1376:
8586:
solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by
Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."
49:
8528:
essentially different from the use of a coordinate frame, whether rectangular or, more generally, oblique. Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The
Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed
9802:
4488:
1721:
9814:
3268:
1311:
intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations (expressed in words) of curves. However, although
Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve
3999:
9838:
9826:
8440:
certainly
Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. It was shortcomings in algebraic notations that, more than anything else, operated against the Greek achievement of a full-fledged coordinate geometry.
1265:
explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the
8537:
derived from a specific geometric situation; That
Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases.
1347:(1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.
8536:
for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed. Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions
8483:
dealt with what might be called an analytic geometry of one dimension. It considered the following general problem, using the typical Greek algebraic analysis in geometric form: Given four points A, B, C, D on a straight line, determine a fifth point P on it such that the rectangle on AP and CP is in
8439:
Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a strong resemblance to the use of coordinates, as illustrated above, it has sometimes been maintained that
Menaechmus had analytic geometry. Such a judgment is warranted only in part, for
4986:
There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the
Knowledge article
1264:
Usually the
Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be
1701:
also laid the groundwork for analytical geometry. The key difference between Fermat's and
Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and
8585:
that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar
Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic
8527:
in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not
1310:
by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and
4461:
7809:
7199:
2316:
In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the
1702:
produced their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was
8484:
a given ratio to the rectangle on BP and DP. Here, too, the problem reduces easily to the solution of a quadratic; and, as in other cases, Apollonius treated the question exhaustively, including the limits of possibility and the number of solutions.
1978:
5028:
has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if
1744:
is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:
4374:
3401:
4357:
3948:
2905:
8815:"Une introduction aux lieux, plans & solides; qui est un traité analytique concernant la solution des problemes plans & solides, qui avoit esté veu devant que M. des Cartes eut rien publié sur ce sujet."
2659:
7699:
7089:
4157:
8817:(An introduction to loci, plane and solid; which is an analytical treatise concerning the solution of plane and solid problems, which was seen before Mr. des Cartes had published anything on this subject.)
1319:. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.
7206:: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get
5854:
2770:
1302:, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the
7692:
7082:
5366:
Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.
1851:
1290:
solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.
1724:
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (â3,1) in red, (â1.5,â2.5) in blue, and the origin (0,0) in purple.
8726:
The person who is popularly credited with being the discoverer of analytic geometry was the philosopher RenĂ© Descartes (1596â1650), one of the most influential thinkers of the modern era.
5726:
2423:
6761:
6827:
3036:
2469:
4202:
7439:
6478:
7608:
7262:
6998:
6028:
3436:
4960:
2351:
5477:
4893:
8699:
6642:
5121:
3699:
3534:
6594:
6421:
6252:
3293:
2989:
2602:
2580:
2511:
3647:
6070:
3485:
3177:
3136:
3095:
7648:
7038:
4218:
3820:
7990:
7506:
7473:
6894:
6861:
5561:
5526:
5402:
5062:
4545:
2582:. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points
2263:
8082:
7316:
7289:
6671:
6310:
6186:
6154:
6102:
5950:
5918:
5886:
5648:
5593:
5026:
3738:
2538:
6278:
4826:
4735:
4664:
4595:
3593:
3567:
5425:
1375:
8102:
8050:
8030:
8010:
7952:
7932:
7912:
7892:
7869:
7849:
7548:
7528:
7376:
7356:
7336:
6936:
6916:
6691:
6546:
6526:
6506:
6373:
6353:
6330:
6206:
6122:
5970:
5746:
5616:
5361:
5341:
5321:
5301:
5281:
5261:
5241:
5221:
5201:
5181:
5161:
5141:
4980:
4913:
4849:
4798:
4775:
4755:
4707:
4684:
4638:
4615:
4569:
2558:
2489:
2786:
8532:
upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down
1678:
in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into
5143:
is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative
2607:
4057:
3290:
in two variables is always a conic section â though it may be degenerate, and all conic sections arise in this way. The equation will be of the form
4456:{\displaystyle \mathbf {A} \cdot \mathbf {B} {\stackrel {\mathrm {def} }{=}}\left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|\cos \theta ,}
1331:
saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and
9262:
8340:, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
8944:
8784:
8802:
8600:
5203:
value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like
2668:
1177:
1306:
further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of
7652:
7042:
1689:
Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of
5363:
values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.
8985:
8896:
8841:
1519:
277:
9625:
6696:
6765:
7804:{\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}
7194:{\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}
5751:
4166:
1615:
1514:
17:
7383:
6425:
9770:
9255:
9108:
9046:
9028:
7553:
6943:
3403:
As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional
1848:). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae:
9842:
9080:
8922:
8719:
8668:
8574:
8516:
8472:
8432:
5856:. The intersection of these two circles is the collection of points which make both equations true. Does the point
5430:
1693:(Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes'
243:
6598:
2170: = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a
9306:
6553:
6380:
2938:
8355:
that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in
9720:
8675:
the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments.
8115:
Also for this may be used the common language use as a: normal (perpendicular) line, otherwise in engineering as
1539:
1298:
3448:
3140:
3099:
3058:
1339:, but the decisive step came later with Descartes. Omar Khayyam is credited with identifying the foundations of
9818:
9126:
7612:
7002:
5653:
2356:
1170:
1124:
730:
189:
4509:
Transformations are applied to a parent function to turn it into a new function with similar characteristics.
9248:
9192:
8829:
7477:
7443:
6865:
6831:
2233:
9830:
2995:
2428:
9745:
9301:
3279:
2041:
1760:
1754:
1467:
1452:
31:
9092:
9864:
9316:
9054:
7209:
5975:
3409:
1973:{\displaystyle x=r\,\cos \theta ,\,y=r\,\sin \theta ;\,r={\sqrt {x^{2}+y^{2}}},\,\theta =\arctan(y/x).}
1567:
1145:
755:
4918:
2327:
1629:
1359:
9730:
9702:
9339:
4854:
2175:
1582:
1437:
1266:
1163:
7831:
One type of intersection which is widely studied is the intersection of a geometric object with the
5067:
3659:
3494:
9775:
8941:
8781:
6211:
1544:
132:
8209:
Tangent is the linear approximation of a spherical or other curved or twisted line of a function.
2585:
2563:
2494:
1658:
Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences
9660:
9650:
9620:
9554:
9289:
9000:
While this discussion is limited to the xy-plane, it can easily be extended to higher dimensions.
8964:
5488:
3607:
2003:
1997:
1981:
1608:
1529:
558:
238:
95:
8799:
8711:
8703:
6033:
9758:
9655:
9635:
9630:
9559:
9284:
4988:
1405:
1258:
634:
345:
223:
108:
8598:
Cooper, Glen M. (2003). "Review: Omar Khayyam, the Mathmetician by R. Rashed, B. Vahabzadeh".
8566:
8508:
8500:
8464:
8456:
9785:
9715:
9592:
9516:
9455:
9440:
9435:
9412:
9294:
8929:
8424:
8416:
8356:
8348:
8154:
4023:
3774:
2171:
1985:
1524:
1442:
1250:
1234:
406:
367:
326:
321:
174:
7960:
5531:
5496:
5372:
5032:
4515:
4163:. Similarly, the angle that a line makes with the horizontal can be defined by the formula
2471:
be a nonzero vector. The plane determined by this point and vector consists of those points
1706:
who first applied the coordinate method in a systematic study of space curves and surfaces.
