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103-713: One-dimensional manifolds include lines and circles , but not self-crossing curves such as a figure 8 . Two-dimensional manifolds are also called surfaces . Examples include the plane , the sphere , and the torus , and also the Klein bottle and real projective plane . The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given

206-400: 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

309-380: 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}

412-612: 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

515-490: 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

618-626: A ) = χ r i g h t ( χ t o p − 1 [ a ] ) = χ r i g h t ( a , 1 − a 2 ) = 1 − a 2 {\displaystyle {\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left[a\right]\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}} Such

721-423: A pure manifold . For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is not a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is locally constant ), each connected component has

824-861: 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

927-417: 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

1030-468: 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,

1133-496: 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

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1236-845: A local coordinate system , coordinate chart , coordinate patch , coordinate map , or local frame . An atlas for a topological space M {\displaystyle M} is an indexed family { ( U α , φ α ) : α ∈ I } {\displaystyle \{(U_{\alpha },\varphi _{\alpha }):\alpha \in I\}} of charts on M {\displaystyle M} which covers M {\displaystyle M} (that is, ⋃ α ∈ I U α = M {\textstyle \bigcup _{\alpha \in I}U_{\alpha }=M} ). If for some fixed n ,

1339-426: 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

1442-419: 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

1545-400: 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

1648-458: A Euclidean space means that every point has a neighborhood homeomorphic to an open subset of the Euclidean space R n , {\displaystyle \mathbb {R} ^{n},} for some nonnegative integer n . This implies that either the point is an isolated point (if n = 0 {\displaystyle n=0} ), or it has a neighborhood homeomorphic to

1751-542: A chart for the plane R 2 {\displaystyle \mathbb {R} ^{2}} minus the positive x -axis and the origin. Another example of a chart is the map χ top mentioned above, a chart for the circle. The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an atlas . An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union

1854-414: A consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold. Line (geometry) In geometry ,

1957-436: A different aspect of the manifold, thereby leading to a slightly different viewpoint. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R 2 {\displaystyle \mathbb {R} ^{2}} is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here

2060-472: A differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions . For symplectic manifolds , the transition functions must be symplectomorphisms . The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible . These notions are made precise in general through

2163-420: A fixed dimension. Sheaf-theoretically , a manifold is a locally ringed space , whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry . The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly,

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2266-665: A function is called a transition map . The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts χ m i n u s ( x , y ) = s = y 1 + x {\displaystyle \chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}} and χ p l u s ( x , y ) = t = y 1 − x {\displaystyle \chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}} Here s

2369-436: 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 ,

2472-430: 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

2575-415: 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

2678-428: 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

2781-496: A line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle , x  +  y  = 1, where the y -coordinate is positive (indicated by the yellow arc in Figure 1 ). Any point of this arc can be uniquely described by its x -coordinate. So, projection onto

2884-630: A manifold M such that U α ∩ U β {\displaystyle U_{\alpha }\cap U_{\beta }} is non-empty . The transition map τ α , β : φ α ( U α ∩ U β ) → φ β ( U α ∩ U β ) {\displaystyle \tau _{\alpha ,\beta }:\varphi _{\alpha }(U_{\alpha }\cap U_{\beta })\to \varphi _{\beta }(U_{\alpha }\cap U_{\beta })}

2987-401: A manifold can be described using mathematical maps , called coordinate charts , collected in a mathematical atlas . It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across

3090-413: A manifold is called differentiable . Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives . If each transition function is a smooth map , then the atlas is called a smooth atlas , and the manifold itself is called smooth . Alternatively, one could require that the transition maps have only k continuous derivatives in which case

3193-482: A manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Russia may both contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an open ball in R n {\displaystyle \mathbb {R} ^{n}} to the manifold and then back to another (or perhaps

Manifold - Misplaced Pages Continue

3296-535: A manifold with boundary. The interior of M {\displaystyle M} , denoted Int ⁡ M {\displaystyle \operatorname {Int} M} , is the set of points in M {\displaystyle M} which have neighborhoods homeomorphic to an open subset of R n {\displaystyle \mathbb {R} ^{n}} . The boundary of M {\displaystyle M} , denoted ∂ M {\displaystyle \partial M} ,

