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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.
In this expanded plane, we define the polar of the point O to be the line at infinity (and O is the pole of the line at infinity), and the poles of the lines through O are the points of infinity where, if a line has slope s (≠ 0) its pole is the infinite point associated to the parallel class of lines with slope −1/s.
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
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 axis to the line). Then there is a unique ...
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 ...
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).
Since a line in the dual plane corresponds to a point in the plane, the degree of the dual is the number of tangents to the X that can be drawn through a given point. The points where these tangents touch the curve are the points of intersection between the curve and the polar curve with respect to the given point.
These four points determine a quadrangle of which P is a diagonal point. 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.