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Related Concept Videos

Curve Sketching and Derivatives01:22

Curve Sketching and Derivatives

Understanding the behavior of a function through its first and second derivatives is essential for analyzing its graph. Derivatives provide insight into where a function increases or decreases, where it attains local maxima or minima, and how its curvature behaves across different intervals.The first derivative of a function reveals the slope of the tangent line at any given point. Points where the derivative is zero or undefined are considered critical, as they often indicate potential extrema...
Elevation of Intermediate Points on Vertical Curves01:20

Elevation of Intermediate Points on Vertical Curves

Vertical curves are essential in roadway design because they provide smooth transitions between varying roadway grades. Designing vertical curves involves calculating intermediate elevations and identifying the curve's highest or lowest point, which is essential for optimal roadway performance.Intermediate elevations on a vertical curve are determined using the tangent offset method. This method considers the initial elevation at the start of the curve, the grades, and the curve's geometry. The...
Curvilinear Motion: Normal and Tangential Components01:27

Curvilinear Motion: Normal and Tangential Components

When a car traverses a curved road, its motion can be elucidated by breaking it down into tangential and normal components. The car-centric coordinates attached to the vehicle move with it.
The positive direction of the t-axis aligns with the increasing position of the car along the curved path, denoted by the unit vector ut. Simultaneously, the n-axis, perpendicular to the t-axis, dissects the curved path into differential arc segments, each forming the arc of a circle with a radius of...
Curvilinear Motion: Rectangular Components01:23

Curvilinear Motion: Rectangular Components

Curvilinear motion characterizes the movement of a particle or object along a curved path, notably evident when envisioning a car navigating a winding road. If the car starts at point A, its position vector is established within a fixed frame of reference, where the ratio of the position vector to its magnitude signifies the unit vector pointing in the position vector's direction.
As the car advances, its position evolves over time. Quantifying the car's velocity involves computing the time...
Step-Growth Polymerization: Overview01:03

Step-Growth Polymerization: Overview

Step-growth or condensation polymerization is a stepwise reaction of bi or multifunctional monomers to form long-chain polymers. As all the monomers are reactive, most of the monomers are consumed at the early stages of the reaction to form small chains of reactive oligomers, which then combine to form long polymer chains in the late stages. Hence, the reaction has to proceed for a long time to achieve high molecular weight polymers.
Many natural and synthetic polymers are produced by...
Introduction to Vertical Curves01:24

Introduction to Vertical Curves

Vertical curves are parabolic transitions that connect different grades on highways and railroads, ensuring a smooth alignment between back and forward tangents. The back tangent represents the initial grade, while the forward tangent defines the subsequent grade. These curves can be symmetrical, with equal tangent lengths, or nonsymmetrical, with varying lengths. The key points defining a vertical curve include the Point of Vertical Intersection (P.V.I.), where the tangents meet; the Point of...

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Related Experiment Video

Updated: May 16, 2026

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics
14:14

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics

Published on: April 16, 2017

Surface growth kinematics via local curve evolution.

Derek E Moulton1, Alain Goriely

  • 1OCCAM, Mathematical Institute, University of Oxford, Oxford, UK, moulton@maths.ox.ac.uk.

Journal of Mathematical Biology
|November 20, 2012
PubMed
Summary

This study introduces a mathematical framework to model surface growth, explaining how simple parameters can generate complex biological structures like seashells and horns. The model links geometric kinematics to the underlying biological growth processes.

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Localizing Protein in 3D Neural Stem Cell Culture: a Hybrid Visualization Methodology
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Localizing Protein in 3D Neural Stem Cell Culture: a Hybrid Visualization Methodology

Published on: December 19, 2010

Related Experiment Videos

Last Updated: May 16, 2026

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics
14:14

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics

Published on: April 16, 2017

Localizing Protein in 3D Neural Stem Cell Culture: a Hybrid Visualization Methodology
21:47

Localizing Protein in 3D Neural Stem Cell Culture: a Hybrid Visualization Methodology

Published on: December 19, 2010

Area of Science:

  • Mathematical Biology
  • Developmental Biology
  • Geometric Modeling

Background:

  • Modeling biological surface growth is complex.
  • Existing models often lack a direct link to underlying biological processes.
  • Understanding the kinematics of evolving shapes is crucial for developmental biology.

Purpose of the Study:

  • To develop a mathematical framework for modeling surface growth of objects generated by evolving curves.
  • To establish a local, elegant mathematical structure for growth kinematics.
  • To connect geometric growth models with biological processes.

Main Methods:

  • Developed a mathematical framework based on a growth velocity vector field defined on a generating curve.
  • Utilized a local orthonormal basis attached to each point of the generating curve.
  • Derived analytical solutions for the kinematics equations.

Main Results:

  • Demonstrated the emergence of biologically relevant structures, such as logarithmic shells and horns, as analytical solutions.
  • Showcased how a small number of parameters in the model can represent the underlying growth process.
  • Established a direct link between geometric kinematics and biological growth through local orientation and cell tracks.

Conclusions:

  • The developed mathematical framework provides an elegant and local model for surface growth kinematics.
  • The model successfully reproduces complex biological structures from simple parameters, offering insights into developmental mechanisms.
  • This approach facilitates connections between geometric modeling and the study of biological growth processes.