70:
5724:
2293:
4640:
5435:
5371:
2035:
4095:
5689:
44:
1879:
1337:) 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 the remainder of the text. In modern geometry, a line is usually either taken as a
2042:
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
2258:
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
5700:
is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same
1896:
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 the other slopes). By extension,
1711:
5306:
3054:
4001:
5356:, 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.
5394:
passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.
4622:
2892:
1553:
4544:
4861:
2763:
4794:
4728:
4321:
3826:
4996:
5241:
3891:
4182:
2897:
5044:
3379:
2250:
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 a
3607:
3527:
4125:
perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by:
2405:
4936:
5344:
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
3886:
1874:{\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}}}
3725:
3666:
3440:
2638:
2582:
5126:
5088:
3293:
3253:
3204:
2463:
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
5650:
The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typically
4459:
4433:
2521:
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,
5236:
5180:
1435:
defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in
2512:
1368:(modern mathematicians added to Euclid's original axioms to fill perceived logical gaps), a line is stated to have certain properties that relate it to other lines and
4889:
4479:
4376:
4343:
4202:
5200:
4252:
3101:
3852:
2267:), 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.
2322:
4080:
Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector.
2784:
1480:
5321:
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in
1372:. 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
2643:
4572:
4743:
4673:
4257:
2202:
share a single point in common. Coincidental lines coincide with each other—every point that is on either one of them is also on the other.
4945:
4098:
Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue.
2243:, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the properties of lines are dictated by the
5390:
of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by
Euclidean
6181:
5739:
are evenly spaced on the line, with positive numbers are on the right, negative numbers on the left. As an extension to the concept, an
5478:, in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required.
4501:
4128:
1198:
4806:
5958:
5382:, the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In
5150:
The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates
5140:
3531:
3451:
1272:
defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several
5701:
relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are
4325:
Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters,
1399:, a line can be represented as a boundary between two regions. Any collection of finitely many lines partitions the plane into
2339:
6311:
5927:
5470:. The point A is considered to be a member of the ray. Intuitively, a ray consists of those points on a line passing through
2518:
of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms.
298:
3730:
5774:
5147:
that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides.
6112:
6256:
6232:
6208:
6156:
6073:
6005:
5901:
264:
5428:
5008:
3298:
1288:
are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as
2239:, meaning it is not being defined by other concepts. In those situations where a line is a defined concept, as in
5779:
5386:
we see a typical example of this. In the spherical representation of elliptic geometry, lines are represented by
4643:
A line on polar coordinates without passing though the origin, with the general parametric equation written above
2470:
1191:
1145:
751:
210:
5374:
A great circle divides the sphere in two equal hemispheres, while also satisfying the "no curvature" property.
6385:
6295:
6282:
5976:
4901:
17:
5301:{\displaystyle r\geq 0,\qquad {\text{and}}\quad \theta =\alpha \quad {\text{or}}\quad \theta =\alpha +\pi .}
3071:
6411:
5740:
2083:
6380:
6277:
5129:
3671:
3612:
2189:
1447:
1358:
52:
3384:
2587:
2531:
6416:
6200:
6104:
3049:{\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}}}\,.}
2263:
falls into this category. Even in the case where a specific geometry is being considered (for example,
1166:
776:
5093:
5055:
4216:
is the (positive) length of the normal segment. The normal form can be derived from the standard form
6392:
3258:
1184:
3209:
3160:
6426:
6197:
The
Student's Introduction to MATHEMATICA: A Handbook for Precalculus, Calculus, and Linear Algebra
4438:
4412:
3062:
153:
6375:
6034:
4204:
is the angle of inclination of the normal segment (the oriented angle from the unit vector of the
5209:
5153:
4631:
is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0.
3873:
2216:
2155:
1416:
579:
259:
116:
3445:
As a note, lines in three dimensions may also be described as the simultaneous solutions of two
1384:. In higher dimensions, two lines that do not intersect are parallel if they are contained in a
6029:
3996:{\displaystyle {\begin{aligned}x&=x_{0}+at\\y&=y_{0}+bt\\z&=z_{0}+ct\end{aligned}}}
2199:
2102:
1289:
655:
366:
244:
129:
6081:
5727:
A number line, with variable x on the left and y on the right. Therefore, x is smaller than y.
5325:, a line in the plane is often defined as the set of points whose coordinates satisfy a given
4874:
4464:
4361:
4328:
4187:
6175:
5995:
5764:
5467:
5345:
5185:
4667:
4219:
2325:
2038:
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
1882:
1420:
427:
388:
347:
342:
195:
37:
6358:
6272:
5789:
4733:
4547:
4122:
3135:
2260:
1706:
1556:
1404:
1313:
1266:
1095:
1018:
866:
771:
293:
188:
102:
48:
8:
5804:
5663:
5403:
5379:
5353:
3867:
3854:). This follows since in three dimensions a single linear equation typically describes a
3831:
3057:
2240:
2205:
2195:
2098:
1381:
1377:
1365:
1293:
1236:
1100:
1044:
957:
811:
791:
716:
606:
477:
467:
330:
205:
200:
183:
158:
146:
98:
93:
74:
5333:, a line may be an independent object, distinct from the set of points which lie on it.
6346:
6145:
6047:
5834:
5799:
5651:
5330:
4061:
2522:
2307:
2301:
2264:
2134:
2023:
2012:
1913:
1909:
1617:
1284:
1059:
786:
626:
254:
178:
168:
139:
124:
1609:
if they lie on the same line. If three points are not collinear, there is exactly one
6421:
6307:
6252:
6228:
6204:
6152:
6126:
6118:
6108:
6069:
6001:
5991:
5933:
5923:
5897:
5838:
5383:
5322:
4652:
4116:
4089:
2105:
is a line that intersects two other lines that may or not be parallel to each other.
1354:
1346:
1309:
1130:
918:
896:
821:
680:
335:
227:
173:
134:
5723:
1120:
1049:
846:
756:
6338:
6039:
5971:
5950:
5809:
5744:
5398:
The "shortness" and "straightness" of a line, interpreted as the property that the
5391:
5341:
3855:
2276:
2236:
2232:
2086:, whose distance from a point helps to establish whether the point is on the conic.
2079:
exterior lines, which do not meet the conic at any point of the
Euclidean plane; or
2019:
1610:
1385:
1369:
1338:
1256:
1110:
851:
561:
439:
374:
232:
217:
82:
6171:
6020:
Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the
Projective Plane",
5850:
On occasion we may consider a ray without its initial point. Such rays are called
5493:. As two points define a unique line, this ray consists of all the points between
6354:
5784:
5655:
5326:
5316:
4648:
3446:
2287:
2109:
2090:
1889:= 2), the above matrix is square and the points are collinear if and only if its
1454:
1396:
1297:
533:
239:
222:
163:
69:
33:
6177:
Plane
Analytic Geometry: With Introductory Chapters on the Differential Calculus
2435:
are not both zero. Using this form, vertical lines correspond to equations with
1105:
1074:
1008:
856:
801:
736:
6326:
5752:
5747:
can be drawn perpendicular to the number line at zero. The two lines forms the
5667:
5605:
5144:
2223:
are lines that are not in the same plane and thus do not intersect each other.
2163:
1400:
1161:
1069:
1013:
978:
886:
796:
766:
726:
631:
3872:
Parametric equations are also used to specify lines, particularly in those in
1276:
as basic unprovable properties on which the rest of geometry was established.
1135:
746:
6405:
5937:
5748:
5659:
4639:
4617:{\displaystyle \mathbf {r} =\mathbf {a} +\lambda (\mathbf {b} -\mathbf {a} )}
2166:
2044:
1140:
1125:
1054:
871:
831:
781:
556:
519:
486:
324:
320:
3883:
In three dimensions lines are frequently described by parametric equations:
6396:
6086:
5794:
5706:
5697:
5683:
5671:
5411:
5387:
5370:
4868:
2067:
1600:
1462:
1334:
1252:
1232:
1228:
1079:
1028:
841:
696:
611:
401:
5917:
5182:
of the points of a line passing through the origin and making an angle of
6303:
6130:
5732:
5718:
5640:). These are not opposite rays since they have different initial points.
4121:
for a given line, which is defined to be the line segment drawn from the
3125:
2424:
2420:
2209:
2188:
and, in the special case where the conic is a pair of lines, we have the
2185:
2170:
2149:
2073:
1890:
1350:
1115:
988:
806:
741:
669:
641:
616:
4871:
perpendicular to the line and delimited by the origin and the line, and
2292:
2076:, which intersect the conic at two points and pass through its interior;
6350:
6051:
4111:
2220:
2142:
1389:
973:
952:
942:
932:
891:
836:
731:
721:
621:
472:
4732:
In polar coordinates, the equation of a line not passing through the
4094:
2887:{\displaystyle y=(x-x_{0})\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+y_{0}}
2126:
2060:
2034:
1373:
1273:
1244:
1240:
1224:
983:
701:
664:
528:
500:
6342:
6043:
5450:
as decomposing this line into two parts. Each such part is called a
5427:"Ray (geometry)" redirects here. For other uses in mathematics, see
2304:, are characterized by linear equations. More precisely, every line
5814:
5702:
5407:
5399:
5365:
5349:
5136:
4498:
The vector equation of the line through points A and B is given by
2174:
2129:, which a curve approaches arbitrarily closely without touching it.
2056:
1548:{\displaystyle L=\left\{(1-t)\,a+tb\mid t\in \mathbb {R} \right\}.}
1364:
In an axiomatic formulation of
Euclidean geometry, such as that of
1212:
1064:
1023:
993:
881:
876:
826:
551:
510:
458:
352:
315:
61:
4539:{\displaystyle \mathbf {r} =\mathbf {OA} +\lambda \,\mathbf {AB} }
3858:
and a line is what is common to two distinct intersecting planes.
