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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.
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 .
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]
The equation of the circle determined by three points (,), (,), (,) not on a line is obtained by a conversion of the 3-point form of a circle equation: () + () () () = () + () () (). Homogeneous form In homogeneous coordinates , each conic section with the equation of a circle has the form x 2 + y 2 − 2 a x z − 2 b y z + c z 2 = 0 ...
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).
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.
Take P to be the origin. For a curve given by the equation F(x, y)=0, if the equation of the tangent line at R=(x 0, y 0) is written in the form + = then the vector (cos α, sin α) is parallel to the segment PX, and the length of PX, which is the distance from the tangent line to the origin, is p.
The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions: (,,) = | | (,) (,) where the () are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary ...