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In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. [1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S 1 because it is a one-dimensional unit n-sphere ...
Two distinct planes are either parallel or they intersect in a line. A line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other. Two distinct planes perpendicular to the same line must be parallel to each other.
Similar to the fact that parallel lines must be located in the same plane, parallel planes must be situated in the same three-dimensional space and contain no point in common. Two distinct planes q and r are parallel if and only if the distance from a point P in plane q to the nearest point in plane r is independent of the location of P in ...
The discriminant B 2 – 4AC of the conic section's quadratic equation (or equivalently the determinant AC – B 2 /4 of the 2 × 2 matrix) and the quantity A + C (the trace of the 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, [14] [15] [16] as is the determinant of the 3 × 3 matrix above.
When, in the model, these lines are considered to be the points and the planes the lines of the projective plane PG(2, R), this association becomes a correlation (actually a polarity) of the projective plane. The sphere model is obtained by intersecting the lines and planes through the origin with a unit sphere centered at the origin.
In the complex plane, the most obvious circle inversion map (i.e., using the unit circle centered at the origin) is the complex conjugate of the complex inverse map taking z to 1/z. The complex analytic inverse map is conformal and its conjugate, circle inversion, is anticonformal.
These non-intersecting lines are divided into two classes: Two of the lines (x and y in the diagram) are limiting parallels (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never ...
One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass through a particular origin point; in this model, lines through the origin are considered to be the "points" of the projective plane, and planes through the origin are considered to be the "lines" in the projective plane.