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Related Concept Videos

Mesh Analysis01:20

Mesh Analysis

Mesh analysis is a valuable method for simplifying circuit analysis using mesh currents as key circuit variables. Unlike nodal analysis, which focuses on determining unknown voltages, mesh analysis applies Kirchhoff's voltage law (KVL) to find unknown currents within a circuit. This method is particularly convenient in reducing the number of simultaneous equations that need to be solved.
A fundamental concept in mesh analysis is the definition of meshes and mesh currents. A mesh is a closed...
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...
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...
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...
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...
Temperature Dependent Deformation01:12

Temperature Dependent Deformation

In a nonhomogeneous rod made up of steel and brass, restrained at both ends and subjected to a temperature change, several steps are involved in calculating the stress and compressive load. Due to the problem's static indeterminacy, one end support is disconnected, allowing the rod to experience the temperature change freely. Next, an unknown force is applied at the free end, triggering deformations in the rod's steel and brass portions. These deformations are then calculated and added together...

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Related Experiment Video

Updated: Jun 24, 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

Quasi-developable mesh surface interpolation via mesh deformation.

Kai Tang1, Ming Chen

  • 1Hong Kong University of Science and Technology, Hong Kong. mektang@ust.hk

IEEE Transactions on Visualization and Computer Graphics
|March 14, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces a novel algorithm for creating developable mesh surfaces that accurately interpolate given points and curves. The method deforms an initial mesh while preserving developability, offering a promising tool for surface interpolation.

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Area of Science:

  • Computer Graphics
  • Computational Geometry
  • Geometric Modeling

Background:

  • Interpolating arbitrary point sets and space curves with smooth surfaces is a fundamental problem in geometric modeling.
  • Existing methods often struggle to maintain developability, a crucial property for manufacturing and design applications.
  • Mesh editing techniques, particularly those preserving Laplacian coordinates, offer new avenues for shape manipulation.

Purpose of the Study:

  • To develop a new algorithm for finding a most developable smooth mesh surface that interpolates given arbitrary points or space curves.
  • To formulate surface interpolation as a mesh deformation process that maintains developability throughout the transformation.

Main Methods:

  • The algorithm formulates interpolation as a mesh deformation, transforming an initial developable mesh to one that interpolates target geometry.
  • Developability is maintained by preserving zero Gaussian curvature on the mesh during deformation.
  • Nonlinear geometric constraints are linearized using Taylor expansion, forming a sparse, over-determined linear system solved via least-squares, with iterative refinement.

Main Results:

  • The iterative procedure smoothly deforms the initial mesh to interpolate the specified points and/or curves.
  • Initial experiments demonstrate the algorithm's effectiveness as a general quasi-developable surface interpolation tool.
  • The method successfully handles the nonlinearities associated with preserving Gaussian curvature.

Conclusions:

  • The proposed algorithm provides a robust method for generating developable mesh surfaces for interpolation.
  • It offers a promising approach for applications requiring smooth, developable surfaces that conform to arbitrary input data.
  • The technique advances mesh deformation and interpolation by incorporating developability constraints.