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Related Concept Videos

Polar and Cylindrical Coordinates01:22

Polar and Cylindrical Coordinates

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The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
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Curvilinear Motion: Polar Coordinates01:27

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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.
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The polar coordinate system offers an alternative to the Cartesian coordinate system for specifying points in a plane, using a distance and an angle instead of x and y coordinates. This system is particularly advantageous in situations involving circular or rotational symmetry, such as in physics or engineering problems involving waves, oscillations, or orbital paths.Defining Polar CoordinatesIn polar coordinates, a point is represented as P(r, ��), where r is the radial distance...
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Graphs of Polar Equations01:17

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The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
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Polar Equations of Conics01:29

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A conic section can be defined in polar coordinates as the set of all points whose distance from a fixed point, known as the focus, bears a constant ratio to their distance from a fixed line, known as the directrix. This constant ratio is called the eccentricity. This definition unifies all types of conic sections—ellipses, parabolas, and hyperbolas—under a single framework. When the focus is positioned at the origin of the polar coordinate system, a single polar equation can...
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Consider a crane whose telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2. Along with the rotation, the boom also extends linearly with a uniform speed of 5 m/s. The extension of the boom is measured at point D, which is measured with respect to the fixed point C on the other end of the boom. For the given instant, the distance between points C and D is 60 meters.
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Dynamic Programming Using Polar Variance for Image Segmentation.

Jose A Rosado-Toro, Maria I Altbach, Jeffrey J Rodriguez

    IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
    |October 11, 2016
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    Summary
    This summary is machine-generated.

    This study introduces polar variance for image segmentation, improving object size analysis without training data. The new method enhances segmentation accuracy for complex shapes and various image conditions.

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    Area of Science:

    • Computer Vision
    • Image Processing
    • Computational Imaging

    Background:

    • Object size is a key feature in polar dynamic programming (PDP) for image segmentation.
    • Unconstrained object size can lead to segmenting extraneous high-gradient regions.
    • Existing methods may require training data or struggle with complex shapes.

    Purpose of the Study:

    • To introduce a novel feature, polar variance, for size-unconstrained image segmentation using PDP.
    • To develop a technique for segmenting complex shapes by identifying and expanding low-gradient regions.
    • To evaluate the proposed method against established active contour segmentation techniques.

    Main Methods:

    • Defined polar variance as the variance within a polar region relative to a user-defined origin.
    • Integrated a novel approach to segment complex shapes by growing low-gradient areas.
    • Conducted comparative experimental analysis against active contour methods.

    Main Results:

    • The proposed polar variance method enables segmentation of objects with varying sizes without prior training.
    • The technique effectively segments complex shapes by leveraging low-gradient region expansion.
    • Experimental results demonstrate favorable performance compared to other active contour segmentation techniques.

    Conclusions:

    • Polar variance offers a robust and versatile feature for PDP-based image segmentation.
    • The combined approach enhances segmentation accuracy and robustness across different image types and noise levels.
    • This method provides a valuable alternative for segmenting objects of diverse sizes and complex geometries.