3399-414: A neighborhood homeomorphic to R n {\displaystyle \mathbb {R} ^{n}} since R n {\displaystyle \mathbb {R} ^{n}} is homeomorphic, and even diffeomorphic to any open ball in it (for n > 0 {\displaystyle n>0} ). The n that appears in the preceding definition is called the local dimension of

3502-786: A neighborhood homeomorphic to the "half" n {\displaystyle n} -ball { ( x 1 , x 2 , … , x n ) | Σ x i 2 < 1  and  x 1 ≥ 0 } {\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}} . Any homeomorphism between half-balls must send points with x 1 = 0 {\displaystyle x_{1}=0} to points with x 1 = 0 {\displaystyle x_{1}=0} . This invariance allows to "define" boundary points; see next paragraph. Let M {\displaystyle M} be

3605-430: 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

3708-461: A point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line segment, since deleting the center point from the "+" gives a space with four components (i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces; topological operations always preserve

3811-419: A second atlas for the circle, with the transition map t = 1 s {\displaystyle t={\frac {1}{s}}} (that is, one has this relation between s and t for every point where s and t are both nonzero). Each chart omits a single point, either (−1, 0) for s or (+1, 0) for t , so neither chart alone is sufficient to cover the whole circle. It can be proved that it

3914-505: 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

4017-419: 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

4120-513: A unique line) that make them suitable representations for lines in this geometry. Atlas (topology) In mathematics , particularly topology , an atlas is a concept used to describe a manifold . An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles . The definition of an atlas depends on

4223-829: A way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.) To be more precise, suppose that ( U α , φ α ) {\displaystyle (U_{\alpha },\varphi _{\alpha })} and ( U β , φ β ) {\displaystyle (U_{\beta },\varphi _{\beta })} are two charts for

Manifold - Misplaced Pages Continue

4326-425: 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

4429-705: Is a circle, a 1-manifold . A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open n {\displaystyle n} -ball { ( x 1 , x 2 , … , x n ) | Σ x i 2 < 1 } {\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1\}} . Every boundary point has

4532-528: Is a manifold with boundary of dimension n {\displaystyle n} , then Int ⁡ M {\displaystyle \operatorname {Int} M} is a manifold (without boundary) of dimension n {\displaystyle n} and ∂ M {\displaystyle \partial M} is a manifold (without boundary) of dimension n − 1 {\displaystyle n-1} . A single manifold can be constructed in different ways, each stressing

4635-471: Is a pair of separate circles. Manifolds need not be closed ; thus a line segment without its end points is a manifold. They are never countable , unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola , a hyperbola , and the locus of points on a cubic curve y = x − x (a closed loop piece and an open, infinite piece). However, excluded are examples like two touching circles that share

4738-409: Is also an atlas. The atlas containing all possible charts consistent with a given atlas is called the maximal atlas (i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations). Charts in an atlas may overlap and a single point of

4841-553: Is an adequate atlas ( U i , φ i ) i ∈ I {\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}} on M {\displaystyle M} , such that ( U i ) i ∈ I {\displaystyle \left(U_{i}\right)_{i\in I}} is a refinement of V {\displaystyle {\mathcal {V}}} . A transition map provides

4944-633: Is another example, applying this method to the construction of a sphere: A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R 3 {\displaystyle \mathbb {R} ^{3}} : S = { ( x , y , z ) ∈ R 3 ∣ x 2 + y 2 + z 2 = 1 } . {\displaystyle S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.} The sphere

5047-461: Is called an adequate atlas if the following conditions hold: Every second-countable manifold admits an adequate atlas. Moreover, if V = ( V j ) j ∈ J {\displaystyle {\mathcal {V}}=\left(V_{j}\right)_{j\in J}} is an open covering of the second-countable manifold M {\displaystyle M} , then there

5150-430: Is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility. The sphere is an example of a surface. The unit sphere of implicit equation may be covered by an atlas of six charts :

