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91-566: [REDACTED] Look up rectilinear in Wiktionary, the free dictionary. Rectilinear means related to a straight line ; it may refer to: Rectilinear grid , a tessellation of the Euclidean plane Rectilinear lens , a photographic lens Rectilinear locomotion , a form of animal locomotion Rectilinear polygon , a polygon whose edges meet at right angles Rectilinear propagation ,

182-400: 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

273-380: 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}

364-612: 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

455-490: 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

546-861: 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

637-417: 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

728-411: 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,

819-496: 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

910-405: A plane is described with Cartesian coordinates : The points are sometimes identified with generalized complex numbers z = x + y ε where ε ∈ { –1, 0, 1}. The Euclidean plane corresponds to the case ε = −1 , an imaginary unit . Since the modulus of z is given by For instance, { z | z z * = 1} is the unit circle . For planar algebra, non-Euclidean geometry arises in

1001-426: 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

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1092-419: 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

1183-410: A complex number z . Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in 1908. Minkowski introduced terms like worldline and proper time into mathematical physics . He realized that the submanifold , of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions. Already in

1274-436: 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 ,

1365-402: A geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter  k . Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of

1456-422: A given line l through a point that is not on l . In hyperbolic geometric models, by contrast, there are infinitely many lines through A parallel to l , and in elliptic geometric models, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.) Euclidean geometry is modelled by our notion of a "flat plane ." The simplest model for elliptic geometry

1547-453: A highly-elliptical orbit around a Lagrangian point of a moon, that due to the moons orbital movement, will be nearly rectilinear in some frames of reference. See also [ edit ] Linear (disambiguation) Straight (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Rectilinear . If an internal link led you here, you may wish to change

1638-455: A limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks to prove all the other results ( propositions ) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate", or simply the parallel postulate , which in Euclid's original formulation is: If a straight line falls on two straight lines in such a manner that

1729-430: 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

1820-415: 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

1911-428: 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

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2002-510: A model of the acute case on a sphere of imaginary radius. He did not carry this idea any further. At this time it was widely believed that the universe worked according to the principles of Euclidean geometry. The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had

2093-401: A new viable geometry, but did not realize it. In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure now known as a Lambert quadrilateral , a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that

2184-485: A non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation . Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways: Models of non-Euclidean geometry are mathematical models of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn parallel to

2275-495: A pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry . As the first 28 propositions of Euclid (in The Elements ) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry. To obtain

2366-481: A possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry. He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered

2457-468: A property of waves Rectilinear Research Corporation , a now defunct manufacturer of high-end loudspeakers Rectilinear style , the third historical division of English Gothic architecture Rectilinear motion or linear motion is motion along a straight line Rectilinear prophecy , where a straight line can be drawn from the prophecy to the fulfillment without any branches as in the case of typological interpretations Near-rectilinear halo orbit ,

2548-505: 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

2639-412: A special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to

2730-432: 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 the end of the 19th century, such as non-Euclidean , projective , and affine geometry . In

2821-425: A term that generally fell out of use ). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry. There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways. There are many kinds of geometry that are quite different from Euclidean geometry but are also not necessarily included in

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2912-531: A third line (in the same plane): Euclidean geometry , named after the Greek mathematician Euclid , includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote Elements . In the Elements , Euclid begins with

3003-469: A thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction , including Ibn al-Haytham (Alhazen, 11th century), Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī (13th century), and Giovanni Girolamo Saccheri (18th century). The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals , including

3094-419: 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

3185-419: A unique line) that make them suitable representations for lines in this geometry. Non-Euclidean geometry The essential difference between the metric geometries is the nature of parallel lines. Euclid 's fifth postulate, the parallel postulate , is equivalent to Playfair's postulate , which states that, within a two-dimensional plane, for any given line l and a point A , which is not on l , there

3276-425: 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

3367-495: Is a sphere, where lines are " great circles " (such as the equator or the meridians on a globe ), and points opposite each other are identified (considered to be the same). The pseudosphere has the appropriate curvature to model hyperbolic geometry. The simplest model for elliptic geometry is a sphere, where lines are " great circles " (such as the equator or the meridians on a globe ), and points opposite each other (called antipodal points ) are identified (considered

3458-447: 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 a part of a line delimited by two points (its endpoints ). Euclid's Elements defines

3549-413: Is exactly one line through A that does not intersect l . In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l , while in elliptic geometry, any line through A intersects l . Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to

3640-423: Is impossible for two convergent straight lines to diverge in the direction in which they converge." Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid, which he didn't realize

3731-442: Is not on l , there are infinitely many lines through A that do not intersect l . In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of

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3822-496: 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

