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The equation defining a plane 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 φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
Construction of the limaçon r = 2 + cos(π – θ) with polar coordinates' origin at (x, y) = ( 1 / 2 , 0). In geometry, a limaçon or limacon / ˈ l ɪ m ə s ɒ n /, also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius.
Conversely, the polar line (or polar) of a point Q in a circle C is the line L such that its closest point P to the center of the circle is the inversion of Q in C. If a point A lies on the polar line q of another point Q, then Q lies on the polar line a of A. More generally, the polars of all the points on the line q must pass through its pole Q.
The locus of points such that the sum of the squares of the distances to the given points is constant is a circle, whose centre is at the centroid of the given points. [22] A generalisation for higher powers of distances is obtained if under n {\displaystyle n} points the vertices of the regular polygon P n {\displaystyle P_{n}} are taken. [ 23 ]
Any two polar circles of two triangles in an orthocentric system are orthogonal. [1]: p. 177 The polar circles of the triangles of a complete quadrilateral form a coaxal system. [1]: p. 179 The most important property of the polar circle is the triangle is self-polar; the polar of each side/point is the opposite side/point.
It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates ( r , θ ) it can be described by the equation r = b ⋅ θ {\displaystyle r=b\cdot \theta } with real number b .
x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, e ix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries.
The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger sets of weighted points. Each circle in the second family (the red circles in the figure) is associated with an angle θ, and is defined as the locus of points X such that the inscribed angle ∠CXD equals θ, { | ^ =}.