of P as measured in S is , which can be evaluated by...
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In mechanical engineering, a three-dimensional force system is a system of forces acting in three dimensions, with forces applied along the x, y, and z coordinate axes. The three-dimensional force system is an important concept in mechanical engineering, as it allows engineers to understand and analyze the behavior of objects and structures in three dimensions. By understanding the forces acting on a system, engineers can design more efficient and effective mechanical systems that can withstand...
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The understanding of the concept of reference frames is essential to discuss relative motion in one or more dimensions. When we say that an object has a certain velocity, we must state the velocity with respect to a given reference frame. In most examples, this reference frame has been Earth. For instance, if a statement reads that a person is sitting in a train moving at 10 m/s east, then it implies that the person on the train is moving relative to the surface of Earth at this velocity,...
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Updated: Jul 9, 2025

Three-dimensional Particle Tracking Velocimetry for Turbulence Applications: Case of a Jet Flow
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u-track3D: Measuring, navigating, and validating dense particle trajectories in three dimensions.

Philippe Roudot1, Wesley R Legant2, Qiongjing Zou3

  • 1Lyda Hill Department of Bioinformatics, UT Southwestern Medical Center, Dallas, TX, USA; Aix Marseille University, CNRS, Centrale Marseille, I2M, Turing Centre for Living Systems, Marseille, France.

Cell Reports Methods
|December 2, 2023
PubMed
Summary

u-track3D software enhances 3D particle tracking for intracellular dynamics. It offers competitive performance, dynamic region of interest visualization, and automated trajectory scoring for unbiased molecular studies.

Keywords:
3D visualizationCP: Imagingcell biologyconfocal microscopyerror inferenceintracellular dynamicslight-sheet microscopymultiple particle trackingparticle tracking challenge

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Area of Science:

  • Biophysics
  • Cell Biology
  • Image Analysis

Background:

  • Accurate 3D particle tracking is crucial for understanding intracellular dynamics.
  • Existing 2D tracking frameworks require adaptation for 3D challenges.
  • Manual validation of 3D trajectories is time-consuming and subjective.

Purpose of the Study:

  • Introduce u-track3D, a software package for 3D particle tracking.
  • Address challenges specific to analyzing volumetric image sequences.
  • Provide a framework for unbiased studies of molecular processes.

Main Methods:

  • Extension of the u-track framework to 3D.
  • Quantification of intracellular dynamics using multiple 3D microscopy platforms.
  • Implementation of dynamic region of interest (dynROI) for interactive 3D visualization.
  • Development of an automated trackability estimator for trajectory validation.

Main Results:

  • u-track3D demonstrates competitive performance against conventional and deep-learning methods.
  • The software successfully quantifies intracellular dynamics on standard 3D datasets.
  • dynROI enables intuitive interaction with 3D processes in 2D views.
  • The trackability estimator automates trajectory scoring, reducing manual inspection.

Conclusions:

  • u-track3D offers a robust and efficient solution for 3D particle tracking.
  • The integrated tools facilitate unbiased analysis of complex volumetric data.
  • This framework supports advanced studies of molecular dynamics in live cells.