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Related Concept Videos

Tangent to a Curve01:30

Tangent to a Curve

The graph of a function where each output is the square of the input creates a smooth curve that bends upward, becoming steeper as one moves further from the center. At any chosen position along this curve, the curve reaches a certain height depending on the input value. This position can be a reference for analyzing how the curve behaves in its immediate vicinity.To understand the change in the curve near a particular position, imagine selecting another point slightly ahead along the curve.
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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...
Introduction to Vertical Curves01:24

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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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Understanding the behavior of a function through its first and second derivatives is essential for analyzing its graph. Derivatives provide insight into where a function increases or decreases, where it attains local maxima or minima, and how its curvature behaves across different intervals.The first derivative of a function reveals the slope of the tangent line at any given point. Points where the derivative is zero or undefined are considered critical, as they often indicate potential extrema...
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The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as the volume tends to zero. More practically, divergence measures how much a vector field spreads out or diverges from a given point. For an outgoing flux, conventionally, the divergence is positive. The diverging point is often called the "source" of the field. Meanwhile, the negative divergence of a vector field at a point means that the vector...
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DTI of the Visual Pathway - White Matter Tracts and Cerebral Lesions
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Published on: August 26, 2014

A Tangent Bundle Theory for Visual Curve Completion.

Guy Ben-Yosef1, Ohad Ben-Shahar

  • 1Department of Computer Science, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel. guybeny@cs.bgu.ac.il

IEEE Transactions on Pattern Analysis and Machine Intelligence
|December 28, 2011
PubMed
Summary
This summary is machine-generated.

This study introduces a new mathematical framework for visual curve completion, modeling it as a least-action principle in the primary visual cortex (V1). This approach successfully predicts perceptual properties and reconstructs curves from incomplete visual information.

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Area of Science:

  • Computational Neuroscience
  • Computer Vision
  • Perceptual Psychology

Background:

  • Visual curve completion is essential for perceiving objects despite occlusions.
  • Previous axiomatic approaches struggle to define and verify perceptual properties.
  • Curve completion is an early visual process, suggesting V1 involvement.

Purpose of the Study:

  • To develop a novel, principle-driven model for visual curve completion.
  • To formalize curve completion using the unit tangent bundle and a least-action principle.
  • To derive perceptual properties from fundamental principles rather than imposing them.

Main Methods:

  • Formalized curve completion in the unit tangent bundle R(2) × S(1), representing V1.
  • Applied the principle of least action to find minimum-length admissible curves.
  • Developed variational methods and algorithms for curve reconstruction and analysis.

Main Results:

  • The least-action principle in R(2) × S(1) provides a theoretical basis for curve completion.
  • The model predicts numerous perceptual properties observed in psychophysical studies.
  • Demonstrated successful curve completions and validated against empirical data and existing models.

Conclusions:

  • A least-action framework offers a powerful, principle-based approach to visual curve completion.
  • This model bridges computational principles in V1 with emergent perceptual phenomena.
  • The findings advance our understanding of early visual processing and contour integration.