3913-402: 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

4004-421: 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

4095-437: 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

4186-528: 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

4277-423: 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

4368-559: 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

4459-550: 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

4550-457: The Cayley–Klein metric because Felix Klein exploited it to describe the non-Euclidean geometries in articles in 1871 and 1873 and later in book form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry. Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry parabolic ,

4641-422: 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 the definitions are never explicitly referenced in

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4732-469: The Hungarian mathematician János Bolyai separately and independently published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such

4823-685: The Lambert quadrilateral and Saccheri quadrilateral , were "the first few theorems of the hyperbolic and the elliptic geometries ". These theorems along with their alternative postulates, such as Playfair's axiom , played an important role in the later development of non-Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo , Levi ben Gerson , Alfonso , John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry, however, provided flawed proofs of

4914-580: 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

5005-1807: 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 ,

5096-404: The mathematical model of space . Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority. The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant 's treatment of human knowledge had

5187-528: 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

5278-457: 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

5369-497: 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

5460-437: 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

5551-426: 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

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5642-431: The 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions , though Macfarlane did not use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry. The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number z = e can represent

5733-477: The approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff , for instance, uses the axiom that says that, "There exists

5824-462: 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)

5915-424: The conventional meaning of "non-Euclidean geometry", such as more general instances of Riemannian geometry . Euclidean geometry can be axiomatically described in several ways. However, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms. Hilbert's system consisting of 20 axioms most closely follows

6006-409: The entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.) In the hyperbolic model, within a two-dimensional plane, for any given line l and a point A , which

6097-452: 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

6188-405: 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

6279-408: The fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of

6370-495: The germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew Franz Taurinus did publish important results of hyperbolic trigonometry in two papers in 1825 and 1826, yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry. Then, in 1829–1830 the Russian mathematician Nikolai Ivanovich Lobachevsky and in 1832

6461-519: The intellectual life of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements . This curriculum issue was hotly debated at the time and was even the subject of a book, Euclid and his Modern Rivals , written by Charles Lutwidge Dodgson (1832–1898) better known as Lewis Carroll , the author of Alice in Wonderland . In analytic geometry

6552-410: The interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears more complicated than Euclid's other postulates : For at least

6643-427: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Rectilinear&oldid=962804259 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Straight line In geometry , a straight line , usually abbreviated line ,

6734-408: 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

6825-494: The nature of parallelism. This commonality is the subject of absolute geometry (also called neutral geometry ). However, the properties that distinguish one geometry from others have historically received the most attention. Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following: Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as

6916-512: The other cases. When ε = +1 , a hyperbolic unit . Then z is a split-complex number and conventionally j replaces epsilon. Then and { z | z z * = 1} is the unit hyperbola . When ε = 0 , then z is a dual number . This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of

7007-523: 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

7098-484: The parallel postulate, depending on assumptions that are now recognized as essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" ( Aristotle ): "Two convergent straight lines intersect and it

7189-406: The physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences. Bernhard Riemann , in a famous lecture in 1854, founded the field of Riemannian geometry , discussing in particular the ideas now called manifolds , Riemannian metric , and curvature . He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on

7280-398: 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

7371-514: 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),

7462-417: The same). This is also one of the standard models of the real projective plane . The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric. In the elliptic model, for any given line l and a point A , which is not on l , all lines through A will intersect l . Even after

7553-408: 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

7644-472: 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

7735-399: The unit ball in Euclidean space . The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines. By formulating the geometry in terms of a curvature tensor , Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative curvature. It

7826-528: The way they are represented. In three dimensions, there are eight models of geometries. There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry that is completely anisotropic (i.e. every direction behaves differently). Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon

7917-463: The work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for hyperbolic geometry ?". The model for hyperbolic geometry was answered by Eugenio Beltrami , in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model , which models

8008-414: The world around it, that was a result of this paradigm shift. Non-Euclidean geometry is an example of a scientific revolution in the history of science , in which mathematicians and scientists changed the way they viewed their subjects. Some geometers called Lobachevsky the " Copernicus of Geometry" due to the revolutionary character of his work. The existence of non-Euclidean geometries impacted

8099-449: Was Gauss who coined the term "non-Euclidean geometry". He was referring to his own work, which today we call hyperbolic geometry or Lobachevskian geometry . Several modern authors still use the generic term non-Euclidean geometry to mean hyperbolic geometry . Arthur Cayley noted that distance between points inside a conic could be defined in terms of logarithm and the projective cross-ratio function. The method has become called

8190-411: Was equivalent to his own postulate. Another example is al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented another hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements ." His work

8281-625: Was published in Rome in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis. Giordano Vitale , in his book Euclide restituo (1680, 1686), used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. In a work titled Euclides ab Omni Naevo Vindicatus ( Euclid Freed from All Flaws ), published in 1733, Saccheri quickly discarded elliptic geometry as

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