9765:
9645:
9640:
9564:
9465:
8055:
7294:
7267:
6649:
6283:
6159:
6127:
6075:
5923:
5891:
5859:
5621:
5566:
4997:
3786:
3711:
2516:
2297:
2226:
1686:
in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.
1549:
1074:
997:
845:
750:
272:
167:
81:
8:
9780:
9690:
9612:
9511:
9445:
9402:
9392:
9372:
9170:
9148:
8960:
8552:
8496:
8452:
8412:
8387:
8352:
6257:
4803:
4712:
4643:
4574:
4215:
In three dimensions, distance is given by the generalization of the Pythagorean theorem:
4160:
3969:
3778:
3572:
3546:
3283:
3272:
3052:
2179:
2100:
2082:
1662:
1572:
1462:
1293:
1079:
1023:
936:
790:
770:
695:
585:
456:
446:
309:
184:
179:
162:
137:
125:
77:
72:
53:
8251:
points on the curve. More precisely, a straight line is said to be a tangent of a curve
8247:
that "just touches" the curve at that point. Informally, it is a line through a pair of
5407:
2224:
equations. In two dimensions, the equation for non-vertical lines is often given in the
1656:, one of the three accompanying essays (appendices) published in 1637 together with his
9806:
9725:
9665:
9597:
9587:
9526:
9501:
9377:
9334:
9329:
9209:
8617:
8087:
8035:
8015:
7995:
7937:
7917:
7897:
7877:
7854:
7834:
7533:
7513:
7361:
7341:
7321:
6921:
6901:
6676:
6531:
6511:
6491:
6358:
6338:
6315:
6191:
6107:
5955:
5731:
5601:
5346:
5326:
5306:
5286:
5266:
5246:
5226:
5206:
5186:
5166:
5146:
5126:
4965:
4898:
4834:
4783:
4760:
4740:
4692:
4669:
4623:
4600:
4554:
4019:
3813:
3782:
3287:
3237:
2543:
2474:
2217:
2140:
1715:
1683:
1601:
1587:
1432:
1340:
1246:
1215:
1038:
765:
605:
233:
157:
147:
118:
103:
30:
This article is about coordinate geometry. For the geometry of analytic varieties, see
9234:
8879:
VujiÄiÄ, Milan; Sanderson, Jeffrey (2008), VujiÄiÄ, Milan; Sanderson, Jeffrey (eds.),
3396:{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0{\text{ with }}A,B,C{\text{ not all zero.}}}
1783:). This system can also be used for three-dimensional geometry, where every point in
9801:
9521:
9450:
9397:
9104:
9088:
9076:
9070:
9042:
9024:
8981:
8918:
8892:
8837:
8715:
8664:
8570:
8557:
8512:
8468:
8428:
8336:
As it passes through the point where the tangent line and the curve meet, called the
8162:
1818:
1812:
1651:
1534:
1477:
1332:
1254:
1211:
1109:
897:
875:
800:
659:
385:
314:
206:
113:
6482:
Traditional methods for finding intersections include substitution and elimination.
1635:
1367:
1099:
1028:
825:
735:
9735:
9710:
9582:
9430:
9367:
9201:
9190:
Coolidge, J. L. (1948), "The Beginnings of Analytic Geometry in Three Dimensions",
9179:
9157:
9146:
Boyer, Carl B. (1944), "Analytic Geometry: The Discovery of Fermat and Descartes",
9135:
9062:
8884:
8609:
8382:
8372:
8240:
8182:
3977:
3404:
2207:
1771:-coordinate representing its vertical position. These are typically written as an
1733:
1639:
1488:
1457:
1421:
1089:
830:
540:
418:
353:
211:
196:
61:
4352:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}
2131:
is said to be the equation for this line. In general, linear equations involving
9675:
9602:
9531:
9324:
8948:
8912:
8806:
8788:
8330:
8178:
3756:
2780:
2213:
2203:
2194:
is the equation for any circle centered at the origin (0, 0) with a radius of r.
2120:
1784:
1741:
1729:
1671:
1472:
1410:
1400:
512:
375:
218:
201:
142:
48:
3943:{\displaystyle \sum _{i,j=1}^{3}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{3}P_{i}x_{i}+R=0.}
1084:
1053:
987:
835:
780:
715:
9753:
9680:
9387:
9058:
8656:
1703:
1577:
1415:
1336:
1283:
1140:
1048:
992:
957:
865:
775:
745:
705:
610:
9124:
Bissell, Christopher C. (1987), "Cartesian geometry: The Dutch contribution",
8888:
4487:
3051:
be described by a single linear equation, so they are frequently described by
2046:
In spherical coordinates, every point in space is represented by its distance
1720:
1114:
725:
9858:
9541:
9473:
9425:
8377:
8360:
8248:
8244:
8198:
8170:
8166:
8131:
8112:
Axis in geometry is the perpendicular line to any line, object or a surface.
3262:
2318:
2144:
1980:
This system may be generalized to three-dimensional space through the use of
1840:-axis. Using this notation, points are typically written as an ordered pair (
1646:, the alternative term used for analytic geometry, is named after Descartes.
1238:
1119:
1104:
1033:
850:
810:
760:
535:
498:
465:
303:
299:
8780:(Toulouse, France: Jean Pech, 1679), "Ad locos planos et solidos isagoge,"
8646:
Cooper, G. (2003). Journal of the American Oriental Society,123(1), 248-249.
9483:
9478:
9382:
8392:
8139:
3441:
2096:
2078:
1772:
1482:
1390:
1328:
1058:
1007:
820:
675:
590:
380:
9240:
3440:
The conic sections described by this equation can be classified using the
9685:
9349:
9272:
9183:
9161:
8322:
7914:-intercept of the object. The intersection of a geometric object and the
7826:
7822:
4360:
3965:
2662:
2287:
1737:
1495:
1395:
1242:
1226:
1191:
1094:
967:
785:
720:
648:
620:
595:
1649:
Descartes made significant progress with the methods in an essay titled
9670:
9549:
9344:
9213:
9139:
8663:(Second ed.). Springer Science + Business Media Inc. p. 105.
8621:
8314:
4018:. These definitions are designed to be consistent with the underlying
4002:
The distance formula on the plane follows from the Pythagorean theorem.
3961:
3267:
2900:{\displaystyle ax+by+cz+d=0,{\text{ where }}d=-(ax_{0}+by_{0}+cz_{0}).}
1427:
1287:
952:
931:
921:
911:
870:
815:
710:
700:
600:
451:
7813:
For conic sections, as many as 4 points might be in the intersection.
8956:
3953:
3741:
3702:
1307:
962:
680:
643:
507:
479:
9205:
8613:
8204:
2654:{\displaystyle \mathbf {n} \cdot (\mathbf {r} -\mathbf {r} _{0})=0.}
2147:, and more complicated equations describe more complicated figures.