5736:
2181:
2052:
1935:
may be used to express the collinearity between three points by:
998:
711:
505:
449:
249:
6122:
5438:
A ray with a terminus at A, with two points B and C on the right
5434:
2467:. If the constant term is put on the left, the equation becomes
5955:
Un nouveau système de définitions pour la géométrie euclidienne
5854:
rays, in contrast to the typical ray which would be said to be
5402:
along the line between any two of its points is minimized (see
4898:
It may be useful to express the equation in terms of the angle
2048:
947:
937:
816:
761:
636:
599:
587:
542:
495:
413:
78:
5643:
In
Euclidean geometry two rays with a common endpoint form an
5340:, the notion of a line is usually left undefined (a so-called
4856:{\displaystyle \varphi -\pi /2<\theta <\varphi +\pi /2.}
4060:
are related to the slope of the line, such that the direction
2758:{\displaystyle (y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0}).}
2528:
The equation of the line passing through two different points
5769:
5644:
5533:. This is, at times, also expressed as the set of all points
5337:
3115:
3111:
2244:
1342:
1325:
1003:
927:
861:
706:
310:
305:
6151:(2nd ed.), New York: John Wiley & Sons, p. 4,
3060:, the equation for non-vertical lines is often given in the
1885:
less than 3. In particular, for three points in the plane (
1349:
of points obeying a linear relationship, for instance when
594:
444:
5688:
5604:, define a line and a decomposition of this line into the
2324:(including vertical lines) is the set of all points whose
2011:
However, there are other notions of distance (such as the
2259:
formal proofs of statements. The "definition" of line in
2231:
The concept of line is often considered in geometry as a
43:
5310:
4385:. On the other hand, if the line is through the origin (
2026:, other methods of determining collinearity are needed.
1575:= 1), or in other words, in the direction of the vector
1223:, is an infinitely long object with no width, depth, or
4942:-axis and the line. In this case, the equation becomes
2296:
Line graphs of linear equations on the
Cartesian plane
1247:
in spaces of dimension two, three, or higher. The word
4789:{\displaystyle r={\frac {p}{\cos(\theta -\varphi )}},}
4723:{\displaystyle x=r\cos \theta ,\quad y=r\sin \theta .}
4316:{\displaystyle {\frac {c}{|c|}}{\sqrt {a^{2}+b^{2}}}.}
3821:{\displaystyle a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}}
1720:
1353:
are taken to be primitive and geometry is established
5666:
nor in a geometry over a non-ordered field, like the
5244:
5212:
5188:
5156:
5096:
5058:
5011:
4991:{\displaystyle r={\frac {p}{\sin(\theta -\alpha )}},}
4948:
4904:
4877:
4809:
4746:
4676:
4575:
4504:
4467:
4441:
4415:
4364:
4331:
4260:
4222:
4190:
4131:
3889:
3834:
3733:
3674:
3615:
3534:
3454:
3387:
3301:
3261:
3212:
3163:
3074:
2900:
2787:
2646:
2590:
2534:
2473:
2342:
2310:
1714:
1483:
6195:
Torrence, Bruce F.; Torrence, Eve A. (29 Jan 2009),
1901:
points in a plane are collinear if and only if any (
1403:(possibly unbounded); this partition is known as an
6294:
6066:
Charming Proofs: A Journey Into
Elegant Mathematics
5894:
Foundations of Euclidean and Non-Euclidean Geometry
5632:is not drawn in the diagram, but is to the left of
5429:
Ray (disambiguation) § Science and mathematics
2173:is the line that connects the midpoints of the two
2018:In the geometries where the concept of a line is a
1905:–1) pairs of points have the same pairwise slopes.
1450:variables define a line under suitable conditions.
6144:
5406:), can be generalized and leads to the concept of
5300:
5230:
5194:
5174:
5120:
5082:
5038:
4990:
4930:
4883:
4855:
4788:
4722:
4616:
4538:
4473:
4453:
4427:
4370:
4337:
4315:
4246:
4196:
4177:{\displaystyle x\cos \varphi +y\sin \varphi -p=0,}
4176:
3995:
3876:or more because in more than two dimensions lines
3846:
3820:
3719:
3660:
3601:
3521:
3434:
3373:
3287:
3247:
3198:
3095:
3048:
2886:
2757:
2632:
2576:
2506:
2399:
2316:
2300:Lines in a Cartesian plane or, more generally, in
1873:
1547:
5489:, they determine a unique ray with initial point
6403:
6194:
5922:, New York: Continuum International Pub. Group,
5577:, will determine another ray with initial point
4569:, then the equation of the line can be written:
1227:, an idealization of such physical objects as a
6227:, New York: McGraw-Hill, p. 59, definition 3,
5990:
5692:Drawing of a line segment "AB" on the line "a"
5596:Thus, we would say that two different points,
5352:(shortest path between points), while in some
4018:are all functions of the independent variable
2198:are lines in the same plane that never cross.
1380:), two lines that do not intersect are called
1329:) is defined as a "breadthless length", and a
6000:, Jones & Bartlett Learning, p. 62,
5751:, a geometrical representation of the set of
5039:{\displaystyle 0<\theta <\alpha +\pi .}
3381:and the equation of this line can be written
3374:{\displaystyle m=(y_{b}-y_{a})/(x_{b}-x_{a})}
1255:, which is a part of a line delimited by two
1192:
6063:
3602:{\displaystyle a_{2}x+b_{2}y+c_{2}z-d_{2}=0}
3522:{\displaystyle a_{1}x+b_{1}y+c_{1}z-d_{1}=0}
2525:, known points on the line and y-intercept.
2391:
2349:
2184:with vertices lying on a conic we have the
6019:
5662:. On the other hand, rays do not exist in
5329:, but in a more abstract setting, such as
3880:be described by a single linear equation.
2070:, which touch the conic at a single point;
1199:
1185:
68:
6325:
6298:; Redlin, Lothar; Watson, Saleem (2008),
6222:
6064:Alsina, Claudi; Nelsen, Roger B. (2010),
6033:
5919:Resources for teaching mathematics, 14–16
5474:and proceeding indefinitely, starting at
5336:When a geometry is described by a set of
4527:
3042:
2910:
2816:
1533:
1510:
27:Straight figure with zero width and depth
5959:International Congress of Mathematicians
5731:A point on number line corresponds to a
5722:
5687:
5433:
5369:
5048:These equations can be derived from the
4638:
4488:
4093:
2400:{\displaystyle L=\{(x,y)\mid ax+by=c\},}
2291:
2033:
42:
6270:
6142:
4931:{\displaystyle \alpha =\varphi +\pi /2}
4254:by dividing all of the coefficients by
2226:
2123:points counted without multiplicity, or
2015:) for which this property is not true.
1410:
1251:may also refer, in everyday life, to a
14:
6404:
6170:
5915:
5569:but not in the ray with initial point
5359:
5311:Generalizations of the Euclidean line
3861:
2336:) satisfy a linear equation; that is,
1559:of the line is from a reference point
299:Straightedge and compass constructions
6246:
6057:
5949:
5891:
3157:The slope of the line through points
2169:with at most two parallel sides, the
1469:passing through two different points
5887:
5885:
5883:
5881:
5879:
5877:
5875:
4634:
4083:
3727:are not proportional (the relations
2781:, this equation may be rewritten as
2119:-secant lines, meeting the curve in
1605:Three or more points are said to be
6098:
5970:
5775:Distance between two parallel lines
5705:and either do not intersect or are
5135:These equations can also be proven
4022:which ranges over the real numbers.
3720:{\displaystyle (a_{2},b_{2},c_{2})}
3661:{\displaystyle (a_{1},b_{1},c_{1})}
1594:
24:
5984:
3435:{\displaystyle y=m(x-x_{a})+y_{a}}
2633:{\displaystyle P_{1}(x_{1},y_{1})}
2577:{\displaystyle P_{0}(x_{0},y_{0})}
2281:
25:
6438:
6368:
6331:The American Mathematical Monthly
6251:, Mineola, NY: Dover, p. 2,
5872:
5348:, a line may be interpreted as a
4891:is the (oriented) angle from the
4110:, after the German mathematician
2514:and this is sometimes called the
1443:−1 first-degree equations in the
265:Noncommutative algebraic geometry
6249:Geometry: A Comprehensive Course
5121:{\displaystyle y=r\sin \theta ,}
5083:{\displaystyle x=r\cos \theta ,}
5052:of the line equation by setting
5049:
4867:is the (positive) length of the
4607:
4599:
4585:
4577:
4532:
4529:
4517:
4514:
4506:
1461:(and analogously in every other
6329:(1941), "The inversive plane",
6319:
6288:
6264:
6240:
6216:
6188:
6184:from the original on 2016-05-13
6164:
6136:
5997:Calculus with Analytic Geometry
5844:
5780:Distance from a point to a line
5677:
5279:
5273:
5263:
5257:
4698:
3288:{\displaystyle x_{a}\neq x_{b}}
2442:One can further suppose either
2093:, a linear coordinate dimension
32:For the graphical concept, see
6393:Equations of the Straight Line
6092:
6013:
5964:
5943:
5909:
5827:
5712:
5225:
5213:
5169:
5157:
4979:
4967:
4777:
4765:
4611:
4595:
4276:
4268:
3714:
3675:
3655:
3616:
3416:
3397:
3368:
3342:
3334:
3308:
3248:{\displaystyle B(x_{b},y_{b})}
3242:
3216:
3199:{\displaystyle A(x_{a},y_{a})}
3193:
3167:
2813:
2794:
2749:
2730:
2727:
2701:
2695:
2669:
2666:
2647:
2627:
2601:
2571:
2545:
2364:
2352:
2097:In the context of determining
1624:-dimensional space the points
1507:
1495:
658:- / other-dimensional
13:
1:
5994:; Protter, Philip E. (1988),
5977:The Principles of Mathematics
5865:
5417:
4454:{\displaystyle \cos \varphi }
4428:{\displaystyle \sin \varphi }
2270:
2022:, as may be the case in some
1951:are collinear if and only if
1303:
5561:, on the line determined by
4736:—the point with coordinates
4670:by the parametric equations:
4210:-axis to this segment), and
2456:, by dividing everything by
2208:are lines that intersect at
7:
6381:Encyclopedia of Mathematics
6278:Encyclopedia of Mathematics
5896:, New York: Marcel Dekker,
5758:
5442:Given a line and any point
5231:{\displaystyle (r,\theta )}
5175:{\displaystyle (r,\theta )}
4378:is uniquely defined modulo
4046:) is any point on the line.
