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The complex plane is two-dimensional when considered to be formed from real-number coordinates, but one-dimensional in terms of complex-number coordinates. A two-dimensional complex space – such as the two-dimensional complex coordinate space , the complex projective plane , or a complex surface – has two complex dimensions, which can ...
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane, since every Euclidean plane is isomorphic to it.
Essentially, he realized an equivalence relation on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane. [9]: 52–4 The term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a real number s (also called scalar) and a 3-dimensional vector.
A plane: the locus of x is a plane if A = P, a vector with a zero n o component. In a homogeneous projective space such a vector P represents a vector on the plane n o =1 that would be infinitely far from the origin (ie infinitely far outside the null cone), so g(x).P =0 corresponds to x on a sphere of infinite radius, a plane. In particular:
A plane duality is a map from a projective plane C = (P, L, I) to its dual plane C ∗ = (L, P, I ∗) (see § Principle of duality above) which preserves incidence. That is, a plane duality σ will map points to lines and lines to points (P σ = L and L σ = P) in such a way that if a point Q is on a line m (denoted by Q I m) then Q I m ⇔ m ...
Reflections, or mirror isometries, denoted by F c,v, where c is a point in the plane and v is a unit vector in R 2. (F is for "flip".) have the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c. The line L is called the reflection axis or the associated mirror.
Special cases are called the real line R 1, the real coordinate plane R 2, and the real coordinate three-dimensional space R 3. With component-wise addition and scalar multiplication, it is a real vector space. The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the ...