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119-423: The simplest example of a coordinate system is the identification of points on a line with real numbers using the number line . In this system, an arbitrary point O (the origin ) is chosen on a given line. The coordinate of a point P is defined as the signed distance from O to P , where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point

238-400: A ≠ x b {\displaystyle x_{a}\neq x_{b}} , is given by m = ( y b − y a ) / ( x b − x a ) {\displaystyle m=(y_{b}-y_{a})/(x_{b}-x_{a})} and the equation of this line can be written y = m ( x − x

357-380: A ) + y a {\displaystyle y=m(x-x_{a})+y_{a}} . As a note, lines in three dimensions may also be described as the simultaneous solutions of two linear equations a 1 x + b 1 y + c 1 z − d 1 = 0 {\displaystyle a_{1}x+b_{1}y+c_{1}z-d_{1}=0}

476-612: A 1 = t a 2 , b 1 = t b 2 , c 1 = t c 2 {\displaystyle a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}} imply t = 0 {\displaystyle t=0} ). This follows since in three dimensions a single linear equation typically describes a plane and a line is what is common to two distinct intersecting planes. Parametric equations are also used to specify lines, particularly in those in three dimensions or more because in more than two dimensions lines cannot be described by

595-490: A 2 x + b 2 y + c 2 z − d 2 = 0 {\displaystyle a_{2}x+b_{2}y+c_{2}z-d_{2}=0} such that ( a 1 , b 1 , c 1 ) {\displaystyle (a_{1},b_{1},c_{1})} and ( a 2 , b 2 , c 2 ) {\displaystyle (a_{2},b_{2},c_{2})} are not proportional (the relations

714-861: A Cartesian plane , polar coordinates ( r , θ ) are related to Cartesian coordinates by the parametric equations: x = r cos ⁡ θ , y = r sin ⁡ θ . {\displaystyle x=r\cos \theta ,\quad y=r\sin \theta .} In polar coordinates, the equation of a line not passing through the origin —the point with coordinates (0, 0) —can be written r = p cos ⁡ ( θ − φ ) , {\displaystyle r={\frac {p}{\cos(\theta -\varphi )}},} with r > 0 and φ − π / 2 < θ < φ + π / 2. {\displaystyle \varphi -\pi /2<\theta <\varphi +\pi /2.} Here, p

833-417: A and b can yield the same line. Three or more points are said to be collinear if they lie on the same line. If three points are not collinear, there is exactly one plane that contains them. In affine coordinates , in n -dimensional space the points X = ( x 1 , x 2 , ..., x n ), Y = ( y 1 , y 2 , ..., y n ), and Z = ( z 1 , z 2 , ..., z n ) are collinear if

952-468: A conic (a circle , ellipse , parabola , or hyperbola ), lines can be: In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other. For more general algebraic curves , lines could also be: With respect to triangles we have: For a convex quadrilateral with at most two parallel sides,

1071-496: A description or mental image of a primitive notion, to give a foundation to build the notion on which would formally be based on the (unstated) axioms. Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation. These are not true definitions, and could not be used in formal proofs of statements. The "definition" of line in Euclid's Elements falls into this category. Even in

1190-422: A dual graph , a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by

1309-440: A dual polyhedron or dual polytope, with an i -dimensional feature of an n -dimensional polytope corresponding to an ( n − i − 1) -dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves order-theoretic duals . Duality of polytopes and order-theoretic duality are both involutions :

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1428-426: A plane , or skew if they are not. On a Euclidean plane , a line can be represented as a boundary between two regions. Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines . In three-dimensional space , a first degree equation in the variables x , y , and z defines a plane, so two such equations, provided

1547-419: A rank less than 3. In particular, for three points in the plane ( n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero. Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope between the remaining pair of points will equal

1666-400: A straight line , usually abbreviated line , is an infinitely long object with no width, depth, or curvature , an idealization of such physical objects as a straightedge , a taut string, or a ray of light . Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment , which is