53:Cartesian coordinate system
10:
6443:
6201:Cambridge University Press
6105:Holt, Rinehart and Winston
5892:Faber, Richard L. (1983),
5716:
5681:
5537:on the line determined by
5426:
5363:
5314:
5143:of sine and cosine to the
5141:right triangle definitions
4493:
4087:
4076:) is parallel to the line.
3865:
2507:{\displaystyle ax+by-c=0,}
2285:
2274:
1598:
31:
6068:, MAA, pp. 108–109,
5735:and vice versa. Usually,
5130:angle difference identity
2247:which they must satisfy.
2101:in Euclidean geometry, a
1591:can yield the same line.
1341:with properties given by
6271:Sidorov, L. A. (2001) ,
6223:Wylie Jr., C.R. (1964),
6147:Introduction to Geometry
5820:
4884:{\displaystyle \varphi }
4627:A ray starting at point
4474:{\displaystyle \varphi }
4371:{\displaystyle \varphi }
4338:{\displaystyle \varphi }
4197:{\displaystyle \varphi }
2029:
406:
154:Non-Archimedean geometry
6225:Foundations of Geometry
6180:, H. Holt, p. 44,
6143:Coxeter, H.S.M (1969),
5446:on it, we may consider
5195:{\displaystyle \alpha }
4895:-axis to this segment.
4481:is only defined modulo
4247:{\displaystyle ax+by=c}
2217:three-dimensional space
2112:, lines could also be:
1705:) are collinear if the
1583:. Different choices of
1417:three-dimensional space
1345:, or else defined as a
260:Noncommutative geometry
51:on the two-dimensional
6099:Kay, David C. (1969),
5916:Foster, Colin (2010),
5728:
5693:
5581:. With respect to the
5481:Given distinct points
5462:. It is also known as
5439:
5422:
5375:
5302:
5232:
5196:
5176:
5128:and then applying the
5122:
5084:
5040:
4992:
4932:
4885:
4857:
4790:
4724:
4644:
4618:
4540:
4475:
4461:, and it follows that
4455:
4429:
4372:
4351:, to be specified. If
4339:
4317:
4248:
4198:
4178:
4099:
3997:
3848:
3822:
3721:
3662:
3603:
3523:
3436:
3375:
3289:
3249:
3200:
3097:
3096:{\displaystyle y=mx+b}
3050:
2888:
2759:
2634:
2578:
2508:
2401:
2318:
2297:
2039:
1875:
1567:= 0) to another point
1549:
1357:in terms of numerical
1312:deductive geometry of
1231:, a taut string, or a
1219:, usually abbreviated
228:Discrete/Combinatorial
55:
36:. For other uses, see
5765:Affine transformation
5726:
5691:
5509:) and all the points
5437:
5373:
5364:Further information:
5354:projective geometries
5346:differential geometry
5303:
5233:
5206:-axis, are the pairs
5197:
5177:
5123:
5085:
5041:
4993:
4933:
4886:
4858:
4791:
4725:
4668:Cartesian coordinates
4642:
4619:
4541:
4489:Other representations
4476:
4456:
4430:
4373:
4340:
4318:
4249:
4199:
4179:
4097:
3998:
3866:Further information:
3849:
3823:
3722:
3663:
3604:
3524:
3437:
3376:
3290:
3250:
3201:
3098:
3051:
2889:
2760:
2635:
2579:
2509:
2402:
2319:
2295:
2037:
1927:) between two points
1876:
1550:
1421:first degree equation
1376:(i.e., the Euclidean
211:Discrete differential
46:
6022:Mathematics Magazine
5841:on the set of lines.
5790:Incidence (geometry)
5513:on the line through
5466:, a one-dimensional
5242:
5210:
5186:
5154:
5132:for sine or cosine.
5094:
5056:
5009:
4946:
4902:
4875:
4807:
4744:
4674:
4573:
4502:
4465:
4439:
4413:
4362:
4329:
4258:
4220:
4188:
4129:
3887:
3832:
3731:
3672:
3613:
3532:
3452:
3385:
3299:
3259:
3210:
3161:
3136:independent variable
3072:
3063:slope–intercept form
2898:
2785:
2644:
2588:
2532:
2471:
2340:
2308:
2227:In axiomatic systems
2024:synthetic geometries
1712:
1613:that contains them.
1481:
1411:In higher dimensions
1405:arrangement of lines
47:A red line near the
6412:Elementary geometry
6247:Pedoe, Dan (1988),
5664:projective geometry
5608:of an open segment
5404:triangle inequality
5380:projective geometry
5360:Projective geometry
4114:), is based on the
3868:Parametric equation
3862:Parametric equation
3847:{\displaystyle t=0}
2460:if it is not zero.
2241:coordinate geometry
2206:Perpendicular lines
1439:-dimensional space
478:Pythagorean theorem
6306:, pp. 13–19,
5992:Protter, Murray H.
5835:collineation group
5800:Generalised circle
5729:
5694:
5652:Euclidean geometry
5589:ray is called the
5440:
5378:In many models of
5376:
5331:incidence geometry
5298:
5228:
5192:
5172:
5118:
5080:
5036:
4988:
4928:
4881:
4853:
4786:
4720:
4645:
4614:
4536:
4471:
4451:
4425:
4368:
4335:
4313:
4244:
4194:
4174:
4100:
3993:
3991:
3844:
3818:
3717:
3658:
3599:
3519:
3432:
3371:
3285:
3245:
3196:
3093:
3046:
2884:
2755:
2640:may be written as
2630:
2574:
2504:
2397:
2314:
2302:affine coordinates
2298:
2265:Euclidean geometry
2200:Intersecting lines
2040:
2013:Manhattan distance
1914:Euclidean distance
1910:Euclidean geometry
1871:
1865:
1618:affine coordinates
1545:
1285:Euclidean geometry
1243:one, which may be
56:
6417:Analytic geometry
6313:978-0-495-56521-5
6296:Stewart, James B.
5972:Russell, Bertrand
5951:Padoa, Alessandro
5929:978-1-4411-3724-1
5833:Technically, the
5745:imaginary numbers
5384:elliptic geometry
5323:analytic geometry
5277:
5261:
4983:
4781:
4653:polar coordinates
4635:Polar coordinates
4396:), one drops the
4308:
4281:
4112:Ludwig Otto Hesse
4108:Hesse normal form
4106:(also called the
4090:Hesse normal form
4084:Hesse normal form
3040:
2963:
2869:
2317:{\displaystyle L}
2261:Euclid's Elements
2237:axiomatic systems
2108:For more general
2063:), lines can be:
1423:in the variables
1392:if they are not.
1209:
1208:
1174:
1173:
897:List of geometers
580:Three-dimensional
569:
568:
16:(Redirected from
6434:
6389:
6362:
6361:
6327:Patterson, B. C.