5253-496: 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

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5356-402: 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

5459-421: 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

5562-506: Is the complement of Int ⁡ M {\displaystyle \operatorname {Int} M} in M {\displaystyle M} . The boundary points can be characterized as those points which land on the boundary hyperplane ( x n = 0 ) {\displaystyle (x_{n}=0)} of R + n {\displaystyle \mathbb {R} _{+}^{n}} under some coordinate chart. If M {\displaystyle M}

5665-437: 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

5768-525: Is the map defined by τ α , β = φ β ∘ φ α − 1 . {\displaystyle \tau _{\alpha ,\beta }=\varphi _{\beta }\circ \varphi _{\alpha }^{-1}.} Note that since φ α {\displaystyle \varphi _{\alpha }} and φ β {\displaystyle \varphi _{\beta }} are both homeomorphisms,

5871-528: 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

5974-745: Is the slope of the line through the point at coordinates ( x ,  y ) and the fixed pivot point (−1, 0); similarly, t is the opposite of the slope of the line through the points at coordinates ( x ,  y ) and (+1, 0). The inverse mapping from s to ( x ,  y ) is given by x = 1 − s 2 1 + s 2 y = 2 s 1 + s 2 {\displaystyle {\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\[5pt]y&={\frac {2s}{1+s^{2}}}\end{aligned}}} It can be confirmed that x  +  y  = 1 for all values of s and t . These two charts provide

6077-423: 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

6180-475: Is two-dimensional, so each chart will map part of the sphere to an open subset of R 2 {\displaystyle \mathbb {R} ^{2}} . Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by χ ( x , y , z ) = ( x , y ) ,   {\displaystyle \chi (x,y,z)=(x,y),\ } maps

6283-559: 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

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6386-550: 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

6489-580: 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

6592-1807: 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 ,

6695-554: The image of each chart is an open subset of n -dimensional Euclidean space , then M {\displaystyle M} is said to be an n -dimensional manifold . The plural of atlas is atlases , although some authors use atlantes . An atlas ( U i , φ i ) i ∈ I {\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}} on an n {\displaystyle n} -dimensional manifold M {\displaystyle M}

6798-528: 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

6901-457: 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

7004-519: The open ball B n = { ( x 1 , x 2 , … , x n ) ∈ R n : x 1 2 + x 2 2 + ⋯ + x n 2 < 1 } . {\displaystyle \mathbf {B} ^{n}=\left\{(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1\right\}.} This implies also that every point has

7107-497: 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

7210-437: 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

7313-426: 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

7416-402: The Euclidean space, this defines coordinates on U {\displaystyle U} : the coordinates of a point P {\displaystyle P} of U {\displaystyle U} are defined as the coordinates of φ ( P ) . {\displaystyle \varphi (P).} The pair formed by a chart and such a coordinate system is called

7519-426: The atlas is said to be C k {\displaystyle C^{k}} . Very generally, if each transition function belongs to a pseudogroup G {\displaystyle {\mathcal {G}}} of homeomorphisms of Euclidean space, then the atlas is called a G {\displaystyle {\mathcal {G}}} -atlas. If the transition maps between charts of an atlas preserve

7622-462: 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)

7725-539: The circle: χ b o t t o m ( x , y ) = x χ l e f t ( x , y ) = y χ r i g h t ( x , y ) = y . {\displaystyle {\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}} Together, these parts cover

7828-430: The co-domain of χ t o p {\displaystyle \chi _{\mathrm {top} }} back to the circle using the inverse, followed by χ r i g h t {\displaystyle \chi _{\mathrm {right} }} back to the interval. If a is any number in ( 0 , 1 ) {\displaystyle (0,1)} , then: T (

7931-565: 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),

8034-554: The desired structure. For a topological manifold, the simple space is a subset of some Euclidean space R n {\displaystyle \mathbb {R} ^{n}} and interest focuses on the topological structure. This structure is preserved by homeomorphisms , invertible maps that are continuous in both directions. In the case of a differentiable manifold, a set of charts called an atlas , whose transition functions (see below) are all differentiable, allows us to do calculus on it. Polar coordinates , for example, form

