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Related Concept Videos

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Curve Equations

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Curves are essential geometric elements characterized by tangent distance, chord length, middle ordinate, and total arc length. These measurements are crucial in understanding a curve's geometric and spatial properties and are defined by the relationship between its radius and its central angle.The tangent distance (T) refers to the straight-line measurement from the intersection point of two tangents to either the start or end of the curve. This distance is influenced by the curve's radius (R)...
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Elevation of Intermediate Points on Vertical Curves01:20

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Vertical curves are essential in roadway design because they provide smooth transitions between varying roadway grades. Designing vertical curves involves calculating intermediate elevations and identifying the curve's highest or lowest point, which is essential for optimal roadway performance.Intermediate elevations on a vertical curve are determined using the tangent offset method. This method considers the initial elevation at the start of the curve, the grades, and the curve's geometry. The...
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Vertical curves are parabolic transitions that connect different grades on highways and railroads, ensuring a smooth alignment between back and forward tangents. The back tangent represents the initial grade, while the forward tangent defines the subsequent grade. These curves can be symmetrical, with equal tangent lengths, or nonsymmetrical, with varying lengths. The key points defining a vertical curve include the Point of Vertical Intersection (P.V.I.), where the tangents meet; the Point of...
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Curvilinear Motion: Rectangular Components01:23

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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.
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A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
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Updated: Oct 7, 2025

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics
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Streamlined variational inference for higher level group-specific curve models.

M Menictas1, T H Nolan1,2, D G Simpson3

  • 1School of Mathematical and Physical Sciences, University of Technology Sydney, Australia.

Statistical Modelling
|January 10, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces streamlined variational inference for complex group-specific curve models. The methods advance analysis of nested data, particularly in ultrasound technology applications.

Keywords:
approximate Bayesian inferencelongitudinal data analysismean field variational Bayesmultilevel modelspanel data

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Area of Science:

  • Statistics
  • Machine Learning

Background:

  • Group-specific curve models analyze individual smooth functions within groups.
  • Extending these models to multiple nested levels presents computational challenges.

Purpose of the Study:

  • To develop streamlined variational inference methods for higher-level group-specific curve models.
  • To address the computational complexity of analyzing nested data structures.

Main Methods:

  • Systematic development of variational inference for two-level and three-level group-specific curve models.
  • Leveraging sparse matrix infrastructure for computational efficiency.

Main Results:

  • Demonstrated a viable approach for streamlined variational inference in hierarchical models.
  • The methods are motivated by and applicable to ultrasound data analysis.

Conclusions:

  • The systematic approach provides a foundation for higher-level group-specific curve models.
  • The developed methods enhance the analysis of complex, nested data structures.