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In planar dynamics a pole is a center of rotation, the polar is the force line of action and the conic is the mass–inertia matrix. [4] The pole–polar relationship is used to define the center of percussion of a planar rigid body. If the pole is the hinge point, then the polar is the percussion line of action as described in planar screw theory.
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.
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 .
where a is the radius of the circle, (,) are the polar coordinates of a generic point on the circle, and (,) are the polar coordinates of the centre of the circle (i.e., r 0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of ...
P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...
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 electrical network will then serve as a discrete model for the Dirichlet problem in the unit disc, where the Laplace equation takes the form of Kirchhoff's law. On the nodes on the boundary of the circle, an electrical potential (Dirichlet data) is defined, which induces an electric current (Neumann data) through the boundary nodes.
The pedal equation can be found by eliminating x and y from these equations and the equation of the curve. The expression for p may be simplified if the equation of the curve is written in homogeneous coordinates by introducing a variable z, so that the equation of the curve is g(x, y, z) = 0. The value of p is then given by [2]