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Related Concept Videos

Tangent Planes to Surfaces01:19

Tangent Planes to Surfaces

In multivariable calculus, the concept of a tangent plane plays a central role in approximating curved surfaces. When dealing with a surface defined by a function of two variables, such as z = f(x, y), the tangent plane at a given point provides the best linear approximation to the surface near that point. This local linearization allows complex, nonlinear geometries to be treated using simpler, planar models.The construction of the tangent plane involves taking vertical slices of the surface...
Quadric Surfaces01:28

Quadric Surfaces

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...
Parametric Surfaces01:30

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...
Tangent Planes to Level Surfaces01:31

Tangent Planes to Level Surfaces

A level surface consists of all points in space where a function of three variables takes the same fixed value. If a point lies on this surface, understanding the surface’s geometry there requires more than just knowing the point’s coordinates; it requires describing how the surface is oriented, or how it tilts, near that point.To probe this local geometry, imagine tracing a path that stays entirely on the level surface and passes through the point of interest. This path can be described as a...
Tangent Planes to a Parametric Surface01:22

Tangent Planes to a Parametric Surface

A tangent plane provides a linear approximation to a curved surface at a specific point, capturing the local behavior of the surface. It can be understood as the plane that just touches the surface at that point and is defined by the tangent directions of curves lying on the surface. These tangent directions arise naturally when the surface is described parametrically, allowing systematic construction of the plane.For a surface expressed in parametric form, the position of any point is...
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Surface Area Calculations

Surface area calculations for a graph z = f(x, y) are fundamental in engineering applications involving curved structures such as satellite dishes. A parabolic dish reflects communication signals efficiently, but engineers must determine its exact curved surface area to estimate coating materials, fabrication costs, and structural requirements. Since the rim of the dish forms a circular boundary, the surface area is calculated over a circular domain in the xy-plane.Parametric Representation of...

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Related Experiment Video

Updated: Jun 18, 2026

Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data
09:37

Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data

Published on: April 26, 2016

Generating triangulated macromolecular surfaces by Euclidean Distance Transform.

Dong Xu1, Yang Zhang

  • 1Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, Michigan, United States of America.

Plos One
|December 4, 2009
PubMed
Summary
This summary is machine-generated.

We developed EDTSurf, a new algorithm for calculating macromolecular surfaces like van der Waals and solvent-accessible surfaces. This method is faster and more accurate than existing techniques, aiding protein structure studies.

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Area of Science:

  • Computational biology
  • Structural bioinformatics
  • Biophysics

Background:

  • Macromolecular surfaces define 3D shape, crucial for protein function studies.
  • Accurate surface calculation aids ligand-protein docking and virtual screening.
  • Triangulated mesh surfaces offer advantages in computer-aided manipulation and visualization.

Purpose of the Study:

  • To develop a novel algorithm, EDTSurf, for generating macromolecular surfaces.
  • To improve the accuracy and efficiency of calculating van der Waals, solvent-accessible, and molecular surfaces.
  • To provide a computationally efficient tool for protein structure analysis.

Main Methods:

  • Utilized fast Euclidean Distance Transform (EDT) for surface generation.
  • Employed a Vertex-Connected Marching Cube algorithm to construct triangulated mesh surfaces from volumetric data.
  • Compared EDTSurf's performance against analytical methods and existing literature approaches.

Main Results:

  • EDTSurf achieved relative errors of <2-4% in surface calculations, outperforming existing methods.
  • The algorithm demonstrated 1.5-4 times lower error rates compared to literature methods.
  • EDTSurf offers improved speed and reduced memory consumption.

Conclusions:

  • EDTSurf provides a highly accurate and efficient method for determining macromolecular surfaces.
  • The algorithm's performance enhancements make it a valuable tool for protein docking and structure prediction.
  • Freely available source code and executable program facilitate broader adoption and research.