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In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole.
In the general projective plane case where duality means plane duality, the definitions of polarity, absolute elements, pole and polar remain the same. Let P denote a projective plane of order n. Counting arguments can establish that for a polarity π of P: [17] The number of non-absolute points (lines) incident with a non-absolute line (point ...
Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3, 60°). In blue, the point (4, 210°). In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
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).
The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem.
In geometry, a polar point group is a point group in which there is more than one point that every symmetry operation leaves unmoved. [1] The unmoved points will constitute a line, a plane, or all of space. While the simplest point group, C 1, leaves all points invariant, most polar point groups will move some, but not all points. To describe ...
Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space .
Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as A is (conventionally) termed the pole of A and it is denoted by A'. The points B' and C' are defined similarly. The triangle A'B'C' is the polar triangle corresponding to triangle ABC.