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Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines.
A plane segment or planar region (or simply "plane", in lay use) is a planar surface region; it is analogous to a line segment. A bivector is an oriented plane segment, analogous to directed line segments. [a] A face is a plane segment bounding a solid object. [1] A slab is a region bounded by two parallel planes.
Since C = 2πr, the circumference of a unit circle is 2π. 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 a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0, 0) and (1, 1)), the unit hyperbola, and so on. The two axes divide the plane into four right angles, called ...
Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane.
The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals . [1] The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry.
The three apex points always define a plane in three dimensions, and all three centers of similarity must lie in the plane containing the circular bases. Hence, the three centers must lie on the intersection of the two planes, which must be a line in three dimensions. [2] Monge's theorem can also be proved by using Desargues' theorem.
The Euclidean plane corresponds to the case ε 2 = −1, an imaginary unit. Since the modulus of z is given by = (+) = +, this quantity is the square of the Euclidean distance between z and the origin. For instance, {z | z z* = 1} is the unit circle.