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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 reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. [1]
Gergonne coined the terms "duality" and "polar" (but "pole" is due to F.-J. Servois) and adopted the style of writing dual statements side by side in his journal. Jean-Victor Poncelet (1788−1867) author of the first text on projective geometry , Traité des propriétés projectives des figures , was a synthetic geometer who systematically ...
Another common coordinate system for the plane is the polar coordinate system. [7] A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the
The line through the other two diagonal points is called the polar of P and P is the pole of this line. [19] Alternatively, the polar line of P is the set of projective harmonic conjugates of P on a variable secant line passing through P and C.
The point P is called the pole of that line of harmonic conjugates, and this line is called the polar line of P with respect to the conic. See the article Pole and polar for more details. Inversive geometry
If the polar line of C with respect to a point Q is a line L, then Q is said to be a pole of L. A given line has (n−1) 2 poles (counting multiplicities etc.) where n is the degree of C. To see this, pick two points P and Q on L. The locus of points whose polar lines pass through P is the first polar of P and this is a curve of degree n−1.
It is possible to choose any projection plane parallel to the equator (except the South pole): the figures will be proportional (property of similar triangles). It is usual to place the projection plane at the North pole. Definition The pole figure is the stereographic projection of the poles used to represent the orientation of an object in space.