8137-518: 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

8240-452: 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

8343-405: 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

8446-436: The first coordinate is a continuous and invertible mapping from the upper arc to the open interval (−1, 1): χ t o p ( x , y ) = x . {\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,} Such functions along with the open regions they map are called charts . Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of

8549-488: The interval ( 0 , 1 ) {\displaystyle (0,1)} , though differently. Thus a function T : ( 0 , 1 ) → ( 0 , 1 ) = χ r i g h t ∘ χ t o p − 1 {\displaystyle T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}} can be constructed, which takes values from

8652-494: The manifold as a whole. Formally, a (topological) manifold is a second countable Hausdorff space that is locally homeomorphic to a Euclidean space. Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense 'too large' such as the long line , while Hausdorff excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in non-Hausdorff manifolds ). Locally homeomorphic to

8755-413: The manifold. Generally manifolds are taken to have a constant local dimension, and the local dimension is then called the dimension of the manifold. This is, in particular, the case when manifolds are connected . However, some authors admit manifolds that are not connected, and where different points can have different dimensions . If a manifold has a fixed dimension, this can be emphasized by calling it

8858-411: The map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure. A coordinate map , a coordinate chart , or simply a chart , of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve

8961-408: 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

9064-677: The need to associate pictures with coordinates (e.g. CT scans ). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds ; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics , while four-dimensional Lorentzian manifolds model spacetime in general relativity . The study of manifolds requires working knowledge of calculus and topology . After

9167-444: The northern hemisphere to the open unit disc by projecting it on the ( x , y ) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the ( x , z ) plane and two charts projecting on the ( y , z ) plane, an atlas of six charts is obtained which covers the entire sphere. This can be easily generalized to higher-dimensional spheres. A manifold can be constructed by gluing together pieces in

9270-404: The notion of a chart . A chart for a topological space M is a homeomorphism φ {\displaystyle \varphi } from an open subset U of M to an open subset of a Euclidean space . The chart is traditionally recorded as the ordered pair ( U , φ ) {\displaystyle (U,\varphi )} . When a coordinate system is chosen in

9373-515: The number of pieces. Informally, a manifold is a space that is "modeled on" Euclidean space. There are many different kinds of manifolds. In geometry and topology , all manifolds are topological manifolds , possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions: homeomorphisms from one region of Euclidean space to another region if they correspond to

9476-467: 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

9579-461: The plane z = 0 divides the sphere into two half spheres ( z > 0 and z < 0 ), which may both be mapped on the disc x + y < 1 by the projection on the xy plane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes. As with the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but

9682-398: 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

9785-420: The same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on

9888-474: The same) open ball in R n {\displaystyle \mathbb {R} ^{n}} . The resultant map, like the map T in the circle example above, is called a change of coordinates , a coordinate transformation , a transition function , or a transition map . An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure,

9991-408: 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

10094-525: 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

10197-486: The sphere cannot be covered by a single chart. This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the Earth cannot have a plane representation consisting of a single map (also called "chart", see nautical chart ), and therefore one needs atlases for covering the whole Earth surface. Manifolds need not be connected (all in "one piece"); an example

10300-410: The structure transfers to the manifold. This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of R n {\displaystyle \mathbb {R} ^{n}} (that is, if they are diffeomorphisms ), the differential structure transfers to the manifold and turns it into

10403-444: The transition map τ α , β {\displaystyle \tau _{\alpha ,\beta }} is also a homeomorphism. One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable . Such

10506-420: The use of pseudogroups . A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an n {\displaystyle n} -manifold with boundary is an ( n − 1 ) {\displaystyle (n-1)} -manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary

10609-566: The whole circle, and the four charts form an atlas for the circle. The top and right charts, χ t o p {\displaystyle \chi _{\mathrm {top} }} and χ r i g h t {\displaystyle \chi _{\mathrm {right} }} respectively, overlap in their domain: their intersection lies in the quarter of the circle where both x {\displaystyle x} and y {\displaystyle y} -coordinates are positive. Both map this part into

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