1785-409: A "principle". The following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. A simple duality arises from considering subsets of a fixed set S . To any subset A ⊆ S , the complement A consists of all those elements in S that are not contained in A . It is again a subset of S . Taking

1904-527: A close relation between objects of seemingly different nature. One example of such a more general duality is from Galois theory . For a fixed Galois extension K / F , one may associate the Galois group Gal( K / E ) to any intermediate field E (i.e., F ⊆ E ⊆ K ). This group is a subgroup of the Galois group G = Gal( K / F ) . Conversely, to any such subgroup H ⊆ G there

2023-815: A consequence of the dimension formula of linear algebra , this space is two-dimensional, i.e., it corresponds to a line in the projective plane associated to ( R 3 ) ∗ {\displaystyle (\mathbb {R} ^{3})^{*}} . The (positive definite) bilinear form ⟨ ⋅ , ⋅ ⟩ : R 3 × R 3 → R , ⟨ x , y ⟩ = ∑ i = 1 3 x i y i {\displaystyle \langle \cdot ,\cdot \rangle :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ,\langle x,y\rangle =\sum _{i=1}^{3}x_{i}y_{i}} yields an identification of this projective plane with

2142-554: A cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from computational geometry : the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S . As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in

2261-436: A geometry is described by a set of axioms , the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry , a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries ,

2380-430: A line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line. In many models of projective geometry , the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In elliptic geometry we see

2499-415: A line is a defined concept, as in coordinate geometry , some other fundamental ideas are taken as primitives. When the line concept is a primitive, the properties of lines are dictated by the axioms which they must satisfy. In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance, it is possible to provide

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2618-428: A line is stated to have certain properties that relate it to other lines and points . For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point. In two dimensions (i.e., the Euclidean plane ), two lines that do not intersect are called parallel . In higher dimensions, two lines that do not intersect are parallel if they are contained in

2737-434: A map That functor may or may not be an equivalence of categories . There are various situations, where such a functor is an equivalence between the opposite category C of C , and D . Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed. Therefore, any duality between categories C and D

2856-454: A natural way. Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction between the colimit functor that assigns to any diagram in C indexed by some category I its colimit and the diagonal functor that maps any object c of C to the constant diagram which has c at all places. Dually, Gelfand duality is a duality between commutative C*-algebras A and compact Hausdorff spaces X

2975-406: A pairing between submanifolds of a given manifold. From a category theory viewpoint, duality can also be seen as a functor , at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f : V → W its dual f : W → V . In the words of Michael Atiyah , Duality in mathematics is not a theorem, but

3094-430: A part of a line delimited by two points (its endpoints ). Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since

3213-402: A point are the signed distances to each of the planes. This can be generalized to create n coordinates for any point in n -dimensional Euclidean space. Depending on the direction and order of the coordinate axes , the three-dimensional system may be a right-handed or a left-handed system. Another common coordinate system for the plane is the polar coordinate system . A point is chosen as

3332-475: A point varies while the other coordinates are held constant, then the resulting curve is called a coordinate curve . If a coordinate curve is a straight line , it is called a coordinate line . A coordinate system for which some coordinate curves are not lines is called a curvilinear coordinate system . Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. A coordinate line with all other constant coordinates equal to zero

3451-437: A point, in an incidence-preserving way. For such planes there arises a general principle of duality in projective planes : given any theorem in such a plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem. A simple example is that the statement "two points determine a unique line, the line passing through these points" has the dual statement that "two lines determine

3570-460: A second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation matrix , which includes, in its three columns, the Cartesian coordinates of three points. These points are used to define the orientation of

3689-505: A single linear equation. In three dimensions lines are frequently described by parametric equations: x = x 0 + a t y = y 0 + b t z = z 0 + c t {\displaystyle {\begin{aligned}x&=x_{0}+at\\y&=y_{0}+bt\\z&=z_{0}+ct\end{aligned}}} where: Parametric equations for lines in higher dimensions are similar in that they are based on

Coordinate system - Misplaced Pages Continue

3808-676: A structure similar to that of X . This is sometimes called internal Hom . In general, this yields a true duality only for specific choices of D , in which case X = Hom ( X , D ) is referred to as the dual of X . There is always a map from X to the bidual , that is to say, the dual of the dual, X → X ∗ ∗ := ( X ∗ ) ∗ = Hom ⁡ ( Hom ⁡ ( X , D ) , D ) . {\displaystyle X\to X^{**}:=(X^{*})^{*}=\operatorname {Hom} (\operatorname {Hom} (X,D),D).} It assigns to some x ∈ X

3927-503: A system of linear constraints (specifying that the point lie in a halfspace ; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize). Every linear program has a dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa. In logic, functions or relations A and B are considered dual if A (¬ x ) = ¬ B ( x ) , where ¬

4046-419: A typical example of this. In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining

4165-410: A unique line) that make them suitable representations for lines in this geometry. Duality (mathematics) In mathematics , a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B , then the dual of B is A . In other cases

4284-512: A unique point, the intersection point of these two lines". For further examples, see Dual theorems . A conceptual explanation of this phenomenon in some planes (notably field planes) is offered by the dual vector space. In fact, the points in the projective plane R P 2 {\displaystyle \mathbb {RP} ^{2}} correspond to one-dimensional subvector spaces V ⊂ R 3 {\displaystyle V\subset \mathbb {R} ^{3}} while

4403-525: Is logical negation . The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃ x .¬ P ( x ) and ¬∀ x . P ( x ) are equivalent for all predicates P in classical logic: if there exists an x for which P fails to hold, then it is false that P holds for all x (but the converse does not hold constructively). From this fundamental logical duality follow several others: Other analogous dualities follow from these: The dual of

4522-403: Is a homeomorphism from an open subset of a space X to an open subset of R . It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an atlas covering the space. A space equipped with such an atlas is called a manifold and additional structure can be defined on a manifold if the structure

4641-425: Is a scalar ). If a is vector OA and b is vector OB , then the equation of the line can be written: r = a + λ ( b − a ) {\displaystyle \mathbf {r} =\mathbf {a} +\lambda (\mathbf {b} -\mathbf {a} )} . A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0. In

4760-420: Is added to the r and θ polar coordinates giving a triple ( r ,  θ ,  z ). Spherical coordinates take this a step further by converting the pair of cylindrical coordinates ( r ,  z ) to polar coordinates ( ρ ,  φ ) giving a triple ( ρ ,  θ ,  φ ). A point in the plane may be represented in homogeneous coordinates by a triple ( x ,  y ,  z ) where x / z and y / z are

4879-638: Is an order automorphism of S ; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a power set S = 2 are induced by permutations of R . A concept defined for a partial order P will correspond to a dual concept on the dual poset P . For instance, a minimal element of P will be a maximal element of P : minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are upper and lower bounds , lower sets and upper sets , and ideals and filters . In topology, open sets and closed sets are dual concepts:

Coordinate system - Misplaced Pages Continue

4998-414: Is called a coordinate axis , an oriented line used for assigning coordinates. In a Cartesian coordinate system , all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise orthogonal . A polar coordinate system is a curvilinear system where coordinate curves are lines or circles . However, one of the coordinate curves

5117-405: Is called a reflexive space : X ≅ X ″ . {\displaystyle X\cong X''.} Examples: The dual lattice of a lattice L is given by Hom ⁡ ( L , Z ) , {\displaystyle \operatorname {Hom} (L,\mathbb {Z} ),} the set of linear functions on the real vector space containing the lattice that map

5236-414: Is closed. The interior of a set is the largest open set contained in it, and the closure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U . A duality in geometry is provided by the dual cone construction. Given a set C {\displaystyle C} of points in

5355-416: Is consistent where the coordinate maps overlap. For example, a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function. In geometry and kinematics , coordinate systems are used to describe the (linear) position of points and the angular position of axes, planes, and rigid bodies . In the latter case, the orientation of

5474-412: Is defined. The three properties of the dual cone carry over to this type of duality by replacing subsets of R 2 {\displaystyle \mathbb {R} ^{2}} by vector space and inclusions of such subsets by linear maps. That is: A particular feature of this duality is that V and V are isomorphic for certain objects, namely finite-dimensional vector spaces. However, this

5593-501: Is described by coordinate transformations , which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates ( x ,  y ) and polar coordinates ( r ,  θ ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x  =  r  cos θ and y  =  r  sin θ . With every bijection from

5712-407: Is formally the same as an equivalence between C and D ( C and D ). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept. A category that is equivalent to its dual is called self-dual . An example of self-dual category is the category of Hilbert spaces . Many category-theoretic notions come in pairs in

5831-405: Is given a unique coordinate and each real number is the coordinate of a unique point. The prototypical example of a coordinate system is the Cartesian coordinate system . In the plane , two perpendicular lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three mutually orthogonal planes are chosen and the three coordinates of

5950-535: Is in a sense a lucky coincidence, for giving such an isomorphism requires a certain choice, for example the choice of a basis of V . This is also true in the case if V is a Hilbert space , via the Riesz representation theorem . In all the dualities discussed before, the dual of an object is of the same kind as the object itself. For example, the dual of a vector space is again a vector space. Many duality statements are not of this kind. Instead, such dualities reveal

6069-495: Is isomorphic to V precisely if V is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degenerate bilinear form φ : V × V → K {\displaystyle \varphi :V\times V\to K} In this case V is called an inner product space . For example, if K is the field of real or complex numbers , any positive definite bilinear form gives rise to such an isomorphism. In Riemannian geometry , V

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6188-496: Is not true. In the geometries where the concept of a line is a primitive notion , as may be the case in some synthetic geometries , other methods of determining collinearity are needed. In a sense, all lines in Euclidean geometry are equal, in that, without coordinates, one can not tell them apart from one another. However, lines may play special roles with respect to other objects in the geometry and be divided into types according to that relationship. For instance, with respect to

6307-402: Is not zero. There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the standard form . If the constant term is put on the left, the equation becomes a x + b y − c = 0 , {\displaystyle ax+by-c=0,} and this is sometimes called

6426-421: Is on either one of them is also on the other. Perpendicular lines are lines that intersect at right angles . In three-dimensional space , skew lines are lines that are not in the same plane and thus do not intersect each other. The concept of line is often considered in geometry as a primitive notion in axiomatic systems , meaning it is not being defined by other concepts. In those situations where

6545-418: Is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero. Many curves can occur as coordinate curves. For example, the coordinate curves of parabolic coordinates are parabolas . In three-dimensional space, if one coordinate is held constant and

6664-482: Is represented by many pairs of coordinates. For example, ( r ,  θ ), ( r ,  θ +2 π ) and (− r ,  θ + π ) are all polar coordinates for the same point. The pole is represented by (0, θ ) for any value of θ . There are two common methods for extending the polar coordinate system to three dimensions. In the cylindrical coordinate system , a z -coordinate with the same meaning as in Cartesian coordinates

6783-466: Is taken to be the tangent space of a manifold and such positive bilinear forms are called Riemannian metrics . Their purpose is to measure angles and distances. Thus, duality is a foundational basis of this branch of geometry. Another application of inner product spaces is the Hodge star which provides a correspondence between the elements of the exterior algebra . For an n -dimensional vector space,

6902-437: Is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and φ {\displaystyle \varphi } is the (oriented) angle from the x -axis to this segment. It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between

7021-420: Is the fixed field K consisting of elements fixed by the elements in H . Compared to the above, this duality has the following features: Given a poset P = ( X , ≤) (short for partially ordered set; i.e., a set that has a notion of ordering but in which two elements cannot necessarily be placed in order relative to each other), the dual poset P = ( X , ≥) comprises the same ground set but

7140-457: Is the same: it assigns to X the space of continuous functions (which vanish at infinity) from X to C , the complex numbers. Conversely, the space X can be reconstructed from A as the spectrum of A . Both Gelfand and Pontryagin duality can be deduced in a largely formal, category-theoretic way. In a similar vein there is a duality in algebraic geometry between commutative rings and affine schemes : to every commutative ring A there

7259-528: Is the set of all points whose coordinates ( x , y ) satisfy a linear equation; that is, L = { ( x , y ) ∣ a x + b y = c } , {\displaystyle L=\{(x,y)\mid ax+by=c\},} where a , b and c are fixed real numbers (called coefficients ) such that a and b are not both zero. Using this form, vertical lines correspond to equations with b = 0. One can further suppose either c = 1 or c = 0 , by dividing everything by c if it

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7378-423: Is the subset L = { ( 1 − t ) a + t b ∣ t ∈ R } . {\displaystyle L=\left\{(1-t)\,a+tb\mid t\in \mathbb {R} \right\}.} The direction of the line is from a reference point a ( t = 0) to another point b ( t = 1), or in other words, in the direction of the vector b  −  a . Different choices of

7497-522: Is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be dualistic . Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the principle of duality . There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems

7616-410: Is weaker than the one above, in that The other two properties carry over without change: A very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V . Its elements are the linear functionals φ : V → K {\displaystyle \varphi :V\to K} , where K is the field over which V

7735-689: The R P 2 {\displaystyle \mathbb {RP} ^{2}} . Concretely, the duality assigns to V ⊂ R 3 {\displaystyle V\subset \mathbb {R} ^{3}} its orthogonal { w ∈ R 3 , ⟨ v , w ⟩ = 0  for all  v ∈ V } {\displaystyle \left\{w\in \mathbb {R} ^{3},\langle v,w\rangle =0{\text{ for all }}v\in V\right\}} . The explicit formulas in duality in projective geometry arise by means of this identification. In

7854-559: The c /| c | term to compute sin ⁡ φ {\displaystyle \sin \varphi } and cos ⁡ φ {\displaystyle \cos \varphi } , and it follows that φ {\displaystyle \varphi } is only defined modulo π . The vector equation of the line through points A and B is given by r = O A + λ A B {\displaystyle \mathbf {r} =\mathbf {OA} +\lambda \,\mathbf {AB} } (where λ

7973-550: The x -axis to this segment), and p is the (positive) length of the normal segment. The normal form can be derived from the standard form a x + b y = c {\displaystyle ax+by=c} by dividing all of the coefficients by a 2 + b 2 . {\displaystyle {\sqrt {a^{2}+b^{2}}}.} and also multiplying through by − 1 {\displaystyle -1} if c < 0. {\displaystyle c<0.} Unlike

8092-400: The ( n − 1) -dimensional spaces resulting from fixing a single coordinate of an n -dimensional coordinate system. The concept of a coordinate map , or coordinate chart is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map

8211-580: The Newton line is the line that connects the midpoints of the two diagonals . For a hexagon with vertices lying on a conic we have the Pascal line and, in the special case where the conic is a pair of lines, we have the Pappus line . Parallel lines are lines in the same plane that never cross. Intersecting lines share a single point in common. Coincidental lines coincide with each other—every point that

8330-449: The converse relation . Familiar examples of dual partial orders include A duality transform is an involutive antiautomorphism f of a partially ordered set S , that is, an order-reversing involution f  : S → S . In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if f 1 , f 2 are two duality transforms then their composition

8449-1807: The general form of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms. These forms are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope, x-intercept , known points on the line and y-intercept. The equation of the line passing through two different points P 0 ( x 0 , y 0 ) {\displaystyle P_{0}(x_{0},y_{0})} and P 1 ( x 1 , y 1 ) {\displaystyle P_{1}(x_{1},y_{1})} may be written as ( y − y 0 ) ( x 1 − x 0 ) = ( y 1 − y 0 ) ( x − x 0 ) . {\displaystyle (y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0}).} If x 0 ≠ x 1 , this equation may be rewritten as y = ( x − x 0 ) y 1 − y 0 x 1 − x 0 + y 0 {\displaystyle y=(x-x_{0})\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+y_{0}} or y = x y 1 − y 0 x 1 − x 0 + x 1 y 0 − x 0 y 1 x 1 − x 0 . {\displaystyle y=x\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+{\frac {x_{1}y_{0}-x_{0}y_{1}}{x_{1}-x_{0}}}\,.} In two dimensions ,

8568-528: The matrix [ 1 x 1 x 2 ⋯ x n 1 y 1 y 2 ⋯ y n 1 z 1 z 2 ⋯ z n ] {\displaystyle {\begin{bmatrix}1&x_{1}&x_{2}&\cdots &x_{n}\\1&y_{1}&y_{2}&\cdots &y_{n}\\1&z_{1}&z_{2}&\cdots &z_{n}\end{bmatrix}}} has

8687-457: The normal form of the line equation by setting x = r cos ⁡ θ , {\displaystyle x=r\cos \theta ,} and y = r sin ⁡ θ , {\displaystyle y=r\sin \theta ,} and then applying the angle difference identity for sine or cosine. These equations can also be proven geometrically by applying right triangle definitions of sine and cosine to

8806-419: The pole and a ray from this point is taken as the polar axis . For a given angle θ , there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r . For a given pair of coordinates ( r ,  θ ) there is a single point, but any point

8925-497: The right triangle that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides. The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates ( r , θ ) {\displaystyle (r,\theta )} of the points of a line passing through the origin and making an angle of α {\displaystyle \alpha } with

9044-470: The strong dual space topology) as a reflexive space . In other cases, showing a relation between the primal and bidual is a significant result, as in Pontryagin duality (a locally compact abelian group is naturally isomorphic to its bidual). A group of dualities can be described by endowing, for any mathematical object X , the set of morphisms Hom ( X , D ) into some fixed object D , with

9163-437: The x -axis and the line. In this case, the equation becomes r = p sin ⁡ ( θ − α ) , {\displaystyle r={\frac {p}{\sin(\theta -\alpha )}},} with r > 0 and 0 < θ < α + π . {\displaystyle 0<\theta <\alpha +\pi .} These equations can be derived from

9282-426: The x -axis, are the pairs ( r , θ ) {\displaystyle (r,\theta )} such that r ≥ 0 , and θ = α or θ = α + π . {\displaystyle r\geq 0,\qquad {\text{and}}\quad \theta =\alpha \quad {\text{or}}\quad \theta =\alpha +\pi .} In modern mathematics, given

9401-421: The "canonical evaluation map". For finite-dimensional vector spaces this is an isomorphism, but these are not identical spaces: they are different sets. In category theory, this is generalized by § Dual objects , and a " natural transformation " from the identity functor to the double dual functor. For vector spaces (considered algebraically), this is always an injection; see Dual space § Injection into

9520-429: The Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane without the use of infinity . In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values. Some other common coordinate systems are

9639-468: The Hodge star operator maps k -forms to ( n − k ) -forms. This can be used to formulate Maxwell's equations . In this guise, the duality inherent in the inner product space exchanges the role of magnetic and electric fields . In some projective planes , it is possible to find geometric transformations that map each point of the projective plane to a line, and each line of the projective plane to

9758-456: The algebraic dual V , with different possible topologies on the dual, each of which defines a different bidual space ⁠ V ″ {\displaystyle V''} ⁠ . In these cases the canonical evaluation map ⁠ V → V ″ {\displaystyle V\to V''} ⁠ is not in general an isomorphism. If it is, this is known (for certain locally convex vector spaces with

9877-539: The axes of the local system; they are the tips of three unit vectors aligned with those axes. The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of the Hellenistic period , a variety of coordinate systems have been developed based on the types above, including: Line (geometry) In geometry ,

9996-401: The bidual is not identical with the primal, though there is often a close connection. For example, the dual cone of the dual cone of a set contains the primal set (it is the smallest cone containing the primal set), and is equal if and only if the primal set is a cone. An important case is for vector spaces, where there is a map from the primal space to the double dual, V → V , known as

10115-462: The case where a specific geometry is being considered (for example, Euclidean geometry ), there is no generally accepted agreement among authors as to what an informal description of a line should be when the subject is not being treated formally. Lines in a Cartesian plane or, more generally, in affine coordinates , are characterized by linear equations. More precisely, every line L {\displaystyle L} (including vertical lines)

10234-480: The circle (with multiplication of complex numbers as group operation). In another group of dualities, the objects of one theory are translated into objects of another theory and the maps between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using the parlance of category theory , this amounts to a contravariant functor between two categories C and D : which for any two objects X and Y of C gives

10353-419: The complement has the following properties: This duality appears in topology as a duality between open and closed subsets of some fixed topological space X : a subset U of X is closed if and only if its complement in X is open. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets

10472-411: The complement of an open set is closed, and vice versa. In matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid . There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of

10591-565: The definitions are never explicitly referenced in the remainder of the text. In modern geometry, a line is usually either taken as a primitive notion with properties given by axioms , or else defined as a set of points obeying a linear relationship, for instance when real numbers are taken to be primitive and geometry is established analytically in terms of numerical coordinates . In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps),

10710-479: The diagram. Unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set C {\displaystyle C} . Instead, C ∗ ∗ {\displaystyle C^{**}} is the smallest cone containing C {\displaystyle C} which may be bigger than C {\displaystyle C} . Therefore this duality

10829-486: The double-dual . This can be generalized algebraically to a dual module . There is still a canonical evaluation map, but it is not always injective; if it is, this is known as a torsionless module ; if it is an isomophism, the module is called reflexive. For topological vector spaces (including normed vector spaces ), there is a separate notion of a topological dual , denoted ⁠ V ′ {\displaystyle V'} ⁠ to distinguish from

10948-729: The dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called primal ). Such involutions sometimes have fixed points , so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry . In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings , bilinear functions from an object of one type and another object of

11067-420: The dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure. From any three-dimensional polyhedron, one can form a planar graph , the graph of its vertices and edges. The dual polyhedron has

11186-416: The dual, called the bidual or double dual , depending on context, is often identical to the original (also called primal ), and duality is an involution. In this case the bidual is not usually distinguished, and instead one only refers to the primal and dual. For example, the dual poset of the dual poset is exactly the original poset, since the converse relation is defined by an involution. In other cases,

11305-469: The duality aspect). Therefore, in some cases, proofs of certain statements can be halved, using such a duality phenomenon. Further notions displaying related by such a categorical duality are projective and injective modules in homological algebra , fibrations and cofibrations in topology and more generally model categories . Two functors F : C → D and G : D → C are adjoint if for all objects c in C and d in D in

11424-518: The end of the 19th century, such as non-Euclidean , projective , and affine geometry . In the Greek deductive geometry of Euclid's Elements , a general line (now called a curve ) is defined as a "breadthless length", and a straight line (now called a line segment ) was defined as a line "which lies evenly with the points on itself". These definitions appeal to readers' physical experience, relying on terms that are not themselves defined, and

11543-452: The equation for non-vertical lines is often given in the slope–intercept form : y = m x + b {\displaystyle y=mx+b} where: The slope of the line through points A ( x a , y a ) {\displaystyle A(x_{a},y_{a})} and B ( x b , y b ) {\displaystyle B(x_{b},y_{b})} , when x

11662-405: The equation of a straight line on the plane is given by: x cos ⁡ φ + y sin ⁡ φ − p = 0 , {\displaystyle x\cos \varphi +y\sin \varphi -p=0,} where φ {\displaystyle \varphi } is the angle of inclination of the normal segment (the oriented angle from the unit vector of

11781-465: The faces of the primal. Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each other, so do the corresponding two parts of the dual polyhedron . More generally, using the concept of polar reciprocation , any convex polyhedron , or more generally any convex polytope , corresponds to

11900-435: The following: There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length . These include: Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes, circles or spheres . For example, Plücker coordinates are used to determine

12019-405: The introduction is an example of such a duality. Indeed, the set of morphisms, i.e., linear maps , forms a vector space in its own right. The map V → V mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if the dimension of V is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis. A vector space V

12138-641: The lines in the projective plane correspond to subvector spaces W {\displaystyle W} of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional V {\displaystyle V} the subspace of ( R 3 ) ∗ {\displaystyle (\mathbb {R} ^{3})^{*}} consisting of those linear maps f : R 3 → R {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} } which satisfy f ( V ) = 0 {\displaystyle f(V)=0} . As

12257-506: The map that associates to any map f  : X → D (i.e., an element in Hom( X , D ) ) the value f ( x ) . Depending on the concrete duality considered and also depending on the object X , this map may or may not be an isomorphism. The construction of the dual vector space V ∗ = Hom ⁡ ( V , K ) {\displaystyle V^{*}=\operatorname {Hom} (V,K)} mentioned in

12376-408: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. When

12495-482: The objects. A classical example of this is the duality of the Platonic solids , in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with

12614-523: The other slopes). By extension, k points in a plane are collinear if and only if any ( k –1) pairs of points have the same pairwise slopes. In Euclidean geometry , the Euclidean distance d ( a , b ) between two points a and b may be used to express the collinearity between three points by: However, there are other notions of distance (such as the Manhattan distance ) for which this property

12733-453: The other two are allowed to vary, then the resulting surface is called a coordinate surface . For example, the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of coordinate planes . Similarly, coordinate hypersurfaces are

12852-855: The plane R 2 {\displaystyle \mathbb {R} ^{2}} (or more generally points in R n {\displaystyle \mathbb {R} ^{n}} ), the dual cone is defined as the set C ∗ ⊆ R 2 {\displaystyle C^{*}\subseteq \mathbb {R} ^{2}} consisting of those points ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} satisfying x 1 c 1 + x 2 c 2 ≥ 0 {\displaystyle x_{1}c_{1}+x_{2}c_{2}\geq 0} for all points ( c 1 , c 2 ) {\displaystyle (c_{1},c_{2})} in C {\displaystyle C} , as illustrated in

12971-398: The planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in n -dimensional space n −1 first-degree equations in the n coordinate variables define a line under suitable conditions. In more general Euclidean space , R (and analogously in every other affine space ), the line L passing through two different points a and b

13090-419: The points of the lattice to the integers Z {\displaystyle \mathbb {Z} } . This is used in the construction of toric varieties . The Pontryagin dual of locally compact topological groups G is given by Hom ⁡ ( G , S 1 ) , {\displaystyle \operatorname {Hom} (G,S^{1}),} continuous group homomorphisms with values in

13209-407: The position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line. It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this

13328-411: The primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in

13447-419: The realm of topological vector spaces , a similar construction exists, replacing the dual by the topological dual vector space. There are several notions of topological dual space, and each of them gives rise to a certain concept of duality. A topological vector space X {\displaystyle X} that is canonically isomorphic to its bidual X ″ {\displaystyle X''}

13566-410: The second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number , viewed as

13685-428: The sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph. A kind of geometric duality also occurs in optimization theory , but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} ),

13804-641: The sense that they correspond to each other while considering the opposite category. For example, Cartesian products Y 1 × Y 2 and disjoint unions Y 1 ⊔ Y 2 of sets are dual to each other in the sense that and for any set X . This is a particular case of a more general duality phenomenon, under which limits in a category C correspond to colimits in the opposite category C ; further concrete examples of this are epimorphisms vs. monomorphism , in particular factor modules (or groups etc.) vs. submodules , direct products vs. direct sums (also called coproducts to emphasize

13923-408: The slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, φ {\displaystyle \varphi } and p , to be specified. If p > 0 , then φ {\displaystyle \varphi } is uniquely defined modulo 2 π . On the other hand, if the line is through the origin ( c = p = 0 ), one drops

14042-399: The space to itself two coordinate transformations can be associated: For example, in 1D , if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more. Given a coordinate system, if one of the coordinates of

14161-525: The specification of one point on the line and a direction vector. The normal form (also called the Hesse normal form , after the German mathematician Ludwig Otto Hesse ), is based on the normal segment for a given line, which is defined to be the line segment drawn from the origin perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of

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