6323:
6317:
6316:
6302:(5th ed.),
6292:
6286:
6285:
6268:
6262:
6261:
6244:
6238:
6237:
6220:
6214:
6213:
6192:
6186:
6185:
6168:
6162:
6161:
6150:
6140:
6134:
6133:
6101:College Geometry
6096:
6090:
6078:
6061:
6055:
6054:
6037:
6017:
6011:
6010:
5988:
5982:
5981:
5968:
5962:
5961:
5947:
5941:
5940:
5913:
5907:
5906:
5889:
5859:
5848:
5842:
5831:
5810:Plane (geometry)
5619:
5307:
5305:
5304:
5299:
5278:
5275:
5262:
5259:
5237:
5235:
5234:
5229:
5205:
5201:
5199:
5198:
5193:
5181:
5179:
5178:
5173:
5127:
5125:
5124:
5119:
5089:
5087:
5086:
5081:
5045:
5043:
5042:
5037:
5004:
4997:
4995:
4994:
4989:
4984:
4982:
4956:
4941:
4937:
4935:
4934:
4929:
4924:
4894:
4890:
4888:
4887:
4882:
4866:
4862:
4860:
4859:
4854:
4849:
4823:
4802:
4795:
4793:
4792:
4787:
4782:
4780:
4754:
4740:—can be written
4739:
4729:
4727:
4726:
4721:
4665:
4623:
4621:
4620:
4615:
4610:
4602:
4588:
4580:
4545:
4543:
4542:
4537:
4535:
4520:
4509:
4484:
4480:
4478:
4477:
4472:
4460:
4458:
4457:
4452:
4434:
4432:
4431:
4426:
4409:term to compute
4408:
4406:
4395:
4384:
4377:
4375:
4374:
4369:
4357:
4350:
4344:
4342:
4341:
4336:
4322:
4320:
4319:
4314:
4309:
4307:
4306:
4294:
4293:
4284:
4282:
4280:
4279:
4271:
4262:
4253:
4251:
4250:
4245:
4215:
4209:
4203:
4201:
4200:
4195:
4183:
4181:
4180:
4175:
4002:
4000:
3999:
3994:
3992:
3979:
3978:
3946:
3945:
3913:
3912:
3874:three dimensions
3853:
3851:
3850:
3845:
3827:
3825:
3824:
3819:
3817:
3816:
3801:
3800:
3788:
3787:
3772:
3771:
3759:
3758:
3743:
3742:
3726:
3724:
3723:
3718:
3713:
3712:
3700:
3699:
3687:
3686:
3667:
3665:
3664:
3659:
3654:
3653:
3641:
3640:
3628:
3627:
3608:
3606:
3605:
3600:
3592:
3591:
3576:
3575:
3560:
3559:
3544:
3543:
3528:
3526:
3525:
3520:
3512:
3511:
3496:
3495:
3480:
3479:
3464:
3463:
3447:linear equations
3441:
3439:
3438:
3433:
3431:
3430:
3415:
3414:
3380:
3378:
3377:
3372:
3367:
3366:
3354:
3353:
3341:
3333:
3332:
3320:
3319:
3294:
3292:
3291:
3286:
3284:
3283:
3271:
3270:
3254:
3252:
3251:
3246:
3241:
3240:
3228:
3227:
3205:
3203:
3202:
3197:
3192:
3191:
3179:
3178:
3152:
3138:of the function
3102:
3100:
3099:
3094:
3055:
3053:
3052:
3047:
3041:
3039:
3038:
3037:
3025:
3024:
3014:
3013:
3012:
3003:
3002:
2990:
2989:
2980:
2979:
2969:
2964:
2962:
2961:
2960:
2948:
2947:
2937:
2936:
2935:
2923:
2922:
2912:
2893:
2891:
2890:
2885:
2883:
2882:
2870:
2868:
2867:
2866:
2854:
2853:
2843:
2842:
2841:
2829:
2828:
2818:
2812:
2811:
2780:
2764:
2762:
2761:
2756:
2748:
2747:
2726:
2725:
2713:
2712:
2694:
2693:
2681:
2680:
2665:
2664:
2639:
2637:
2636:
2631:
2626:
2625:
2613:
2612:
2600:
2599:
2583:
2581:
2580:
2575:
2570:
2569:
2557:
2556:
2544:
2543:
2513:
2511:
2510:
2505:
2459:
2455:
2448:
2406:
2404:
2403:
2398:
2323:
2321:
2320:
2315:
2277:Line coordinates
2233:primitive notion
2133:With respect to
2110:algebraic curves
2020:primitive notion
1880:
1878:
1877:
1872:
1870:
1869:
1862:
1861:
1845:
1844:
1833:
1832:
1814:
1813:
1797:
1796:
1785:
1784:
1766:
1765:
1749:
1748:
1737:
1736:
1595:Collinear points
1554:
1552:
1551:
1546:
1541:
1537:
1536:
1453:In more general
1339:primitive notion
1201:
1194:
1187:
915:
914:
434:
433:
367:Zero-dimensional
72:
58:
57:
21:
6442:
6441:
6437:
6436:
6435:
6433:
6432:
6431:
6427:Line (geometry)
6402:
6401:
6374:
6371:
6366:
6365:
6343:10.2307/2303867
6324:
6320:
6314:
6300:College Algebra
6293:
6289:
6269:
6265:
6259:
6245:
6241:
6235:
6221:
6217:
6211:
6203:, p. 314,
6193:
6189:
6169:
6165:
6159:
6141:
6137:
6115:
6107:, p. 114,
6097:
6093:
6076:
6062:
6058:
6044:10.2307/2690881
6018:
6014:
6008:
5989:
5985:
5969:
5965:
5948:
5944:
5930:
5914:
5910:
5904:
5890:
5873:
5868:
5863:
5862:
5849:
5845:
5832:
5828:
5823:
5785:Flat (geometry)
5761:
5753:complex numbers
5721:
5715:
5686:
5680:
5668:complex numbers
5656:affine geometry
5609:
5549:is not between
5432:
5425:
5420:
5368:
5362:
5327:linear equation
5319:
5317:Geometric space
5313:
5274:
5258:
5243:
5240:
5239:
5211:
5208:
5207:
5203:
5187:
5184:
5183:
5155:
5152:
5151:
5095:
5092:
5091:
5057:
5054:
5053:
5010:
5007:
5006:
4999:
4960:
4955:
4947:
4944:
4943:
4939:
4920:
4903:
4900:
4899:
4892:
4876:
4873:
4872:
4864:
4845:
4819:
4808:
4805:
4804:
4797:
4758:
4753:
4745:
4742:
4741:
4737:
4675:
4672:
4671:
4666:are related to
4655:
4649:Cartesian plane
4637:
4606:
4598:
4584:
4576:
4574:
4571:
4570:
4528:
4513:
4505:
4503:
4500:
4499:
4496:
4491:
4482:
4466:
4463:
4462:
4440:
4437:
4436:
4414:
4411:
4410:
4402:
4397:
4386:
4379:
4363:
4360:
4359:
4352:
4346:
4330:
4327:
4326:
4302:
4298:
4289:
4285:
4283:
4275:
4267:
4266:
4261:
4259:
4256:
4255:
4221:
4218:
4217:
4211:
4205:
4189:
4186:
4185:
4130:
4127:
4126:
4092:
4086:
4045:
4038:
4031:
3990:
3989:
3974:
3970:
3963:
3957:
3956:
3941:
3937:
3930:
3924:
3923:
3908:
3904:
3897:
3890:
3888:
3885:
3884:
3870:
3864:
3833:
3830:
3829:
3812:
3808:
3796:
3792:
3783:
3779:
3767:
3763:
3754:
3750:
3738:
3734:
3732:
3729:
3728:
3708:
3704:
3695:
3691:
3682:
3678:
3673:
3670:
3669:
3649:
3645:
3636:
3632:
3623:
3619:
3614:
3611:
3610:
3587:
3583:
3571:
3567:
3555:
3551:
3539:
3535:
3533:
3530:
3529:
3507:
3503:
3491:
3487:
3475:
3471:
3459:
3455:
3453:
3450:
3449:
3426:
3422:
3410:
3406:
3386:
3383:
3382:
3362:
3358:
3349:
3345:
3337:
3328:
3324:
3315:
3311:
3300:
3297:
3296:
3279:
3275:
3266:
3262:
3260:
3257:
3256:
3236:
3232:
3223:
3219:
3211:
3208:
3207:
3187:
3183:
3174:
3170:
3162:
3159:
3158:
3139:
3073:
3070:
3069:
3033:
3029:
3020:
3016:
3015:
3008:
3004:
2998:
2994:
2985:
2981:
2975:
2971:
2970:
2968:
2956:
2952:
2943:
2939:
2938:
2931:
2927:
2918:
2914:
2913:
2911:
2899:
2896:
2895:
2878:
2874:
2862:
2858:
2849:
2845:
2844:
2837:
2833:
2824:
2820:
2819:
2817:
2807:
2803:
2786:
2783:
2782:
2779:
2772:
2766:
2743:
2739:
2721:
2717:
2708:
2704:
2689:
2685:
2676:
2672:
2660:
2656:
2645:
2642:
2641:
2621:
2617:
2608:
2604:
2595:
2591:
2589:
2586:
2585:
2565:
2561:
2552:
2548:
2539:
2535:
2533:
2530:
2529:
2472:
2469:
2468:
2457:
2450:
2443:
2341:
2338:
2337:
2309:
2306:
2305:
2290:
2288:Linear equation
2284:
2282:Linear equation
2279:
2273:
2229:
2091:coordinate line
2032:
1864:
1863:
1857:
1853:
1851:
1846:
1840:
1836:
1834:
1828:
1824:
1822:
1816:
1815:
1809:
1805:
1803:
1798:
1792:
1788:
1786:
1780:
1776:
1774:
1768:
1767:
1761:
1757:
1755:
1750:
1744:
1740:
1738:
1732:
1728:
1726:
1716:
1715:
1713:
1710:
1709:
1704:
1695:
1688:
1677:
1668:
1661:
1650:
1641:
1634:
1603:
1597:
1532:
1494:
1490:
1482:
1479:
1478:
1455:Euclidean space
1413:
1401:convex polygons
1397:Euclidean plane
1306:
1298:affine geometry
1205:
1176:
1175:
912:
911:
902:
901:
692:
691:
675:
674:
660:
659:
647:
646:
583:
582:
571:
570:
431:
430:
428:Two-dimensional
419:
418:
392:
391:
389:One-dimensional
380:
379:
370:
369:
358:
357:
291:
290:
289:
272:
271:
120:
119:
108:
85:
41:
34:Line (graphics)
28:
23:
22:
15:
12:
11:
5:
6440:
6430:
6429:
6424:
6419:
6414:
6400:
6399:
6390:
6376:"Line (curve)"
6370:
6369:External links
6367:
6364:
6363:
6337:(9): 589–599,
6318:
6312:
6287:
6263:
6257:
6239:
6233:
6215:
6209:
6187:
6172:Bôcher, Maxime
6163:
6157:
6135:
6114:978-0030731006
6113:
6091:
6074:
6056:
6028:(3): 183–192,
6012:
6006:
5983:
5963:
5942:
5928:
5908:
5902:
5870:
5869:
5867:
5864:
5861:
5860:
5843:
5825:
5824:
5822:
5819:
5818:
5817:
5812:
5807:
5802:
5797:
5792:
5787:
5782:
5777:
5772:
5767:
5760:
5757:
5741:imaginary line
5717:Main article:
5714:
5711:
5682:Main article:
5679:
5676:
5620:and two rays,
5606:disjoint union
5573:determined by
5458:is called its
5454:and the point
5424:
5421:
5419:
5416:
5361:
5358:
5312:
5309:
5297:
5294:
5291:
5288:
5285:
5282:
5272:
5269:
5266:
5256:
5253:
5250:
5247:
5227:
5224:
5221:
5218:
5215:
5191:
5171:
5168:
5165:
5162:
5159:
5145:right triangle
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5035:
5032:
5029:
5026:
5023:
5020:
5017:
5014:
4987:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4959:
4954:
4951:
4927:
4923:
4919:
4916:
4913:
4910:
4907:
4880:
4852:
4848:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4822:
4818:
4815:
4812:
4785:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4757:
4752:
4749:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4636:
4633:
4613:
4609:
4605:
4601:
4597:
4594:
4591:
4587:
4583:
4579:
4546:(where λ is a
4534:
4531:
4526:
4523:
4519:
4516:
4512:
4508:
4495:
4492:
4490:
4487:
4470:
4450:
4447:
4444:
4424:
4421:
4418:
4367:
4334:
4312:
4305:
4301:
4297:
4292:
4288:
4278:
4274:
4270:
4265:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4193:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4088:Main article:
4085:
4082:
4078:
4077:
4047:
4043:
4036:
4029:
4023:
3988:
3985:
3982:
3977:
3973:
3969:
3966:
3964:
3962:
3959:
3958:
3955:
3952:
3949:
3944:
3940:
3936:
3933:
3931:
3929:
3926:
3925:
3922:
3919:
3916:
3911:
3907:
3903:
3900:
3898:
3896:
3893:
3892:
3863:
3860:
3843:
3840:
3837:
3815:
3811:
3807:
3804:
3799:
3795:
3791:
3786:
3782:
3778:
3775:
3770:
3766:
3762:
3757:
3753:
3749:
3746:
3741:
3737:
3716:
3711:
3707:
3703:
3698:
3694:
3690:
3685:
3681:
3677:
3657:
3652:
3648:
3644:
3639:
3635:
3631:
3626:
3622:
3618:
3598:
3595:
3590:
3586:
3582:
3579:
3574:
3570:
3566:
3563:
3558:
3554:
3550:
3547:
3542:
3538:
3518:
3515:
3510:
3506:
3502:
3499:
3494:
3490:
3486:
3483:
3478:
3474:
3470:
3467:
3462:
3458:
3429:
3425:
3421:
3418:
3413:
3409:
3405:
3402:
3399:
3396:
3393:
3390:
3370:
3365:
3361:
3357:
3352:
3348:
3344:
3340:
3336:
3331:
3327:
3323:
3318:
3314:
3310:
3307:
3304:
3295:, is given by
3282:
3278:
3274:
3269:
3265:
3244:
3239:
3235:
3231:
3226:
3222:
3218:
3215:
3195:
3190:
3186:
3182:
3177:
3173:
3169:
3166:
3155:
3154:
3129:
3119:
3092:
3089:
3086:
3083:
3080:
3077:
3058:two dimensions
3045:
3036:
3032:
3028:
3023:
3019:
3011:
3007:
3001:
2997:
2993:
2988:
2984:
2978:
2974:
2967:
2959:
2955:
2951:
2946:
2942:
2934:
2930:
2926:
2921:
2917:
2909:
2906:
2903:
2881:
2877:
2873:
2865:
2861:
2857:
2852:
2848:
2840:
2836:
2832:
2827:
2823:
2815:
2810:
2806:
2802:
2799:
2796:
2793:
2790:
2777:
2770:
2754:
2751:
2746:
2742:
2738:
2735:
2732:
2729:
2724:
2720:
2716:
2711:
2707:
2703:
2700:
2697:
2692:
2688:
2684:
2679:
2675:
2671:
2668:
2663:
2659:
2655:
2652:
2649:
2629:
2624:
2620:
2616:
2611:
2607:
2603:
2598:
2594:
2573:
2568:
2564:
2560:
2555:
2551:
2547:
2542:
2538:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2313:
2286:Main article:
2283:
2280:
2275:Main article:
2272:
2269:
2228:
2225:
2196:Parallel lines
2160:
2159:
2153:
2146:
2131:
2130:
2124:
2095:
2094:
2087:
2080:
2077:
2071:
2031:
2028:
2009:
2008:
1868:
1860:
1856:
1852:
1850:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1821:
1818:
1817:
1812:
1808:
1804:
1802:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1773:
1770:
1769:
1764:
1760:
1756:
1754:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1725:
1722:
1721:
1719:
1700:
1693:
1686:
1673:
1666:
1659:
1646:
1639:
1632:
1599:Main article:
1596:
1593:
1544:
1540:
1535:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1509:
1506:
1503:
1500:
1497:
1493:
1489:
1486:
1477:is the subset
1412:
1409:
1333:(now called a
1323:(now called a
1305:
1302:
1279:Euclidean line
1207:
1206:
1204:
1203:
1196:
1189:
1181:
1178:
1177:
1172:
1171:
1170:
1169:
1164:
1156:
1155:
1151:
1150:
1149:
1148:
1143:
1138:
1133:
1128:
1123:
1118:
1113:
1108:
1103:
1098:
1090:
1089:
1085:
1084:
1083:
1082:
1077:
1072:
1067:
1062:
1057:
1052:
1047:
1039:
1038:
1034:
1033:
1032:
1031:
1026:
1021:
1016:
1011:
1006:
1001:
996:
991:
986:
981:
976:
968:
967:
963:
962:
961:
960:
955:
950:
945:
940:
935:
930:
922:
921:
913:
909:
908:
907:
904:
903:
900:
899:
894:
889:
884:
879:
874:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
693:
689:
688:
687:
684:
683:
677:
676:
673:
672:
667:
661:
654:
653:
652:
649:
648:
645:
644:
639:
634:
632:Platonic Solid
629:
624:
619:
614:
609:
604:
603:
602:
591:
590:
584:
578:
577:
576:
573:
572:
567:
566:
565:
564:
559:
554:
546:
545:
539:
538:
537:
536:
531:
523:
522:
516:
515:
514:
513:
508:
503:
498:
490:
489:
483:
482:
481:
480:
475:
470:
462:
461:
455:
454:
453:
452:
447:
442:
432:
426:
425:
424:
421:
420:
417:
416:
411:
410:
409:
404:
393:
387:
386:
385:
382:
381:
378:
377:
371:
365:
364:
363:
360:
359:
356:
355:
350:
345:
339:
338:
333:
328:
318:
313:
308:
302:
301:
292:
288:
287:
284:
280:
279:
278:
277:
274:
273:
270:
269:
268:
267:
257:
252:
247:
242:
237:
236:
235:
225:
220:
215:
214:
213:
208:
203:
193:
192:
191:
186:
176:
171:
166:
161:
156:
151:
150:
149:
144:
143:
142:
127:
121:
115:
114:
113:
110:
109:
107:
106:
96:
90:
87:
86:
73:
65:
64:
26:
9:
6:
4:
3:
2:
6439:
6428:
6425:
6423:
6420:
6418:
6415:
6413:
6410:
6409:
6407:
6398:
6394:
6391:
6387:
6383:
6382:
6377:
6373:
6372:
6360:
6356:
6352:
6348:
6344:
6340:
6336:
6332:
6328:
6322:
6315:
6309:
6305:
6301:
6297:
6291:
6284:
6280:
6279:
6274:
6267:
6260:
6258:0-486-65812-0
6254:
6250:
6243:
6236:
6234:0-07-072191-2
6230:
6226:
6219:
6212:
6210:9781139473736
6206:
6202:
6198:
6191:
6183:
6179:
6178:
6173:
6167:
6160:
6158:0-471-18283-4
6154:
6149:
6148:
6139:
6132:
6128:
6124:
6120:
6116:
6110:
6106:
6102:
6095:
6088:
6085:, p. 108, at
6084:
6083:
6077:
6075:9780883853481
6071:
6067:
6060:
6053:
6049:
6045:
6041:
6036:
6035:10.1.1.502.72
6031:
6027:
6023:
6016:
6009:
6007:9780867200935
6003:
5999:
5998:
5993:
5987:
5980:, p. 410
5979:
5978:
5973:
5967:
5960:
5957:(in French),
5956:
5952:
5946:
5939:
5935:
5931:
5925:
5921:
5920:
5912:
5905:
5903:0-8247-1748-1
5899:
5895:
5888:
5886:
5884:
5882:
5880:
5878:
5876:
5871:
5857:
5853:
5847:
5840:
5836:
5830:
5826:
5816:
5813:
5811:
5808:
5806:
5803:
5801:
5798:
5796:
5793:
5791:
5788:
5786:
5783:
5781:
5778:
5776:
5773:
5771:
5768:
5766:
5763:
5762:
5756:
5754:
5750:
5749:complex plane
5746:
5743:representing
5742:
5738:
5734:
5725:
5720:
5710:
5708:
5704:
5699:
5690:
5685:
5675:
5673:
5669:
5665:
5661:
5660:ordered field
5657:
5653:
5648:
5646:
5641:
5639:
5635:
5631:
5627:
5623:
5617:
5613:
5607:
5603:
5599:
5594:
5592:
5588:
5584:
5580:
5576:
5572:
5568:
5564:
5560:
5556:
5552:
5548:
5544:
5540:
5536:
5532:
5528:
5524:
5520:
5516:
5512:
5508:
5504:
5500:
5496:
5492:
5488:
5484:
5479:
5477:
5473:
5469:
5465:
5461:
5460:initial point
5457:
5453:
5449:
5445:
5436:
5430:
5415:
5413:
5412:metric spaces
5409:
5405:
5401:
5396:
5393:
5389:
5388:great circles
5385:
5381:
5372:
5367:
5357:
5355:
5351:
5347:
5343:
5339:
5334:
5332:
5328:
5324:
5318:
5308:
5295:
5292:
5289:
5286:
5283:
5280:
5270:
5267:
5264:
5254:
5251:
5248:
5245:
5222:
5219:
5216:
5189:
5166:
5163:
5160:
5148:
5146:
5142:
5138:
5137:geometrically
5133:
5131:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5077:
5074:
5071:
5068:
5065:
5062:
5059:
5051:
5046:
5033:
5030:
5027:
5024:
5021:
5018:
5015:
5012:
5002:
4985:
4976:
4973:
4970:
4964:
4961:
4957:
4952:
4949:
4925:
4921:
4917:
4914:
4911:
4908:
4905:
4896:
4878:
4870:
4850:
4846:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4820:
4816:
4813:
4810:
4800:
4783:
4774:
4771:
4768:
4762:
4759:
4755:
4750:
4747:
4735:
4730:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4669:
4663:
4659:
4654:
4650:
4641:
4632:
4630:
4625:
4603:
4592:
4589:
4581:
4568:
4564:
4560:
4556:
4551:
4549:
4524:
4521:
4510:
4486:
4468:
4448:
4445:
4442:
4422:
4419:
4416:
4405:
4400:
4393:
4389:
4383:
4365:
4355:
4349:
4332:
4323:
4310:
4303:
4299:
4295:
4290:
4286:
4272:
4263:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4214:
4208:
4191:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4124:
4120:
4118:
4113:
4109:
4105:
4096:
4091:
4081:
4075:
4071:
4067:
4063:
4059:
4055:
4051:
4048:
4042:
4035:
4028:
4024:
4021:
4017:
4013:
4009:
4006:
4005:
4004:
3986:
3983:
3980:
3975:
3971:
3967:
3965:
3960:
3953:
3950:
3947:
3942:
3938:
3934:
3932:
3927:
3920:
3917:
3914:
3909:
3905:
3901:
3899:
3894:
3881:
3879:
3875:
3869:
3859:
3857:
3841:
3838:
3835:
3813:
3809:
3805:
3802:
3797:
3793:
3789:
3784:
3780:
3776:
3773:
3768:
3764:
3760:
3755:
3751:
3747:
3744:
3739:
3735:
3709:
3705:
3701:
3696:
3692:
3688:
3683:
3679:
3650:
3646:
3642:
3637:
3633:
3629:
3624:
3620:
3596:
3593:
3588:
3584:
3580:
3577:
3572:
3568:
3564:
3561:
3556:
3552:
3548:
3545:
3540:
3536:
3516:
3513:
3508:
3504:
3500:
3497:
3492:
3488:
3484:
3481:
3476:
3472:
3468:
3465:
3460:
3456:
3448:
3443:
3427:
3423:
3419:
3411:
3407:
3403:
3400:
3394:
3391:
3388:
3363:
3359:
3355:
3350:
3346:
3338:
3329:
3325:
3321:
3316:
3312:
3305:
3302:
3280:
3276:
3272:
3267:
3263:
3237:
3233:
3229:
3224:
3220:
3213:
3188:
3184:
3180:
3175:
3171:
3164:
3150:
3146:
3142:
3137:
3133:
3130:
3127:
3123:
3120:
3117:
3113:
3109:
3106:
3105:
3104:
3090:
3087:
3084:
3081:
3078:
3075:
3067:
3065:
3064:
3059:
3043:
3034:
3030:
3026:
3021:
3017:
3009:
3005:
2999:
2995:
2991:
2986:
2982:
2976:
2972:
2965:
2957:
2953:
2949:
2944:
2940:
2932:
2928:
2924:
2919:
2915:
2907:
2904:
2901:
2879:
2875:
2871:
2863:
2859:
2855:
2850:
2846:
2838:
2834:
2830:
2825:
2821:
2808:
2804:
2800:
2797:
2791:
2788:
2776:
2769:
2752:
2744:
2740:
2736:
2733:
2722:
2718:
2714:
2709:
2705:
2698:
2690:
2686:
2682:
2677:
2673:
2661:
2657:
2653:
2650:
2622:
2618:
2614:
2609:
2605:
2596:
2592:
2566:
2562:
2558:
2553:
2549:
2540:
2536:
2526:
2524:
2519:
2517:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2466:
2465:standard form
2461:
2453:
2446:
2440:
2438:
2434:
2430:
2426:
2422:
2418:
2414:
2410:
2394:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2361:
2358:
2355:
2346:
2343:
2335:
2331:
2327:
2311:
2303:
2294:
2289:
2278:
2268:
2266:
2262:
2257:
2253:
2248:
2246:
2242:
2238:
2234:
2224:
2222:
2218:
2213:
2211:
2207:
2203:
2201:
2197:
2193:
2191:
2187:
2183:
2178:
2176:
2172:
2168:
2167:quadrilateral
2165:
2157:
2156:central lines
2154:
2151:
2147:
2144:
2140:
2139:
2138:
2136:
2128:
2125:
2122:
2118:
2115:
2114:
2113:
2111:
2106:
2104:
2100:
2092:
2088:
2085:
2081:
2078:
2075:
2072:
2069:
2068:tangent lines
2066:
2065:
2064:
2062:
2058:
2054:
2050:
2046:
2036:
2027:
2025:
2021:
2016:
2014:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1937:
1936:
1934:
1930:
1926:
1922:
1918:
1915:
1911:
1906:
1904:
1900:
1894:
1892:
1888:
1884:
1866:
1858:
1854:
1848:
1841:
1837:
1829:
1825:
1819:
1810:
1806:
1800:
1793:
1789:
1781:
1777:
1771:
1762:
1758:
1752:
1745:
1741:
1733:
1729:
1723:
1717:
1708:
1703:
1699:
1692:
1685:
1681:
1676:
1672:
1665:
1658:
1654:
1649:
1645:
1638:
1631:
1627:
1623:
1619:
1614:
1612:
1608:
1602:
1592:
1590:
1586:
1582:
1579: −
1578:
1574:
1570:
1566:
1562:
1558:
1542:
1538:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1504:
1501:
1498:
1491:
1487:
1484:
1476:
1472:
1468:
1464:
1460:
1456:
1451:
1449:
1446:
1442:
1438:
1434:
1430:
1426:
1422:
1418:
1408:
1406:
1402:
1398:
1393:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1362:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1331:straight line
1328:
1327:
1322:
1318:
1317:
1311:
1301:
1299:
1295:
1291:
1290:non-Euclidean
1287:
1286:
1281:
1280:
1275:
1271:
1270:
1264:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1217:straight line
1214:
1202:
1197:
1195:
1190:
1188:
1183:
1182:
1180:
1179:
1168:
1165:
1163:
1160:
1159:
1158:
1157:
1153:
1152:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1127:
1124:
1122:
1119:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1093:
1092:
1091:
1087:
1086:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1042:
1041:
1040:
1036:
1035:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
975:
972:
971:
970:
969:
965:
964:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
925:
924:
923:
920:
917:
916:
906:
905:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
694:
686:
685:
682:
679:
678:
671:
668:
666:
663:
662:
657:
651:
650:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
608:
605:
601:
598:
597:
596:
593:
592:
589:
586:
585:
581:
575:
574:
563:
560:
558:
557:Circumference
555:
553:
550:
549:
548:
547:
544:
541:
540:
535:
532:
530:
527:
526:
525:
524:
521:
520:Quadrilateral
518:
517:
512:
509:
507:
504:
502:
499:
497:
494:
493:
492:
491:
488:
487:Parallelogram
485:
484:
479:
476:
474:
471:
469:
466:
465:
464:
463:
460:
457:
456:
451:
448:
446:
443:
441:
438:
437:
436:
435:
429:
423:
422:
415:
412:
408:
405:
403:
400:
399:
398:
395:
394:
390:
384:
383:
376:
373:
372:
368:
362:
361:
354:
351:
349:
346:
344:
341:
340:
337:
334:
332:
329:
326:
325:Perpendicular
322:
321:Orthogonality
319:
317:
314:
312:
309:
307:
304:
303:
300:
297:
296:
295:
285:
282:
281:
276:
275:
266:
263:
262:
261:
258:
256:
253:
251:
248:
246:
245:Computational
243:
241:
238:
234:
231:
230:
229:
226:
224:
221:
219:
216:
212:
209:
207:
204:
202:
199:
198:
197:
194:
190:
187:
185:
182:
181:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
148:
145:
141:
138:
137:
136:
133:
132:
131:
130:Non-Euclidean
128:
126:
123:
122:
118:
112:
111:
104:
100:
97:
95:
92:
91:
89:
88:
84:
80:
76:
71:
67:
66:
63:
60:
59:
54:
50:
45:
39:
35:
30:
19:
18:Straight line
6397:Cut-the-Knot
6379:
6334:
6330:
6321:
6299:
6290:
6276:
6266:
6248:
6242:
6224:
6218:
6196:
6190:
6176:
6166:
6146:
6138:
6103:, New York:
6100:
6094:
6087:Google Books
6080:
6065:
6059:
6025:
6021:
6015:
5996:
5986:
5975:
5966:
5954:
5945:
5918:
5911:
5893:
5855:
5851:
5846:
5839:transitively
5829:
5795:Line segment
5730:
5698:line segment
5695:
5684:Line segment
5678:Line segment
5672:finite field
5649:
5642:
5637:
5636:on the line
5633:
5629:
5625:
5621:
5615:
5611:
5601:
5597:
5595:
5591:opposite ray
5590:
5586:
5582:
5578:
5574:
5570:
5566:
5562:
5558:
5554:
5550:
5546:
5542:
5538:
5534:
5530:
5526:
5522:
5518:
5514:
5510:
5506:
5502:
5498:
5494:
5490:
5486:
5482:
5480:
5475:
5471:
5463:
5459:
5455:
5451:
5447:
5443:
5441:
5397:
5377:
5335:
5320:
5149:
5139:by applying
5134:
5047:
5000:
4938:between the
4897:
4869:line segment
4798:
4731:
4661:
4657:
4646:
4628:
4626:
4566:
4562:
4558:
4554:
4552:
4497:
4403:
4398:
4391:
4387:
4381:
4353:
4347:
4324:
4212:
4206:
4115:
4107:
4103:
4101:
4079:
4073:
4069:
4065:
4057:
4053:
4049:
4040:
4033:
4026:
4019:
4015:
4011:
4007:
3882:
3877:
3871:
3444:
3156:
3148:
3144:
3140:
3131:
3128:of the line.
