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From Wikipedia

Projective plane

In mathematics, the projective plane is a geometric construction that extends the concept of a plane. In the ordinary plane, two lines typically intersect in a single point, but there are some pairs of lines — namely, parallel lines — that do not intersect. The projective plane is, in one view, the ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two lines in the projective plane intersect in one and only one point.

The projective plane has two common definitions. The first comes from linear algebra; it produces planes that are homogeneous spaces for some of the classical groups. Important examples include the real projective plane \mathbb{RP}^2 and the complex projective plane \mathbb{CP}^2. The second, more general definition comes from axiomatic geometry and finite geometry; it is suitable for study of the incidence properties of plane geometry.

The projective plane generalizes to higher-dimensional projective spaces; that is, a projective plane is a 2-dimensional projective space.

Linear-algebraic definition

In one view, the projective plane is the set of lines through the origin in 3-dimensional space, and a line in the projective plane arises from a plane through the origin in 3-dimensional space. This idea can be made precise as follows.

Let K be any division ring. Let K3 denote the set of all triples x = (x0, x1, x2) of elements of K (a Cartesian product). For any nonzero x in K3, the line in K3 through the origin and x is the subset

\{k x : k \in K\}

of K3. Similarly, let x and y be linearly independent elements of K3, meaning that if k x + l y = 0 then k = l = 0. The plane through the origin, x, and y in K3 is the subset

\{k x + l y : k, l \in K\}

of K3. The plane contains various lines.

The projective plane over K, denoted K\mathbb{P}^2, is the set of all lines in K3. A subset L of K\mathbb{P}^2 is a line in K\mathbb{P}^2 if there exists a plane in K3 whose set of lines is exactly L.

A slightly different definition is as follows. The projective plane is the set K3 - {(0, 0, 0)} modulo the equivalence relation

x \sim k x, k \in K.

Lines in the projective plane are defined exactly as above. If K is a topological space, then K\mathbb{P}^2 inherits a topology via the product, subspace, and quotient topologies.

The coordinates (x0, x1, x2) on K\mathbb{P}^2 are called homogeneous coordinates. Each triple (x0, x1, x2) represents a well-defined point in K\mathbb{P}^2, except for the triple (0, 0, 0), which represents no point. Each point in K\mathbb{P}^2 is potentially represented by many triples.

Examples

The real projective plane \mathbb{RP}^2 arises when K is taken to be the real numbers. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.

The complex projective plane \mathbb{CP}^2 arises when K is taken to be the complex numbers. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other fields serve as fundamental examples in algebraic geometry.

The quaternionic projective plane is also of independent interest. The Cayley plane is considered to be a projective plane over the octonions, but the preceding construction does not suffice to describe it, because the octonions do not form a division ring.

Taking K to be the finite field of pn elements produces a projective plane of p2 n + pn + 1 points. The Fano plane, discussed below, is the example with pn = 2.

Relationship to the ordinary plane

The ordinary plane K2 over K embeds into K\mathbb{P}^2 via the map

(x_1, x_2) \mapsto (1, x_1, x_2).

The complement of the image is the set of points of the form (0, x1, x2). From the point of view of the embedding just given, these points are points at infinity. They constitute a line in K\mathbb{P}^2 — namely, the line arising from the plane

\{k (0, 0, 1) + l (0, 1, 0) : k, l \in K\}

in K3. Intuitively, the points at infinity are the "extra" points where parallel lines intersect; the point (0, x1, x2) is where all lines of slope x2 / x1 intersect. Consider for example the two lines

a = \{(x_1, 0) : x_1 \in K\},
b = \{(x_1, 1) : x_1 \in K\}

in the ordinary plane K2. These lines have slope 0 and do not intersect. They can be regarded as subsets of K\mathbb{P}^2 via the embedding above, but these subsets are not lines in K\mathbb{P}^2. Add the point (0, 1, 0) to each subset; that is, let

\bar a = \{(1, x_1, 0) : x_1 \in K\} \cup \{(0, 1, 0)\},
\bar b = \{

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768-1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745-1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates– the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.