1674:
tongue, and its philosophical principles, provided a foundation for
1245:. It is the foundation of most modern fields of geometry, including
9574:
9493:
9420:
8975:
8547:
8545:
8224:
8123:
4007:
3998:
3989:
3650:
2088:
1675:
1230:
1207:
1043:
1002:
972:
860:
855:
805:
530:
489:
437:
331:
294:
40:
9359:
8951:
in "Geometry Formulas and Facts", excerpted from 30th Edition of
8218:
8134:
to a given object. For example, in the two-dimensional case, the
4015:
3537:
2154:
on the plane. This is not always the case: the trivial equation
1222:
977:
690:
484:
428:
228:
9059:
Analytic Geometry of the Point, Line, Circle, and Conic Sections
8542:
4152:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}},}
2061:-plane makes with respect to the horizontal axis, and the angle
5493:
For two geometric objects P and Q represented by the relations
3957:
3596:
2092:
2011:
926:
916:
795:
740:
615:
578:
566:
521:
474:
392:
57:
8778:
Varia Opera Mathematica d. Petri de Fermat, Senatoris Tolosani
8635:
Mathematical Masterpieces: Further Chronicles by the Explorers
7358:
has been eliminated. We then solve the remaining equation for
2069:-axis. The names of the angles are often reversed in physics.
1748:
9168:
Boyer, Carl B. (1965), "Johann Hudde and space coordinates",
8236:
8186:
8138:
to a curve at a given point is the line perpendicular to the
5482:
4468:
4209:
4011:
3993:
3973:
2277:
2273:
2151:
2051:
2022:
1826:
1788:
1679:
982:
906:
840:
685:
289:
284:
2111:
corresponds to the set of all the points on the plane whose
573:
423:
1821:, every point of the plane is represented by its distance
8581:
Omar Khayyam (ca. 1050â1123), the "tent-maker," wrote an
8748:
8746:
8507:(Second ed.). John Wiley & Sons, Inc. pp.
8463:(Second ed.). John Wiley & Sons, Inc. pp.
8423:(Second ed.). John Wiley & Sons, Inc. pp.
1767:-coordinate representing its horizontal position, and a
1991:
8743:
8090:
8058:
8038:
8018:
7998:
7963:
7940:
7920:
7900:
7880:
7857:
7837:
7702:
7655:
7615:
7556:
7536:
7516:
7480:
7446:
7386:
7364:
7344:
7324:
7297:
7270:
7212:
7092:
7045:
7005:
6946:
6924:
6904:
6868:
6834:
6768:
6699:
6679:
6652:
6601:
6556:
6534:
6514:
6494:
6428:
6383:
6361:
6341:
6318:
6286:
6260:
6214:
6194:
6162:
6130:
6110:
6078:
6036:
5978:
5958:
5926:
5894:
5862:
5754:
5734:
5656:
5624:
5604:
5569:
5534:
5499:
5433:
5410:
5375:
5349:
5329:
5309:
5289:
5269:
5249:
5229:
5209:
5189:
5169:
5149:
5129:
5070:
5035:
5000:
4968:
4921:
4901:
4857:
4837:
4806:
4786:
4763:
4743:
4715:
4695:
4672:
4646:
4626:
4603:
4577:
4557:
4518:
4377:
4221:
4169:
4060:
3823:
3714:
3662:
3610:
3575:
3549:
3497:
3451:
3412:
3296:
3143:
3102:
3061:
2998:
2941:
2789:
2671:
2610:
2588:
2566:
2546:
2519:
2497:
2477:
2431:
2359:
2330:
2236:
2035:
1854:
1836:
normally measured counterclockwise from the positive
1642:, although Descartes is sometimes given sole credit.
8177:. The word "normal" is also used as an adjective: a
6375:
can be found by solving the simultaneous equations:
4359:
while the angle between two vectors is given by the
3236:
are related to the slope of the line, such that the
2665:, not scalar multiplication.) Expanded this becomes
2006:, every point of space is represented by its height
9072:
A History of Mathematics: An Introduction (2nd Ed.)
5849:{\displaystyle (1,0):Q=\{(x,y)|(x-1)^{2}+y^{2}=1\}}
4547:is changed by standard transformations as follows:
2765:{\displaystyle a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0,}
8852:Percey Franklyn Smith, Arthur Sullivan Gale (1905)
8556:
8276:on the curve if the line passes through the point
8096:
8076:
8044:
8024:
8004:
7984:
7946:
7926:
7906:
7886:
7863:
7843:
7803:
7686:
7642:
7602:
7542:
7530:in either of the original equations and solve for
7522:
7500:
7467:
7433:
7370:
7350:
7330:
7310:
7283:
7256:
7193:
7076:
7032:
6992:
6930:
6918:in either of the original equations and solve for
6910:
6888:
6855:
6821:
6755:
6685:
6665:
6636:
6588:
6540:
6520:
6500:
6472:
6415:
6367:
6347:
6324:
6304:
6272:
6246:
6200:
6180:
6148:
6116:
6096:
6064:
6022:
5964:
5944:
5912:
5880:
5848:
5740:
5720:
5642:
5610:
5587:
5555:
5520:
5471:
5419:
5396:
5355:
5335:
5315:
5295:
5275:
5255:
5235:
5215:
5195:
5175:
5155:
5135:
5115:
5056:
5020:
4974:
4954:
4907:
4887:
4843:
4820:
4792:
4769:
4749:
4729:
4701:
4678:
4658:
4632:
4609:
4589:
4563:
4539:
4455:
4351:
4196:
4151:
3942:
3732:
3693:
3641:
3587:
3561:
3528:
3479:
3430:
3395:
3171:
3130:
3089:
3030:
2983:
2899:
2764:
2653:
2596:
2574:
2552:
2532:
2505:
2483:
2463:
2417:
2345:
2257:
2032:-plane makes with respect to the horizontal axis.
1972:
9093:Lectures in Geometry Semester I Analytic Geometry
8813:, 9 February 1665, pp. 69â72. From p. 70:
8205:Spherical and nonlinear planes and their tangents
7378:, in the same way as in the substitution method:
6673:into the other equation and proceed to solve for
5563:the intersection is the collection of all points
3040:This familiar equation for a plane is called the
2935:are not all zero, then the graph of the equation
1806:
1345:Treatise on Demonstrations of Problems of Algebra
9856:
8976:M.R. Spiegel; S. Lipschutz; D. Spellman (2009).
8878:
5479:is the relation that describes the unit circle.
4737:stretches the graph horizontally by a factor of
4006:In analytic geometry, geometric notions such as
1759:The most common coordinate system to use is the
1634:Analytic geometry was independently invented by
8012:specifies the point where the line crosses the
7874:The intersection of a geometric object and the
7687:{\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}
7077:{\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}
4026:on the plane, the distance between two points (
2178:of two surfaces (see below), or as a system of
8130:is an object such as a line or vector that is
3194:are all functions of the independent variable
9256:
8953:CRC Standard Mathematical Tables and Formulas
7291:in the first equation is subtracted from the
5748:might be the circle with radius 1 and center
5618:might be the circle with radius 1 and center
2162:specifies the entire plane, and the equation
1732:is given a coordinate system, by which every
1609:
1171:
8601:The Journal of the American Oriental Society
5843:
5779:
5715:
5663:
2150:Usually, a single equation corresponds to a
2119:-coordinate are equal. These points form a
9270:
8969:
8914:Math refresher for scientists and engineers
8212:
5483:Finding intersections of geometric objects
4363:. The dot product of two Euclidean vectors
1749:Cartesian coordinates (in a plane or space)
1335:with his geometric solution of the general
32:Algebraic geometry § Analytic geometry
27:Study of geometry using a coordinate system
9263:
9249:
8825:
8823:
8708:The History of Mathematics: A Brief Course
8359:and has been extensively generalized; see
7754:
7753:
7747:
7746:
7144:
7143:
7137:
7136:
1616:
1602:
1178:
1164:
47:
8836:, 6th ed., Brooks Cole Cengage Learning.
