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Related Concept Videos

Polar Coordinates01:24

Polar Coordinates

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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

Graphs of Polar Equations

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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

Polar Equations of Conics

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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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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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Polar and Cylindrical Coordinates01:22

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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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The second moment of area, also known as the moment of inertia of area, is a crucial factor in understanding an object's resistance against bending deformation, or stiffness. To accurately estimate the second moment of area along any axis, one needs to concentrate all areas associated with that object into a thin strip, which should be placed parallel to that particular axis.
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Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence
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Simple and efficient algorithm for the roundness error from polar coordinate measurement data.

Xiuming Li1, Hanbing Zhu1, Zhen Guo1

  • 1School of Mechanical Electronic and Information Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China.

The Review of Scientific Instruments
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Summary
This summary is machine-generated.

This study presents a new algorithm for finding the minimum circumscribed circle and maximum inscribed circle. The algorithm transforms Cartesian coordinate conditions into polar coordinates for efficient evaluation.

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Area of Science:

  • Computational Geometry
  • Geometric Algorithms

Background:

  • Evaluating minimum circumscribed and maximum inscribed circles is crucial in geometric analysis.
  • Transforming between Cartesian and polar coordinates is a common technique in computational geometry.

Purpose of the Study:

  • To develop and validate an efficient algorithm for determining the minimum circumscribed circle and maximum inscribed circle.
  • To adapt existing geometric evaluation methods from Cartesian to polar coordinates.

Main Methods:

  • Transformation of Cartesian coordinate conditions to polar coordinates.
  • Categorization of candidate points into three cases based on measured radii.
  • Development of specific algorithms tailored to each case for circle evaluation.

Main Results:

  • Successfully transformed the evaluation conditions for minimum circumscribed and maximum inscribed circles into polar coordinates.
  • Developed a case-based algorithmic approach for efficient point evaluation.
  • Validated the proposed algorithm's effectiveness through several examples.

Conclusions:

  • The proposed algorithm provides a valid and efficient method for evaluating minimum circumscribed and maximum inscribed circles.
  • The transformation to polar coordinates simplifies the evaluation process.
  • The case-based approach enhances the algorithm's applicability to different point distributions.