Notational conventions

In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this:

z = x + iy\, for example: z = 4 + i5,

where x and y are real numbers, and i is the imaginary unit. In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane.

In the Cartesian plane the point (x, y) can also be represented in polar coordinates as

(x, y) = (r\cos\theta, r\sin\theta)\qquad\left(r = \sqrt{x^2+y^2}; \quad \theta=\arctan\frac{y}{x}\right).\,

In the Cartesian plane it may be assumed that the arctangent takes values from −Ï€/2 to Ï€/2 (in radians), and some care must be taken to define the real arctangent function for points (x, y) when x≤ 0. In the complex plane these polar coordinates take the form

z = x + iy = |z|\left(\cos\theta + i\sin\theta\right) = |z|e^{i\theta}\,

where

|z| = \sqrt{x^2+y^2}; \quad \theta = \arg(z) = -i\ln\frac{z}.\,

Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ< 2π; and the last equality (to |z|eiθ) is taken from Euler's formula. Notice that the argument of z is multi-valued, because the complex exponential function is periodic, with period 2πi. Thus, if θ is one value of arg(z), the other values are given by arg(z) = θ + 2nπ, where n is any integer ≠ 0.

The theory of contour integration comprises a major part of complex analysis. In this context the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then down and to the left through −1, then down and to the right through −i, and finally up and to the right to z = 1, where we started.

Almost all of complex analysis is concerned with complex functions– that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here it is customary to speak of the domain of f(z) as lying in the z-plane, while referring to the range or image of f(z) as a set of points in the w-plane. In symbols we write

z = x + iy;\qquad f(z) = w = u + iv\,

and often think of the function f as a transformation of the z-plane (with coordinates (x, y)) into the w-plane (with coordinates (u, v)).

Stereographic projections

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place it's center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane.

We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped


From Yahoo Answers

Question:I need an example of a plane in math. In case you don't know it's a endless flat surface. Please help. :(

Answers:a piece of paper, the floor of a room, the earth back when people thought the earth was flat are all examples. There are no infinite, endless plane examples since a result of physics is that space and time are curved.

Question:for example pt a lies on the line but outside the plane. is it still considered to be coplanar?

Answers:yes the point of intersection

Question:My Answer: - Line can lie in the plane - Line can be parrallel to the plane - Line can intersect the plane at a point

Answers:The line can be contained by the plane, thus the intersection would be a line. If the line is not contained by the plane and the line is parallel to the plane then there is no intersection. If the line is not contained by the plane and the line is not parallel to the plane then the line intersects the plane in a point. I like your answers but I did some rewording in case you are at risk for losing points as follows: Your first two answers don't mention the shape of the intersection and you could lose points for this. You might lose points if you don't write it with a format: we can have ___, we can have ____, and if it isn't either of those then it has to be _____. (I'm hoping your teacher isn't this cruel, but such piranhas are out there... Good luck!)

Question:I'm studying for a test, but the teacher didn't include answers for a few questions. Would you be able to answer T/F to these? 1. Two points can lie in each of two different lines. 2. Three non collinear points can lie in each of two different planes. 3. Two intersecting lines are contained in exactly one plane. 4. Three collinear points lie in only one plane. Thank you so much!

Answers:1. Go back to the definition of a line and pay attention to the adjective "unique". 2. Go back to the definition of a plane and pay attention to the adjective "unique". 3. True. 4. Go back to the definition of intersecting planes and pay attention to the qualifier "two".

From Youtube

Geometry: Point, Line, and Plane Postulates :Watch more free lectures and examples of Geometry at www.educator.com Other subjects include Algebra 1/2, Pre Calculus, Pre Algebra, Calculus, Statistics, Biology, Chemistry, Physics, and Computer Science. -All lectures are broken down by individual topics -No more wasted time -Just search and jump directly to the answer

Points, Lines, and Planes :www.gdawgenterprises.com A five minute review with background music to definitions and descriptions of points, lines, and planes. Collinear and noncollinear points are explained as well as coplanar and non coplanar points. The background music is by Kurt Bestor, Man Among Men.