3121:
3118:of the line.
3107:
3068:
3061:
2774:
2767:
2527:
2520:
2516:general form
2515:
2464:
2462:
2451:
2444:
2441:
2436:
2432:
2428:
2427:) such that
2425:coefficients
2421:real numbers
2416:
2412:
2408:
2333:
2329:
2299:
2256:mental image
2255:
2251:
2249:
2230:
2214:
2210:right angles
2204:
2194:
2179:
2161:
2150:Simson lines
2132:
2120:
2116:
2107:
2096:
2074:secant lines
2041:
2017:
2010:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1944:
1940:
1932:
1928:
1924:
1920:
1916:
1907:
1902:
1898:
1895:
1886:
1701:
1697:
1690:
1683:
1679:
1674:
1670:
1663:
1656:
1652:
1647:
1643:
1636:
1629:
1625:
1621:
1615:
1606:
1604:
1601:Collinearity
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1474:
1470:
1466:
1465:), the line
1463:affine space
1458:
1452:
1444:
1440:
1436:
1432:
1428:
1424:
1414:
1394:
1363:
1355:analytically
1351:real numbers
1335:line segment
1330:
1324:
1320:
1319:, a general
1315:
1307:
1283:
1278:
1277:
1268:
1265:
1260:
1253:line segment
1248:
1235:. Lines are
1233:ray of light
1229:straightedge
1220:
1216:
1210:
1029:Parameshvara
842:Parameshvara
612:Dodecahedron
396:
196:Differential
29:
6304:Brooks Cole
6082:online copy
5733:real number
5719:Number line
5713:Number line
5628:(the point
5525:is between
5501:(including
5238:such that
5050:normal form
4104:normal form
3126:y-intercept
2523:x-intercept
2326:coordinates
2252:description
2190:Pappus line
2186:Pascal line
2171:Newton line
2103:transversal
2099:parallelism
1939:The points
1891:determinant
1359:coordinates
1154:Present day
1101:Lobachevsky
1088:1700s–1900s
1045:Jyeṣṭhadeva
1037:1400s–1700s
989:Brahmagupta
812:Lobachevsky
792:Jyeṣṭhadeva
742:Brahmagupta
670:Hypersphere
642:Tetrahedron
617:Icosahedron
189:Diophantine
6406:Categories
5866:References
5557:. A point
5545:such that
5521:such that
5468:half-space
5418:Extensions
5315:See also:
4565:is vector
4557:is vector
3609:such that
2419:are fixed
2271:Definition
2221:skew lines
2143:Euler line
2127:asymptotes
1999:) implies
1448:coordinate
1374:dimensions
1304:Properties
1294:projective
1274:postulates
1014:al-Yasamin
958:Apollonius
953:Archimedes
943:Pythagoras
933:Baudhayana
887:al-Yasamin
837:Pythagoras
732:Baudhayana
722:Archimedes
717:Apollonius
622:Octahedron
473:Hypotenuse
348:Similarity
343:Congruence
255:Incidence
206:Symplectic
201:Riemannian
184:Arithmetic
159:Projective
147:Hyperbolic
75:Projecting
6386:EMS Press
6283:EMS Press
6030:CiteSeerX
5938:747274805
5707:collinear
5585:ray, the
5464:half-line
5408:geodesics
5342:primitive
5293:π
5287:α
5281:θ
5271:α
5265:θ
5249:≥
5223:θ
5202:with the
5190:α
5167:θ
5113:θ
5110:
5075:θ
5072:
5031:π
5025:α
5019:θ
4977:α
4974:−
4971:θ
4965:
4918:π
4912:φ
4906:α
4879:φ
4843:π
4837:φ
4831:θ
4817:π
4814:−
4811:φ
4775:φ
4772:−
4769:θ
4763:
4715:θ
4712:
4693:θ
4690:
4604:−
4593:λ
4525:λ
4469:φ
4449:φ
4446:
4423:φ
4420:
4366:φ
4333:φ
4192:φ
4160:−
4157:φ
4154:
4142:φ
4139:
3581:−
3501:−
3404:−
3356:−
3322:−
3273:≠
3027:−
2992:−
2950:−
2925:−
2856:−
2831:−
2801:−
2737:−
2715:−
2683:−
2654:−
2490:−
2368:∣
2175:diagonals
2137:we have:
2135:triangles
2084:directrix
2061:hyperbola
1893:is zero.
1849:⋯
1801:⋯
1753:⋯
1607:collinear
1557:direction
1530:∈
1524:∣
1502:−
1314:Euclid's
1267:Euclid's
1261:endpoints
1241:dimension
1225:curvature
1131:Minkowski
1050:Descartes
984:Aryabhata
979:Kātyāyana
910:by period
822:Minkowski
797:Kātyāyana
757:Descartes
702:Aryabhata
681:Geometers
665:Tesseract
529:Trapezoid
501:Rectangle
294:Dimension
179:Algebraic
169:Synthetic
140:Spherical
125:Euclidean
6422:Infinity
6182:archived
6174:(1915),
6123:69-12075
5953:(1900),
5815:Polyline
5759:See also
5737:integers
5703:coplanar
5658:over an
5400:distance
5366:Geodesic
5350:geodesic
3116:gradient
2423:(called
2057:parabola
1382:parallel
1316:Elements
1269:Elements
1245:embedded
1213:geometry
1121:Poincaré
1065:Minggatu
1024:Yang Hui
994:Virasena
882:Yang Hui
877:Virasena
847:Poincaré
827:Minggatu
607:Cylinder
552:Diameter
511:Rhomboid
468:Altitude
459:Triangle
353:Symmetry
331:Parallel
316:Diagonal
286:Features
283:Concepts
174:Analytic
135:Elliptic
117:Branches
103:Timeline
62:Geometry
6388:, 2001
6359:0006034
6351:2303867
6273:"Angle"
6052:2690881
5670:or any
4494:Vectors
4401:/|
4358:, then
4119:segment
4003:where:
3255:, when
3134:is the
3124:is the
3110:is the
3103:where:
2182:hexagon
2053:ellipse
1696:, ...,
1678:), and
1669:, ...,
1642:, ...,
1366:Hilbert
1308:In the
1146:Coxeter
1126:Hilbert
1111:Riemann
1060:Huygens
1019:al-Tusi
1009:Khayyám
999:Alhazen
966:1–1400s
867:al-Tusi
852:Riemann
802:Khayyám
787:Huygens
782:Hilbert
752:Coxeter
712:Alhazen
690:by name
627:Pyramid
506:Rhombus
450:Polygon
402:segment
250:Fractal
233:Digital
218:Complex
99:History
94:Outline
6357:
6349:
6310:
6255:
6231:
6207:
6155:
6129:
6121:
6111:
6072:
6050:
6032:
6004:
5936:
5926:
5900:
5856:closed
5392:planes
5338:axioms
5003:> 0
4863:Here,
4801:> 0
4738:(0, 0)
4734:origin
4548:scalar
4407:|
4356:> 0
4184:where
4123:origin
4117:normal
4062:vector
4056:, and
4014:, and
3878:cannot
3828:imply
2407:where
2245:axioms
2180:For a
2164:convex
2162:For a
2049:circle
1975:) and
1912:, the
1881:has a
1707:matrix
1431:, and
1370:points
1343:axioms
1296:, and
1257:points
1237:spaces
1167:Gromov
1162:Atiyah
1141:Veblen
1136:Cartan
1106:Bolyai
1075:Sakabe
1055:Pascal
948:Euclid
938:Manava
872:Veblen
857:Sakabe
832:Pascal
817:Manava
777:Gromov
762:Euclid
747:Cartan
737:Bolyai
727:Atiyah
637:Sphere
600:cuboid
588:Volume
543:Circle
496:Square
414:Length
336:Vertex
240:Convex
223:Finite
164:Affine
79:sphere
49:origin
6347:JSTOR
6131:47870
6048:JSTOR
5837:acts
5821:Notes
5805:Locus
5770:Curve
5645:angle
4998:with
4796:with
4647:In a
3856:plane
3112:slope
2439:= 0.
