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Related Concept Videos

Spherical Coordinates01:23

Spherical Coordinates

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Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
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Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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Curvilinear Motion: Normal and Tangential Components01:27

Curvilinear Motion: Normal and Tangential Components

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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...
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Curvilinear Motion: Rectangular Components01:23

Curvilinear Motion: Rectangular Components

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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...
1.1K
Curvilinear Motion: Polar Coordinates01:27

Curvilinear Motion: Polar Coordinates

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In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
The particle's location is described using a unit vector along the radial direction. Deriving the particle's position...
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Bending of Curved Members - Neutral Surface01:16

Bending of Curved Members - Neutral Surface

488
In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the...
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Related Experiment Video

Updated: Jan 18, 2026

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

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TSSS: A Novel Triangulated Spherical Spline Smoothing for Surface-Based Data.

Zhiling Gu1,2, Shan Yu3, Guannan Wang4

  • 1Department of Biostatistics, Yale University, New Haven, CT, 06510, USA.

Journal of Nonparametric Statistics
|September 11, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a new nonparametric method for signal discovery on complex surfaces. The approach effectively handles boundary effects and sparse data, offering optimal convergence rates for surface-based data analysis.

Keywords:
62G0562G0862G20Complex domainnonparametric smoothingsphere-like surfacespherical splinestriangulation

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Area of Science:

  • Geophysics
  • Neuroimaging
  • Computational Statistics

Background:

  • Surface-based data are increasingly utilized in fields like neuroimaging and atmospheric science.
  • Analyzing data on complex surfaces presents challenges such as boundary effects and irregular distributions.
  • Existing methods often struggle with the intricacies of surface-based data analysis.

Purpose of the Study:

  • To develop a novel nonparametric method for discovering underlying signals in surface-based data.
  • To address limitations of current methods in handling complex domains, boundary effects, and sparse data.
  • To provide a statistically rigorous framework with theoretical guarantees for signal recovery on surfaces.

Main Methods:

  • A penalized spline estimator defined on a triangulation of surface patches is proposed.
  • The method is designed for enhanced computational efficiency and analysis of sparse, irregularly distributed data.
  • Theoretical guarantees, including convergence rates and asymptotic normality, are established.

Main Results:

  • The proposed method demonstrates superior handling of boundary effects on complex surfaces.
  • Optimal convergence rates are achieved within the nonparametric estimation framework.
  • A bootstrap method is introduced for uncertainty quantification and confidence interval estimation.

Conclusions:

  • The novel nonparametric method provides effective signal extraction and recovery for surface-based data.
  • The approach is robust for analyzing sparse and irregularly distributed data on complex objects.
  • Applications in neuroimaging and atmospheric data highlight the method's practical utility and advantages.