8655:
5721:{\displaystyle P=\{(x,y)|x^{2}+y^{2}=1\}}
5163:values, the function is reflected in the
2418:{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}
1937:
1900:
1887:
1877:
1864:
1697:. Clearly written and well received, the
9189:
8869:Courier Dover Publications, Jan 27, 2012
4486:
4159:which can be viewed as a version of the
3997:
3812:, the general quadric is defined by the
3599:, which is a special case of an ellipse;
3266:
1719:
9123:
9101:A Source Book in Mathematics, 1200-1800
8917:, John Wiley and Sons, pp. 44â45,
8820:
8032:axis. Depending on the context, either
6528:and then substitute the expression for
2907:Conversely, it is easily shown that if
2174:, and a curve must be specified as the
2072:
1630:René Descartes § Analytic geometry
1327:The 11th-century Persian mathematician
14:
9857:
9098:
9036:
8910:
8597:
3777:in 3-dimensional space defined as the
3487:If the conic is non-degenerate, then:
2091:involving the coordinates specifies a
278:Straightedge and compass constructions
9244:
9167:
9145:
9018:
8867:Analytic Geometry of Three Dimensions
8846:
8737:
8698:
8686:
8640:
8551:
8495:
8451:
8411:
7816:
6756:{\displaystyle (x-1)^{2}+(1-x^{2})=1}
3983:
2353:be the position vector of some point
9825:
9068:
8934:
8764:
8752:
7696:So our intersection has two points:
7086:So our intersection has two points:
6822:{\displaystyle x^{2}-2x+1+1-x^{2}=1}
3031:{\displaystyle \mathbf {n} =(a,b,c)}
2464:{\displaystyle \mathbf {n} =(a,b,c)}
2220:, can be described algebraically by
1992:Cylindrical coordinates (in a space)
9837:
9075:, Reading: Addison Wesley Longman,
8978:Vector Analysis (Schaum's Outlines)
8881:Linear Algebra Thoroughly Explained
5123:. In the new transformed function,
4197:{\displaystyle \theta =\arctan(m),}
3750:
3198:which ranges over the real numbers.
2197:
1515:Rules for the Direction of the Mind
24:
8634:
8321:. A similar definition applies to
7434:{\displaystyle x^{2}-2x+1-x^{2}=0}
7318:in the second equation leaving no
6646:We then substitute this value for
6473:{\displaystyle (x-1)^{2}+y^{2}=1.}
6332:so it is not in the intersection.
6124:. On the other hand, still using
5888:make both equations true? Using
5064:, then it can be transformed into
4482:
4407:
4404:
4401:
2513:, such that the vector drawn from
2065:that it makes with respect to the
2036:Spherical coordinates (in a space)
1691:Ad locos planos et solidos isagoge
1682:and the addition of commentary by
25:
9876:
9228:
8854:Introduction to Analytic Geometry
8107:
7603:{\displaystyle (1/2)^{2}+y^{2}=1}
7257:{\displaystyle (x-1)^{2}-x^{2}=0}
6993:{\displaystyle (1/2)^{2}+y^{2}=1}
6023:{\displaystyle (0-1)^{2}+0^{2}=1}
4994:For example, the parent function
3431:{\displaystyle \mathbf {P} ^{5}.}
3256:
2321:) to indicate its "inclination".
1350:
1277:
244:Noncommutative algebraic geometry
9836:
9824:
9813:
9812:
9800:
9219:
8523:The method of Apollonius in the
8417:"The Age of Plato and Aristotle"
8145:In the three-dimensional case a
4955:{\displaystyle -x\sin A+y\cos A}
4433:
4420:
4387:
4379:
3415:
3000:
2777:form of the equation of a plane.
2632:
2623:
2612:
2590:
2568:
2499:
2433:
2346:{\displaystyle \mathbf {r} _{0}}
2333:
1374:
9721:Computational complexity theory
8994:
8904:
8872:
8859:
8834:Calculus: Early Transcendentals
8793:
8770:
8758:
8731:
8710:. Wiley-Interscience. pp.
8692:
5472:{\displaystyle x^{2}+y^{2}-1=0}
5283:values introduce translations,
5243:-axis when it is negative. The
5223:, reflects the function in the
4888:{\displaystyle x\cos A+y\sin A}
4828:stretches the graph vertically.
3047:In three dimensions, lines can
1540:Meditations on First Philosophy
9127:The Mathematical Intelligencer
8680:
8649:
8628:
8591:
8489:
8445:
8405:
8071:
8059:
7572:
7557:
7226:
7213:
6962:
6947:
6744:
6725:
6713:
6700:
6637:{\displaystyle y^{2}=1-x^{2}.}
6442:
6429:
6299:
6287:
6175:
6163:
6143:
6131:
6091:
6079:
6047:
6037:
5992:
5979:
5939:
5927:
5907:
5895:
5875:
5863:
5815:
5802:
5798:
5794:
5782:
5767:
5755:
5682:
5678:
5666:
5637:
5625:
5582:
5570:
5550:
5538:
5515:
5503:
5391:
5379:
5116:{\displaystyle y=af(b(x-k))+h}
5104:
5101:
5089:
5083:
5051:
5045:
4962:rotates the graph by an angle
4534:
4522:
4502:
4498:
4494:
4437:
4429:
4424:
4416:
4335:
4308:
4296:
4269:
4257:
4230:
4188:
4182:
4135:
4108:
4096:
4069:
3694:{\displaystyle B^{2}-4AC>0}
3529:{\displaystyle B^{2}-4AC<0}
3044:of the equation of the plane.
3025:
3007:
2891:
2843:
2750:
2731:
2722:
2703:
2694:
2675:
2642:
2619:
2458:
2440:
2412:
2373:
1964:
1950:
1807:Polar coordinates (in a plane)
1709:
637:- / other-dimensional
13:
1:
9193:American Mathematical Monthly
9007:
8980:(2nd ed.). McGraw Hill.
8800:"Eloge de Monsieur de Fermat"
8659:(2004). "Analytic Geometry".
6898:Next, we place this value of
6589:{\displaystyle x^{2}+y^{2}=1}
6488:Solve the first equation for
6416:{\displaystyle x^{2}+y^{2}=1}
6247:{\displaystyle 0^{2}+0^{2}=1}
5595:which are in both relations.
4597:moves the graph to the right
3536:, the equation represents an
2992:is a plane having the vector
2984:{\displaystyle ax+by+cz+d=0,}
2103:. For example, the equation
1221:Analytic geometry is used in
9222:Newton and analytic geometry
9103:, Harvard University Press,
9021:History of Analytic Geometry
8185:, the normal component of a
7510:We then place this value of
4054:) is defined by the formula
3740:, the equation represents a
3701:, the equation represents a
3649:, the equation represents a
3595:, the equation represents a
2597:{\displaystyle \mathbf {r} }
2575:{\displaystyle \mathbf {n} }
2506:{\displaystyle \mathbf {r} }
7:
9237:with interactive animations
9117:
8809:(Eulogy of Mr. de Fermat),
8661:Mathematics and its History
8366:
8292:on the curve and has slope
8142:to the curve at the point.
3642:{\displaystyle B^{2}-4AC=0}
3280:Cartesian coordinate system
3222:) is any point on the line.
2042:Spherical coordinate system
1761:Cartesian coordinate system
1755:Cartesian coordinate system
10:
9881:
9771:Films about mathematicians
9235:Coordinate Geometry topics
8216:
7954:-intercept of the object.
7820:
6548:into the second equation:
6065:{\displaystyle (-1)^{2}=1}
5486:
4014:measure are defined using
3987:
3754:
3480:{\displaystyle B^{2}-4AC.}
3260:
3252:) is parallel to the line.
3172:{\displaystyle z=z_{0}+ct}
3131:{\displaystyle y=y_{0}+bt}
3090:{\displaystyle x=x_{0}+at}
2201:
2087:In analytic geometry, any
2076:
2039:
1995:
1810:
1763:, where each point has an
1752:
1728:In analytic geometry, the
1713:
1660:, commonly referred to as
1627:
1568:Christina, Queen of Sweden
1272:
29:
9794:
9744:
9701:
9611:
9573:
9540:
9492:
9464:
9411:
9358:
9340:Philosophy of mathematics
9315:
9280:
8889:10.1007/978-3-540-74639-3
7643:{\displaystyle y^{2}=3/4}
7033:{\displaystyle y^{2}=3/4}
3952:Quadric surfaces include
2095:of the plane, namely the
1583:Gottfried Wilhelm Leibniz
1438:Causal adequacy principle
1322:
9776:Recreational mathematics
9069:Katz, Victor J. (1998),
9039:A History of Mathematics
9037:Cajori, Florian (1999),
9019:Boyer, Carl B. (2004) ,
9012:
8911:Fanchi, John R. (2006),
8883:, Springer, p. 27,
8563:A History of Mathematics
8505:A History of Mathematics
8479:The Apollonian treatise
8461:A History of Mathematics
8421:A History of Mathematics
8398:
8351:at a given point is the
8213:Tangent lines and planes
2216:, or more generally, in
1825:from the origin and its
1740:coordinates. Similarly,
1670:, written in his native
1545:Principles of Philosophy
133:Non-Archimedean geometry
9661:Mathematical statistics
9651:Mathematical psychology
9621:Engineering mathematics
9555:Algebraic number theory
8965:University of Minnesota
5489:Intersection (geometry)
2491:, with position vector
2004:cylindrical coordinates
1998:Cylindrical coordinates
1530:Discourse on the Method
239:Noncommutative geometry
9807:Mathematics portal
9656:Mathematical sociology
9636:Mathematical economics
9631:Mathematical chemistry
9560:Analytic number theory
9441:Differential equations
9099:Struik, D. J. (1969),
9023:, Dover Publications,
8811:Le Journal des Scavans
8481:On Determinate Section
8193:, etc. The concept of
8098:
8078:
8046:
8026:
8006:
7986:
7985:{\displaystyle y=mx+b}
7948:
7928:
7908:
7888:
7865:
7845:
7805:
7688:
7644:
7604:
7544:
7524:
7502:
7501:{\displaystyle x=1/2.}
7469:
7468:{\displaystyle -2x=-1}
7435:
7372:
7352:
7332:
7312:
7285:
7258:
7195:
7078:
7034:
6994:
6932:
6912:
6890:
6889:{\displaystyle x=1/2.}
6857:
6856:{\displaystyle -2x=-1}
6823:
6757:
6687:
6667:
6638:
6590:
6542:
6522:
6502:
6474:
6417:
6369:
6349:
6326:
6306:
6274:
6248:
6202:
6182:
6150:
6118:
6098:
6066:
6024:
5966:
5946:
5914:
5882:
5850:
5742:
5722:
5644:
5612:
5589:
5557:
5556:{\displaystyle Q(x,y)}
5522:
5521:{\displaystyle P(x,y)}
5473:
5421:
5398:
5397:{\displaystyle R(x,y)}
5357:
5337:
5317:
5297:
5277:
5257:
5237:
5217:
5197:
5177:
5157:
5137:
5117:
5058:
5057:{\displaystyle y=f(x)}
5022:
4989:affine transformations
4976:
4956:
4909:
4889:
4845:
4822:
4794:
4771:
4751:
4731:
4703:
4680:
4660:
4634:
4611:
4591:
4565:
4541:
4540:{\displaystyle R(x,y)}
4506:
4457:
4353:
4198:
4153:
4022:. For example, using
4003:
3944:
3907:
3850:
3734:
3695:
3643:
3589:
3563:
3530:
3481:
3432:
3397:
3275:
3173:
3132:
3091:
3032:
2985:
2901:
2766:
2661:(The dot here means a
2655:
2598:
2576:
2554:
2534:
2507:
2485:
2465:
2419:
2347:
2259:
2258:{\displaystyle y=mx+b}
2057:its projection on the
2028:its projection on the
1974:
1725:
1299:On Determinate Section
1259:computational geometry
1214:. This contrasts with
207:Discrete/Combinatorial
9786:Mathematics education
9716:Theory of computation
9436:Hypercomplex analysis
8558:"The Arabic Hegemony"
8501:"Apollonius of Perga"
8457:"Apollonius of Perga"
8357:differential geometry
8099:
8079:
8077:{\displaystyle (0,b)}
8047:
8027:
8007:
7987:
7949:
7929:
7909:
7889:
7866:
7846:
7806:
7689:
7645:
7605:
7545:
7525:
7503:
7470:
7436:
7373:
7353:
7333:
7313:
7311:{\displaystyle y^{2}}
7286:
7284:{\displaystyle y^{2}}
7259:
7196:
7079:
7035:
6995:
6933:
6913:
6891:
6858:
6824:
6758:
6688:
6668:
6666:{\displaystyle y^{2}}
6639:
6591:
6543:
6523:
6503:
6475:
6418:
6370:
6350:
6327:
6307:
6305:{\displaystyle (0,0)}
6275:
6249:
6203:
6183:
6181:{\displaystyle (x,y)}
6151:
6149:{\displaystyle (0,0)}
6119:
6099:
6097:{\displaystyle (0,0)}
6067:
6025:
5967:
5947:
5945:{\displaystyle (x,y)}
5915:
5913:{\displaystyle (0,0)}
5883:
5881:{\displaystyle (0,0)}
5851:
5743:
5723:
5645:
5643:{\displaystyle (0,0)}
5613:
5590:
5588:{\displaystyle (x,y)}
5558:
5523:
5474:
5427:plane. For example,
5422:
5404:is a relation in the
5399:
5358:
5338:
5323:horizontal. Positive
5318:
5298:
5278:
5258:
5238:
5218:
5198:
5178:
5158:
5138:
5118:
5059:
5023:
5021:{\displaystyle y=1/x}
4977:
4957:
4910:
4890:
4846:
4823:
4795:
4772:
4752:
4732:
4704:
4681:
4661:
4635:
4612:
4592:
4566:
4542:
4490:
4458:
4354:
4199:
4154:
4024:Cartesian coordinates
4001:
3945:
3887:
3824:
3742:rectangular hyperbola
3735:
3733:{\displaystyle A+C=0}
3696:
3644:
3590:
3564:
3531:
3482:
3433:
3398:
3270:
3174:
3133:
3092:
3033:
2986:
2902:
2767:
2656:
2599:
2577:
2555:
2535:
2533:{\displaystyle P_{0}}
2508:
2486:
2466:
2420:
2348:
2260:
2099:for the equation, or
2050:from the origin, the
1975:
1787:is represented by an
1723:
1267:CantorâDedekind axiom
190:Discrete differential
9766:Informal mathematics
9646:Mathematical physics
9641:Mathematical finance
9626:Mathematical biology
9565:Diophantine geometry
9184:10.5951/MT.58.1.0033
9162:10.5951/MT.37.3.0099
9095:via Internet Archive
8930:Section 3.2, page 45
8088:
8056:
8036:
8016:
7996:
7961:
7938:
7934:-axis is called the
7918:
7898:
7894:-axis is called the
7878:
7855:
7835:
7700:
7653:
7613:
7554:
7534:
7514:
7478:
7444:
7384:
7362:
7342:
7322:
7295:
7268:
7210:
7090:
7043:
7003:
6944:
6922:
6902:
6866:
6832:
6766:
6697:
6677:
6650:
6599:
6554:
6532:
6512:
6492:
6426:
6381:
6359:
6339:
6335:The intersection of
6316:
6284:
6258:
6212:
6192:
6160:
6128:
6108:
6076:
6034:
5976:
5956:
5924:
5892:
5860:
5752:
5732:
5654:
5622:
5602:
5567:
5532:
5497:
5431:
5408:
5373:
5347:
5327:
5307:
5287:
5267:
5247:
5227:
5207:
5187:
5167:
5147:
5127:
5068:
5033:
4998:
4966:
4919:
4899:
4855:
4835:
4804:
4784:
4761:
4741:
4713:
4693:
4670:
4644:
4624:
4601:
4575:
4555:
4516:
4375:
4219:
4167:
4058:
3821:
3787:quadratic polynomial
3712:
3660:
3608:
3573:
3547:
3495:
3449:
3410:
3294:
3271:A hyperbola and its
3141:
3100:
3059:
3053:parametric equations
2996:
2939:
2787:
2669:
2608:
2586:
2564:
2560:is perpendicular to
2544:
2517:
2495:
2475:
2429:
2357:
2328:
2298:independent variable
2234:
2227:slope-intercept form
2180:parametric equations
2073:Equations and curves
1852:
1550:Passions of the Soul
1520:The Search for Truth
9781:Mathematics and art
9691:Operations research
9446:Functional analysis
9171:Mathematics Teacher
9149:Mathematics Teacher
8961:The Geometry Center
8865:William H. McCrea,
8388:Translation of axes
8173:to that surface at
7338:term. The variable
6273:{\displaystyle 0=1}
6104:is in the relation
5952:, the equation for
4821:{\displaystyle y/a}
4730:{\displaystyle x/b}
4666:moves the graph up
4659:{\displaystyle y-k}
4590:{\displaystyle x-h}
4161:Pythagorean theorem
3588:{\displaystyle B=0}
3562:{\displaystyle A=C}
3390: not all zero.
3273:conjugate hyperbola
2141:quadratic equations
2083:Locus (mathematics)
1663:Discourse on Method
1573:Nicolas Malebranche
1443:Mindâbody dichotomy
1411:Doubt and certainty
1294:Apollonius of Perga
1200:coordinate geometry
457:Pythagorean theorem
18:Analytical geometry
9726:Numerical analysis
9335:Mathematical logic
9330:Information theory
9140:10.1007/BF03023730
8947:2018-07-18 at the
8805:2015-08-04 at the
8787:2015-08-04 at the
8776:Pierre de Fermat,
8094:
8074:
8042:
8022:
8002:
7982:
7944:
7924:
7904:
7884:
7861:
7841:
7817:Finding intercepts
7801:
7684:
7640:
7600:
7540:
7520:
7498:
7465:
7431:
7368:
7348:
7328:
7308:
7281:
7254:
7191:
7074:
7030:
6990:
6928:
6908:
6886:
6853:
6819:
6753:
6683:
6663:
6634:
6586:
6538:
6518:
6498:
6470:
6413:
6365:
6345:
6322:
6302:
6270:
6244:
6198:
6178:
6146:
6114:
6094:
6072:which is true, so
6062:
6020:
5962:
5942:
5910:
5878:
5846:
5738:
5718:
5640:
5608:
5585:
5553:
5518:
5469:
5420:{\displaystyle xy}
5417:
5394:
5353:
5333:
5313:
5293:
5273:
5253:
5233:
5213:
5193:
5173:
5153:
5133:
5113:
5054:
5018:
4972:
4952:
4905:
4885:
4841:
4818:
4790:
4767:
4747:
4727:
4699:
4676:
4656:
4630:
4607:
4587:
4561:
4537:
4507:
4492:a) y = f(x) = |x|
4453:
4349:
4194:
4149:
4020:Euclidean geometry
4004:
3984:Distance and angle
3940:
3814:algebraic equation
3730:
3691:
3639:
3585:
3559:
3526:
3477:
3428:
3393:
3288:quadratic equation
3276:
3169:
3128:
3087:
3028:
2981:
2923:are constants and
2897:
2762:
2651:
2594:
2572:
2550:
2530:
2503:
2481:
2461:
2415:
2343:
2324:Specifically, let
2255:
2218:affine coordinates
1970:
1726:
1716:Coordinate systems
1644:Cartesian geometry
1588:Francine Descartes
1433:Trademark argument
1341:algebraic geometry
1216:synthetic geometry
1206:, is the study of
1204:Cartesian geometry
9865:Analytic geometry
9852:
9851:
9451:Harmonic analysis
9089:Mikhail Postnikov
8987:978-0-07-161545-7
8898:978-3-540-74637-9
8856:, Athaeneum Press
8842:978-0-495-01166-8
8338:point of tangency
8097:{\displaystyle y}
8045:{\displaystyle b}
8025:{\displaystyle y}
8005:{\displaystyle b}
7947:{\displaystyle x}
7927:{\displaystyle x}
7907:{\displaystyle y}
7887:{\displaystyle y}
7871:coordinate axes.
7864:{\displaystyle y}
7844:{\displaystyle x}
7791:
7785:
7751:
7739:
7733:
7679:
7673:
7543:{\displaystyle y}
7523:{\displaystyle x}
7371:{\displaystyle x}
7351:{\displaystyle y}
7331:{\displaystyle y}
7181:
7175:
7141:
7129:
7123:
7069:
7063:
6931:{\displaystyle y}
6911:{\displaystyle x}
6686:{\displaystyle x}
6541:{\displaystyle y}
6521:{\displaystyle x}
6501:{\displaystyle y}
6368:{\displaystyle Q}
6348:{\displaystyle P}
6325:{\displaystyle P}
6280:which is false.
6201:{\displaystyle P}
6188:the equation for
6117:{\displaystyle Q}
5965:{\displaystyle Q}
5741:{\displaystyle Q}
5611:{\displaystyle P}
5356:{\displaystyle k}
5336:{\displaystyle h}
5316:{\displaystyle k}
5296:{\displaystyle h}
5276:{\displaystyle h}
5256:{\displaystyle k}
5236:{\displaystyle y}
5216:{\displaystyle a}
5196:{\displaystyle b}
5176:{\displaystyle x}
5156:{\displaystyle a}
5136:{\displaystyle a}
4975:{\displaystyle A}
4908:{\displaystyle y}
4844:{\displaystyle x}
4793:{\displaystyle y}
4777:as being dilated)
4770:{\displaystyle x}
4757:. (think of the
4750:{\displaystyle b}
4702:{\displaystyle x}
4679:{\displaystyle k}
4633:{\displaystyle y}
4610:{\displaystyle h}
4564:{\displaystyle x}
4412:
4344:
4144:
3789:. In coordinates
3391:
3371:
2832:
2831: where
2553:{\displaystyle P}
2484:{\displaystyle P}
1932:
1819:polar coordinates
1813:Polar coordinates
1626:
1625:
1478:Balloonist theory
1453:Coordinate system
1448:Analytic geometry
1333:geometric algebra
1212:coordinate system
1196:analytic geometry
1188:
1187:
1153:
1152:
876:List of geometers
559:Three-dimensional
548:
547:
16:(Redirected from
9872:
9840:
9839:
9828:
9827:
9816:
9815:
9805:
9804:
9736:Computer algebra
9711:Computer science
9431:Complex analysis
9265:
9258:
9251:
9242:
9241:
9224:
9216:
9186:
9164:
9142:
9113:
9085:
9063:Internet Archive
9051:
9033:
9001:
8998:
8992:
8991:
8973:
8967:
8938:
8932:
8927:
8908:
8902:
8901:
8876:
8870:
8863:
8857:
8850:
8844:
8827:
8818:
8797:
8791:
8782:pp. 91â103.