2152:, and
2059:, or
2045:conic
2043:to a
2030:Types
1620:, in
1611:plane
1395:On a
1388:, or
1386:plane
1378:plane
1326:curve
1310:Greek
1259:(its
1116:Klein
1096:Gauss
1070:Euler
1004:Sijzi
974:Zhang
928:Ahmes
892:Zhang
862:Sijzi
807:Klein
772:Gauss
767:Euler
707:Ahmes
440:Plane
375:Point
311:Curve
306:Angle
83:plane
81:to a
6308:ISBN
6253:ISBN
6229:ISBN
6205:ISBN
6153:ISBN
6127:OCLC
6119:LCCN
6109:ISBN
6070:ISBN
6002:ISBN
5934:OCLC
5924:ISBN
5898:ISBN
5852:open
5624:and
5600:and
5565:and
5553:and
5541:and
5529:and
5517:and
5505:and
5497:and
5485:and
5090:and
5022:<
5016:<
5005:and
4834:<
4828:<
4803:and
4561:and
4435:and
4345:and
4102:The
3668:and
3206:and
2584:and
2431:and
2415:and
2148:the
2141:the
1987:) =
1963:) =
1947:and
1931:and
1883:rank
1587:and
1555:The
1473:and
1419:, a
1390:skew
1321:line
1282:and
1249:line
1221:line
1215:, a
1080:Aida
697:Aida
656:Four
595:Cube
562:Area
534:Kite
445:Area
397:Line
38:Line
6395:at
6339:doi
6040:doi
5654:or
5452:ray
5423:Ray
5410:in
5260:and
5107:sin
5069:cos
4962:sin
4760:cos
4709:sin
4687:cos
4553:If
4550:).
4443:cos
4417:sin
4394:= 0
4151:sin
4136:cos
3114:or
3056:In
2894:or
2765:If
2454:= 0
2449:or
2447:= 1
2254:or
2235:in
2215:In
2047:(a
1908:In
1682:= (
1655:= (
1651:),
1628:= (
1616:In
1415:In
1347:set
1263:).
1239:of
1211:In
919:BCE
407:ray
6408::
6384:,
6378:,
6355:MR
6353:,
6345:,
6335:48
6333:,
6281:,
6275:,
6199:,
6125:,
6117:,
6046:,
6038:,
6026:72
6024:,
5974:,
5932:,
5874:^
5755:.
5709:.
5696:A
5674:.
5647:.
5638:AB
5626:AD
5622:BC
5614:,
5593:.
5587:AD
5583:AB
5414:.
5276:or
4851:2.
4660:,
4651:,
4624:.
4567:OB
4559:OA
4485:.
4390:=
4072:,
4068:,
4052:,
4039:,
4032:,
4010:,
3442:.
3143:=
3066::
2773:≠
2411:,
2332:,
2219:,
2212:.
2192:.
2177:.
2089:a
2082:a
2055:,
2051:,
2003:=
1943:,
1689:,
1662:,
1635:,
1457:,
1427:,
1407:.
1361:.
1300:.
1292:,
77:a
6341::
6089:)
6079:(
6042::
5858:.
5634:A
5630:D
5618:)
5616:B
5612:A
5610:(
5602:B
5598:A
5579:A
5575:B
5571:A
5567:B
5563:A
5559:D
5555:C
5551:B
5547:A
5543:B
5539:A
5535:C
5531:C
5527:A
5523:B
5519:B
5515:A
5511:C
5507:B
5503:A
5499:B
5495:A
5491:A
5487:B
5483:A
5476:A
5472:A
5456:A
5448:A
5444:A
5431:.
5296:.
5290:+
5284:=
5268:=
5255:,
5252:0
5246:r
5226:)
5220:,
5217:r
5214:(
5204:x
5170:)
5164:,
5161:r
5158:(
5116:,
5104:r
5101:=
5098:y
5078:,
5066:r
5063:=
5060:x
5034:.
5028:+
5013:0
5001:r
4986:,
4980:)
4968:(
4958:p
4953:=
4950:r
4940:x
4926:2
4922:/
4915:+
4909:=
4893:x
4865:p
4847:/
4840:+
4825:2
4821:/
4799:r
4784:,
4778:)
4766:(
4756:p
4751:=
4748:r
4718:.
4706:r
4703:=
4700:y
4696:,
4684:r
4681:=
4678:x
4664:)
4662:θ
4658:r
4656:(
4629:A
4612:)
4608:a
4600:b
4596:(
4590:+
4586:a
4582:=
4578:r
4563:b
4555:a
4533:B
4530:A
4522:+
4518:A
4515:O
4511:=
4507:r
4483:π
4404:c
4399:c
4392:p
4388:c
4382:π
4380:2
4354:p
4348:p
4311:.
4304:2
4300:b
4296:+
4291:2
4287:a
4277:|
4273:c
4269:|
4264:c
4242:c
4239:=
4236:y
4233:b
4230:+
4227:x
4224:a
4213:p
4207:x
4172:,
4169:0
4166:=
4163:p
4148:y
4145:+
4133:x
4074:c
4070:b
4066:a
4064:(
4058:c
4054:b
4050:a
4044:0
4041:z
4037:0
4034:y
4030:0
4027:x
4025:(
4020:t
4016:z
4012:y
4008:x
3987:t
3984:c
3981:+
3976:0
3972:z
3968:=
3961:z
3954:t
3951:b
3948:+
3943:0
3939:y
3935:=
3928:y
3921:t
3918:a
3915:+
3910:0
3906:x
3902:=
3895:x
3842:0
3839:=
3836:t
3814:2
3810:c
3806:t
3803:=
3798:1
3794:c
3790:,
3785:2
3781:b
3777:t
3774:=
3769:1
3765:b
3761:,
3756:2
3752:a
3748:t
3745:=
3740:1
3736:a
3715:)
3710:2
3706:c
3702:,
3697:2
3693:b
3689:,
3684:2
3680:a
3676:(
3656:)
3651:1
3647:c
3643:,
3638:1
3634:b
3630:,
3625:1
3621:a
3617:(
3597:0
3594:=
3589:2
3585:d
3578:z
3573:2
3569:c
3565:+
3562:y
3557:2
3553:b
3549:+
3546:x
3541:2
3537:a
3517:0
3514:=
3509:1
3505:d
3498:z
3493:1
3489:c
3485:+
3482:y
3477:1
3473:b
3469:+
3466:x
3461:1
3457:a
3428:a
3424:y
3420:+
3417:)
3412:a
3408:x
3401:x
3398:(
3395:m
3392:=
3389:y
3369:)
3364:a
3360:x
3351:b
3347:x
3343:(
3339:/
3335:)
3330:a
3326:y
3317:b
3313:y
3309:(
3306:=
3303:m
3281:b
3277:x
3268:a
3264:x
3243:)
3238:b
3234:y
3230:,
3225:b
3221:x
3217:(
3214:B
3194:)
3189:a
3185:y
3181:,
3176:a
3172:x
3168:(
3165:A
3153:.
3151:)
3149:x
3147:(
3145:f
3141:y
3132:x
3122:b
3108:m
3091:b
3088:+
3085:x
3082:m
3079:=
3076:y
3044:.
3035:0
3031:x
3022:1
3018:x
3010:1
3006:y
3000:0
2996:x
2987:0
2983:y
2977:1
2973:x
2966:+
2958:0
2954:x
2945:1
2941:x
2933:0
2929:y
2920:1
2916:y
2908:x
2905:=
2902:y
2880:0
2876:y
2872:+
2864:0
2860:x
2851:1
2847:x
2839:0
2835:y
2826:1
2822:y
2814:)
2809:0
2805:x
2798:x
2795:(
2792:=
2789:y
2778:1
2775:x
2771:0
2768:x
2753:.
2750:)
2745:0
2741:x
2734:x
2731:(
2728:)
2723:0
2719:y
2710:1
2706:y
2702:(
2699:=
2696:)
2691:0
2687:x
2678:1
2674:x
2670:(
2667:)
2662:0
2658:y
2651:y
2648:(
2628:)
2623:1
2619:y
2615:,
2610:1
2606:x
2602:(
2597:1
2593:P
2572:)
2567:0
2563:y
2559:,
2554:0
2550:x
2546:(
2541:0
2537:P
2502:,
2499:0
2496:=
2493:c
2487:y
2484:b
2481:+
2478:x
2475:a
2458:c
2452:c
2445:c
2437:b
2433:b
2429:a
2417:c
2413:b
2409:a
2395:,
2392:}
2389:c
2386:=
2383:y
2380:b
2377:+
2374:x
2371:a
2365:)
2362:y
2359:,
2356:x
2353:(
2350:{
2347:=
2344:L
2334:y
2330:x
2328:(
2312:L
2158:.
2145:,
2121:i
2117:i
2007:.
2005:c
2001:x
1997:b
1995:,
1993:c
1991:(
1989:d
1985:b
1983:,
1981:x
1979:(
1977:d
1973:a
1971:,
1969:c
1967:(
1965:d
1961:a
1959:,
1957:x
1955:(
1953:d
1949:c
1945:b
1941:a
1933:b
1929:a
1925:b
1923:,
1921:a
1919:(
1917:d
1903:k
1899:k
1887:n
1867:]
1859:n
1855:z
1842:2
1838:z
1830:1
1826:z
1820:1
1811:n
1807:y
1794:2
1790:y
1782:1
1778:y
1772:1
1763:n
1759:x
1746:2
1742:x
1734:1
1730:x
1724:1
1718:[
1702:n
1698:z
1694:2
1691:z
1687:1
1684:z
1680:Z
1675:n
1671:y
1667:2
1664:y
1660:1
1657:y
1653:Y
1648:n
1644:x
1640:2
1637:x
1633:1
1630:x
1626:X
1622:n
1589:b
1585:a
1581:a
1577:b
1573:t
1571:(
1569:b
1565:t
1563:(
1561:a
1543:.
1539:}
1534:R
1527:t
1521:b
1518:t
1515:+
1512:a
1508:)
1505:t
1499:1
1496:(
1492:{
1488:=
1485:L
1475:b
1471:a
1467:L
1459:R
1445:n
1441:n
1437:n
1433:z
1429:y
1425:x
1200:e
1193:t
1186:v
327:)
323:(
105:)
101:(
40:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
