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102-412: In geometry , 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

204-400: 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

306-380: 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}

408-612: 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

510-490: 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

612-860: 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

714-412: A Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity

816-417: 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

918-463: 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,

1020-490: 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

1122-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

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1224-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

1326-424: 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

1428-411: A point on a circle or sphere , the intersection between the sphere and a ray in that direction emanating from the sphere's center; the tips of unit vectors emanating from a common origin point lie on the unit sphere . A Cartesian coordinate system is defined in terms of several oriented reference lines, called coordinate axes ; any arbitrary direction can be represented numerically by finding

1530-419: 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

1632-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

1734-448: A Cartesian coordinate system, can be represented numerically by its slope . A direction is used to represent linear objects such as axes of rotation and normal vectors . A direction may be used as part of the representation of a more complicated object 's orientation in physical space (e.g., axis–angle representation ). Two directions are said to be opposite if the unit vectors representing them are additive inverses , or if

1836-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

1938-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

2040-436: 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 ,

2142-430: 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

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2244-415: 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

2346-426: 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

2448-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

2550-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

2652-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in

2754-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

2856-518: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

2958-413: A polar angle relative to a fixed polar axis and an azimuthal angle about the polar axis: the angular components of spherical coordinates . Non-oriented straight lines can also be considered to have a direction, the common characteristic of all parallel lines , which can be made to coincide by translation to pass through a common point. The direction of a non-oriented line in a two-dimensional plane, given

3060-469: A problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in

3162-598: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

Line (geometry) - Misplaced Pages Continue

3264-505: 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

3366-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

3468-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

3570-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

3672-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

3774-417: 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

3876-492: A unique line) that make them suitable representations for lines in this geometry. Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure') is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of

3978-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

4080-425: 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

4182-563: Is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

Line (geometry) - Misplaced Pages Continue

4284-433: Is 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

4386-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

4488-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood

4590-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

4692-495: 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

4794-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

4896-402: 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

4998-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

5100-417: 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

5202-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

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5304-437: 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

5406-528: 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

5508-726: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

5610-559: 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

5712-550: 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

5814-667: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

5916-518: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

6018-567: 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

6120-506: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

6222-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

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6324-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

6426-450: The direction cosines (a list of cosines of the angles) between the given direction and the directions of the axes; the direction cosines are the coordinates of the associated unit vector. A two-dimensional direction can also be represented by its angle , measured from some reference direction, the angular component of polar coordinates (ignoring or normalizing the radial component). A three-dimensional direction can be represented using

6528-1807: 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 ,

6630-528: 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

6732-457: 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

6834-497: 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

6936-437: 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

7038-426: 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

7140-569: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing

7242-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by

7344-584: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

7446-462: 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)

7548-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

7650-504: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

7752-563: 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),

7854-465: 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

7956-452: 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

8058-405: 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

8160-512: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

8262-523: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

8364-588: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

8466-533: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

8568-546: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

8670-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of

8772-429: 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. In differential geometry,

8874-408: 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

8976-465: The nature of geometric structures modelled on, or arising out of, the complex plane . Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables , and has found applications to string theory and mirror symmetry . Direction (geometry) Two vectors sharing the same direction are said to be codirectional or equidirectional . All co directional line segments sharing

9078-755: The oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem ,

9180-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

9282-460: 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

9384-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

9486-401: 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 is

9588-401: The points on a sphere representing them are antipodal , at the two opposite ends of a common diameter. Two directions are parallel (as in parallel lines ) if they can be brought to lie on the same straight line without rotations; parallel directions are either codirectional or opposite. Two directions are obtuse or acute if they form, respectively, an obtuse angle (greater than

9690-478: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

9792-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid

9894-440: The same size (length) are said to be equipollent . Two equipollent segments are not necessarily coincident; for example, a given direction can be evaluated at different starting positions , defining different unit directed line segments (as a bound vector instead of a free vector ). A direction is often represented as a unit vector , the result of dividing a vector by its length. A direction can alternately be represented by

9996-408: 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

10098-516: 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

10200-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

10302-420: 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

10404-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

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