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Parametric Surfaces

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...

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Related Experiment Video

Updated: Jun 18, 2026

An Analytical Tool that Quantifies Cellular Morphology Changes from Three-dimensional Fluorescence Images

An Analytical Tool that Quantifies Cellular Morphology Changes from Three-dimensional Fluorescence Images

Published on: August 31, 2012

Quantification of spatial parameters in 3D cellular constructs using graph theory.

A W Lund¹, C C Bilgin, M A Hasan

¹Department of Biology, Rensselaer Polytechnic Institute, Troy, NY 12180, USA.

Journal of Biomedicine & Biotechnology

|November 19, 2009

Summary

This summary is machine-generated.

This study introduces a novel graph theory method to track 3D biological structures over time. This technique quantitatively analyzes tissue construct changes during collagen gel compaction, offering new insights into structure-function relationships.

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Area of Science:

Biotechnology
Bioengineering
Computational Biology

Background:

Traditional methods lack spatial information for dynamic 3D biological structures.
Quantifying temporal-spatial changes in biological tissues is crucial for understanding complex processes.

Purpose of the Study:

To present a novel methodology for tracking 3D biological structures over time using graph theory.
To quantitatively analyze structural changes in biological tissues during compaction events.

Main Methods:

Generated cell-graphs based on 3D Euclidean distances between nuclei during collagen I gel compaction.
Extracted quantitative features from graphs to measure global topography and local structures.
Compared feature trends to random graphs to assess significance.

Main Results:

Developed a method to track 3D biological structures and quantify changes over time.
Demonstrated that feature trends are controllable by cell density during compaction.
Showed significant differences between cell-graphs and random graphs.

Conclusions:

This work presents a novel methodology for tracking simple 3D biological events and quantitatively analyzing structural changes.
The method enables the study of complex biological problems requiring 3D temporal-spatial quantification.
Establishes a new paradigm for understanding structure-function relationships in biological systems.