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Related Concept Videos

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...
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...
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...
Interpretations of Partial Derivatives01:14

Interpretations of Partial Derivatives

A surface defined by a function of two variables can be visualized as a vast, uneven terrain, where each point is identified using Cartesian coordinates. The elevation of the terrain at any point is determined by a function that assigns a height value to every pair of horizontal coordinates. This representation allows the surface to be studied in terms of how its height varies across different directions.At a specific point on this terrain, understanding how the height changes requires...
Calculus with Parametric Curves: Surface Areas01:30

Calculus with Parametric Curves: Surface Areas

A parametric curve is a description of a path in the plane where both the x and y coordinates are functions of a single parameter, typically denoted t. When such a curve is revolved about an external axis lying in the same plane, it generates a surface of revolution in three dimensions. The surface area of this rotated shape depends fundamentally on two aspects: the geometry of the original curve and how far it lies from the chosen axis of rotation.A torus is a classical surface of revolution...
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Curves Defined by Parametric Equations

A baseball hit into the air follows a parabolic trajectory when air resistance is neglected. The motion can be described within a two-dimensional coordinate system, where both the horizontal displacement and vertical height are functions of time. Instead of expressing the trajectory as a single function of position, the motion is modeled using parametric equations: one function for the horizontal position and another for the vertical position as time progresses. Let the horizontal position be...

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Related Experiment Video

Updated: Jun 13, 2026

How to Measure Cortical Folding from MR Images: a Step-by-Step Tutorial to Compute Local Gyrification Index
09:57

How to Measure Cortical Folding from MR Images: a Step-by-Step Tutorial to Compute Local Gyrification Index

Published on: January 2, 2012

Parametric representation of cortical surface folding based on polynomials.

Tuo Zhang1, Lei Guo, Gang Li

  • 1School of Automation, Northwestern Polytechnical University, Xi'an, China.

Medical Image Computing and Computer-Assisted Intervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention
|April 30, 2010
PubMed
Summary

This study introduces a polynomial-based parametric representation for human cerebral cortex surface patches. This method effectively describes cortical folding patterns and enables classification for improved surface segmentation.

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Last Updated: Jun 13, 2026

How to Measure Cortical Folding from MR Images: a Step-by-Step Tutorial to Compute Local Gyrification Index
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Published on: January 22, 2018

Area of Science:

  • Neuroscience
  • Computational Geometry
  • Medical Imaging

Background:

  • Describing the geometrical complexity of the human cerebral cortex is crucial for understanding brain structure and function.
  • Existing methods for analyzing cortical folding patterns often lack compactness or sufficient shape information.

Purpose of the Study:

  • To present a novel parametric representation of cortical surface patches using polynomials.
  • To enable classification of cortical folding patterns through a model-driven or data-driven approach.
  • To apply this representation for effective segmentation of the cortical surface.

Main Methods:

  • Developed a parametric representation using four coefficients to describe primitive cortical surface patches.
  • Classified cortical patches into eight primitive shape patterns (peak, pit, ridge, valley, saddle ridge, saddle valley, flat, inflection) using a model-driven approach based on parameter sub-spaces.
  • Utilized a data-driven clustering approach for pattern classification.
  • Applied the polynomial representation for cortical surface segmentation.

Main Results:

  • Achieved a compact and effective parametric description of cortical folding patterns.
  • Demonstrated the ability to classify cortical patches into predefined shape categories.
  • Obtained promising results in cortical surface segmentation using the proposed method.

Conclusions:

  • The polynomial-based parametric representation offers a compact, effective, and information-rich method for describing cortical folding.
  • This approach facilitates the classification and segmentation of the human cerebral cortex.
  • The findings suggest potential for advanced analysis of brain morphology and function.