Plane (geometry)

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry; when working in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, trigonometry, graph theory, graphing are performed in a two-dimensional space, or, in other words, in the plane. Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry, he selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.

Euclid never used numbers to measure angle, or area. In this way the Euclidean plane is not quite the same as the Cartesian plane. A plane is a ruled surface; this section is concerned with planes embedded in three dimensions: in R3. In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: Three non-collinear points. A line and a point not on that line. Two distinct but intersecting lines. Two parallel lines; the following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues: Two distinct planes are either parallel or they intersect in a line. A line 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. In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it to indicate its "inclination".

Let r0 be the position vector of some point P0 =, let n = be a nonzero vector. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that n ⋅ = 0. Expanded this becomes a + b + c = 0, the point-normal form of the equation of a plane; this is just a linear equation a x + b y + c z + d = 0, where d = −. Conversely, it is shown that if a, b, c and d are constants and a, b, c are not all zero the graph of the equation a x + b y + c z + d = 0, is a plane having the vector n = as a normal; this familiar equation for a plane is called the general form of the equation of the plane. Thus for example a regression equation of the form y = d + ax + cz establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Alternatively, a plane may be described parametrically as the set of all points of the form r = r 0 + s v + t w, where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, r0 is the vector representing the position of an arbitrary point on the plane. The vectors v and w can be visualized as vectors starting at r0 and pointing in different directions along the plane. Note that v and w can be perpendicular, but cannot be parallel. Let p1=, p2=, p3= be non-collinear points; the plane passing through p1, p2, p3 can be described as the set of all points that satisfy the following determinant equations: | x − x 1 y − y 1 z − z 1 x 2 − x 1 y 2 − y

Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.

Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.

Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.

Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.

Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv

Affine transformation

In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Sets of parallel lines remain parallel after an affine transformation. An affine transformation does not preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, homothety, similarity transformation, rotation, shear mapping, compositions of them in any combination and sequence. If X and Y are affine spaces every affine transformation f: X → Y is of the form x ↦ M x + b, where M is a linear transformation on the space X, x is a vector in X, b is a vector in Y. Unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine. All Euclidean spaces are affine. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, each output coordinate of an affine map is a linear function of all input coordinates.

Another way to deal with affine transformations systematically is to select a point as the origin. An affine map f: A → B between two affine spaces is a map on the points that acts linearly on the vectors. In symbols, f determines a linear transformation φ such that, for any pair of points P, Q ∈ A: f f → = φ or f − f = φ. We can interpret this definition in a few other ways. If an origin O ∈ A is chosen, B denotes its image f ∈ B this means that for any vector x →: f: ↦. If an origin O ′ ∈ B is chosen, this can be decomposed as an affine transformation g: A → B that sends O ↦ O ′, namely g: ↦,followed by the translation by a vector b → = O ′ B →; the conclusion is that, intuitively, f consists of a linear map. Given two affine spaces A and B, over the same field, a function f: A → B is an affine map if and only if for every family i ∈ I of weighted points in A such that ∑ i ∈ I λ i = 1,we have f = ∑ i ∈ I λ i f. I