Polar Equation of A Curve
The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of θ. The resulting curve then consists of points of the form (r(θ), θ) and can be regarded as the graph of the polar function r.
Different forms of symmetry can be deduced from the equation of a polar function r. If r(−θ) = r(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π − θ) = r(θ) it will be symmetric about the vertical (90°/270°) ray, and if r(θ − α) = r(θ) it will be rotationally symmetric α counterclockwise about the pole.
Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.
For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.
Read more about this topic: Polar Coordinate System
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