8774:
8768:
8762:
8756:
8750:
8741:
8735:
8729:
8728:
8696:
8690:
8684:
8678:
8677:
8653:
8647:
8644:
8638:
8632:
8626:
8625:
8595:
8589:
8588:
8560:
8549:
8540:
8539:
8493:
8487:
8486:
8449:
8443:
8442:
8409:
8383:Rotation of axes
8373:Applied geometry
8312:
8305:
8299:
8291:
8275:
8265:
8249:infinitely close
8103:
8101:
8100:
8095:
8083:
8081:
8080:
8075:
8051:
8049:
8048:
8043:
8031:
8029:
8028:
8023:
8011:
8009:
8008:
8003:
7992:, the parameter
7991:
7989:
7988:
7983:
7953:
7951:
7950:
7945:
7933:
7931:
7930:
7925:
7913:
7911:
7910:
7905:
7893:
7891:
7890:
7885:
7870:
7868:
7867:
7862:
7850:
7848:
7847:
7842:
7810:
7808:
7807:
7802:
7797:
7793:
7792:
7787:
7786:
7781:
7775:
7767:
7752:
7749:
7745:
7741:
7740:
7735:
7734:
7729:
7723:
7715:
7693:
7691:
7690:
7685:
7680:
7675:
7674:
7669:
7663:
7649:
7647:
7646:
7641:
7636:
7625:
7624:
7609:
7607:
7606:
7601:
7593:
7592:
7580:
7579:
7567:
7549:
7547:
7546:
7541:
7529:
7527:
7526:
7521:
7507:
7505:
7504:
7499:
7494:
7474:
7472:
7471:
7466:
7440:
7438:
7437:
7432:
7424:
7423:
7396:
7395:
7377:
7375:
7374:
7369:
7357:
7355:
7354:
7349:
7337:
7335:
7334:
7329:
7317:
7315:
7314:
7309:
7307:
7306:
7290:
7288:
7287:
7282:
7280:
7279:
7263:
7261:
7260:
7255:
7247:
7246:
7234:
7233:
7200:
7198:
7197:
7192:
7187:
7183:
7182:
7177:
7176:
7171:
7165:
7157:
7142:
7139:
7135:
7131:
7130:
7125:
7124:
7119:
7113:
7105:
7083:
7081:
7080:
7075:
7070:
7065:
7064:
7059:
7053:
7039:
7037:
7036:
7031:
7026:
7015:
7014:
6999:
6997:
6996:
6991:
6983:
6982:
6970:
6969:
6957:
6937:
6935:
6934:
6929:
6917:
6915:
6914:
6909:
6895:
6893:
6892:
6887:
6882:
6862:
6860:
6859:
6854:
6828:
6826:
6825:
6820:
6812:
6811:
6778:
6777:
6762:
6760:
6759:
6754:
6743:
6742:
6721:
6720:
6692:
6690:
6689:
6684:
6672:
6670:
6669:
6664:
6662:
6661:
6643:
6641:
6640:
6635:
6630:
6629:
6611:
6610:
6595:
6593:
6592:
6587:
6579:
6578:
6566:
6565:
6547:
6545:
6544:
6539:
6527:
6525:
6524:
6519:
6507:
6505:
6504:
6499:
6479:
6477:
6476:
6471:
6463:
6462:
6450:
6449:
6422:
6420:
6419:
6414:
6406:
6405:
6393:
6392:
6374:
6372:
6371:
6366:
6354:
6352:
6351:
6346:
6331:
6329:
6328:
6323:
6311:
6309:
6308:
6303:
6279:
6277:
6276:
6271:
6253:
6251:
6250:
6245:
6237:
6236:
6224:
6223:
6207:
6205:
6204:
6199:
6187:
6185:
6184:
6179:
6155:
6153:
6152:
6147:
6123:
6121:
6120:
6115:
6103:
6101:
6100:
6095:
6071:
6069:
6068:
6063:
6055:
6054:
6029:
6027:
6026:
6021:
6013:
6012:
6000:
5999:
5971:
5969:
5968:
5963:
5951:
5949:
5948:
5943:
5919:
5917:
5916:
5911:
5887:
5885:
5884:
5879:
5855:
5853:
5852:
5847:
5836:
5835:
5823:
5822:
5801:
5747:
5745:
5744:
5739:
5727:
5725:
5724:
5719:
5708:
5707:
5695:
5694:
5685:
5649:
5647:
5646:
5641:
5617:
5615:
5614:
5609:
5594:
5592:
5591:
5586:
5562:
5560:
5559:
5554:
5527:
5525:
5524:
5519:
5478:
5476:
5475:
5470:
5456:
5455:
5443:
5442:
5426:
5424:
5423:
5418:
5403:
5401:
5400:
5395:
5362:
5360:
5359:
5354:
5342:
5340:
5339:
5334:
5322:
5320:
5319:
5314:
5303:, vertical, and
5302:
5300:
5299:
5294:
5282:
5280:
5279:
5274:
5262:
5260:
5259:
5254:
5242:
5240:
5239:
5234:
5222:
5220:
5219:
5214:
5202:
5200:
5199:
5194:
5182:
5180:
5179:
5174:
5162:
5160:
5159:
5154:
5142:
5140:
5139:
5134:
5122:
5120:
5119:
5114:
5063:
5061:
5060:
5055:
5027:
5025:
5024:
5019:
5014:
4981:
4979:
4978:
4973:
4961:
4959:
4958:
4953:
4914:
4912:
4911:
4906:
4894:
4892:
4891:
4886:
4850:
4848:
4847:
4842:
4827:
4825:
4824:
4819:
4814:
4799:
4797:
4796:
4791:
4776:
4774:
4773:
4768:
4756:
4754:
4753:
4748:
4736:
4734:
4733:
4728:
4723:
4708:
4706:
4705:
4700:
4685:
4683:
4682:
4677:
4665:
4663:
4662:
4657:
4639:
4637:
4636:
4631:
4616:
4614:
4613:
4608:
4596:
4594:
4593:
4588:
4570:
4568:
4567:
4562:
4546:
4544:
4543:
4538:
4503:
4499:
4495:
4462:
4460:
4459:
4454:
4440:
4436:
4427:
4423:
4414:
4413:
4411:
4410:
4398:
4393:
4390:
4382:
4358:
4356:
4355:
4350:
4345:
4343:
4342:
4333:
4332:
4320:
4319:
4304:
4303:
4294:
4293:
4281:
4280:
4265:
4264:
4255:
4254:
4242:
4241:
4229:
4203:
4201:
4200:
4195:
4158:
4156:
4155:
4150:
4145:
4143:
4142:
4133:
4132:
4120:
4119:
4104:
4103:
4094:
4093:
4081:
4080:
4068:
3949:
3947:
3946:
3941:
3927:
3926:
3917:
3916:
3906:
3901:
3883:
3882:
3873:
3872:
3860:
3859:
3849:
3844:
3811:
3751:Quadric surfaces
3739:
3737:
3736:
3731:
3708:if we also have
3700:
3698:
3697:
3692:
3672:
3671:
3648:
3646:
3645:
3640:
3620:
3619:
3594:
3592:
3591:
3586:
3568:
3566:
3565:
3560:
3535:
3533:
3532:
3527:
3507:
3506:
3486:
3484:
3483:
3478:
3461:
3460:
3437:
3435:
3434:
3429:
3424:
3423:
3418:
3405:projective space
3402:
3400:
3399:
3394:
3392:
3389:
3372:
3370: with
3369:
3337:
3336:
3309:
3308:
3178:
3176:
3175:
3170:
3159:
3158:
3137:
3135:
3134:
3129:
3118:
3117:
3096:
3094:
3093:
3088:
3077:
3076:
3039:
3037:
3035:
3034:
3029:
3003:
2990:
2988:
2987:
2982:
2906:
2904:
2903:
2898:
2890:
2889:
2874:
2873:
2858:
2857:
2833:
2830:
2778:
2771:
2769:
2768:
2763:
2749:
2748:
2721:
2720:
2693:
2692:
2660:
2658:
2657:
2652:
2641:
2640:
2635:
2626:
2615:
2603:
2601:
2600:
2595:
2593:
2581:
2579:
2578:
2573:
2571:
2559:
2557:
2556:
2551:
2539:
2537:
2536:
2531:
2529:
2528:
2512:
2510:
2509:
2504:
2502:
2490:
2488:
2487:
2482:
2470:
2468:
2467:
2462:
2436:
2424:
2422:
2421:
2416:
2411:
2410:
2398:
2397:
2385:
2384:
2369:
2368:
2352:
2350:
2349:
2344:
2342:
2341:
2336:
2300:of the function
2264:
2262:
2261:
2256:
2208:Plane (geometry)
2198:Lines and planes
2115:-coordinate and
1979:
1977:
1976:
1971:
1960:
1933:
1931:
1930:
1918:
1917:
1908:
1791:of coordinates (
1640:Pierre de Fermat
1618:
1611:
1604:
1458:Cartesian circle
1422:Cogito, ergo sum
1378:
1355:
1354:
1198:, also known as
1180:
1173:
1166:
894:
893:
413:
412:
346:Zero-dimensional
51:
37:
36:
21:
9880:
9879:
9875:
9874:
9873:
9871:
9870:
9869:
9855:
9854:
9853:
9848:
9799:
9790:
9740:
9697:
9676:Systems science
9607:
9603:Homotopy theory
9569:
9536:
9488:
9460:
9407:
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9325:Category theory
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9231:
9206:10.2307/2305740
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8657:Stillwell, John
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8343:Similarly, the
8331:Euclidean space
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8221:
8215:
8207:
8197:generalizes to
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4504:d) y = 1/2 f(x)
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4483:Transformations
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2781:linear equation
2779:This is just a
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2214:Cartesian plane
2210:
2204:Line (geometry)
2202:Main articles:
2200:
2182:. The equation
2139:specify lines,
2085:
2077:Main articles:
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1994:
1956:
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1785:Euclidean space
1757:
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1742:Euclidean space
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1473:Cartesian diver
1401:Foundationalism
1386:
1353:
1343:, and his book
1337:cubic equations
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9754:Mathematicians
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9746:Related topics
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9681:Control theory
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9456:Measure theory
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9229:External links
9227:
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9110:978-0674823556
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8830:Stewart, James
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8704:"The Calculus"
8691:
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8608:(1): 248â249.
8590:
8575:
8541:
8517:
8497:Boyer, Carl B.
8488:
8473:
8453:Boyer, Carl B.
8444:
8433:
8413:Boyer, Carl B.
8403:
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8325:and curves in
8217:Main article:
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8147:surface normal
8109:
8108:Geometric axis
8106:
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8084:is called the
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5487:Main article:
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4521:
4500:c) y = f(x)-3
4496:b) y = f(x+3)
4491:
4484:
4481:
4452:
4449:
4446:
4443:
4439:
4435:
4431:
4426:
4422:
4418:
4409:
4406:
4403:
4397:
4389:
4385:
4381:
4371:is defined by
4348:
4341:
4337:
4331:
4327:
4323:
4318:
4314:
4310:
4307:
4302:
4298:
4292:
4288:
4284:
4279:
4275:
4271:
4268:
4263:
4259:
4253:
4249:
4245:
4240:
4236:
4232:
4227:
4224:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4148:
4141:
4137:
4131:
4127:
4123:
4118:
4114:
4110:
4107:
4102:
4098:
4092:
4088:
4084:
4079:
4075:
4071:
4066:
4063:
4051:
4044:
4037:
4030:
3985:
3982:
3939:
3936:
3933:
3930:
3925:
3921:
3915:
3911:
3905:
3900:
3897:
3894:
3890:
3886:
3881:
3877:
3871:
3868:
3864:
3858:
3854:
3848:
3843:
3840:
3837:
3834:
3831:
3827:
3808:
3801:
3794:
3755:Main article:
3752:
3749:
3748:
3747:
3746:
3745:
3729:
3726:
3723:
3720:
3717:
3690:
3687:
3684:
3681:
3678:
3675:
3670:
3666:
3654:
3638:
3635:
3632:
3629:
3626:
3623:
3618:
3614:
3602:
3601:
3600:
3584:
3581:
3578:
3558:
3555:
3552:
3525:
3522:
3519:
3516:
3513:
3510:
3505:
3501:
3476:
3473:
3470:
3467:
3464:
3459:
3455:
3427:
3422:
3417:
3387:
3384:
3381:
3378:
3375:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3335:
3331:
3327:
3324:
3321:
3318:
3315:
3312:
3307:
3303:
3299:
3261:Main article:
3258:
3257:Conic sections
3255:
3254:
3253:
3223:
3219:
3212:
3205:
3199:
3168:
3165:
3162:
3157:
3153:
3149:
3146:
3127:
3124:
3121:
3116:
3112:
3108:
3105:
3086:
3083:
3080:
3075:
3071:
3067:
3064:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3006:
3002:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2896:
2893:
2888:
2884:
2880:
2877:
2872:
2868:
2864:
2861:
2856:
2852:
2848:
2845:
2842:
2839:
2836:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2761:
2758:
2755:
2752:
2747:
2743:
2739:
2736:
2733:
2730:
2727:
2724:
2719:
2715:
2711:
2708:
2705:
2702:
2699:
2696:
2691:
2687:
2683:
2680:
2677:
2674:
2650:
2647:
2644:
2639:
2634:
2629:
2625:
2621:
2618:
2614:
2592:
2570:
2549:
2527:
2523:
2501:
2480:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2435:
2414:
2409:
2405:
2401:
2396:
2392:
2388:
2383:
2379:
2375:
2372:
2367:
2363:
2340:
2335:
2314:
2313:
2291:
2281:
2254:
2251:
2248:
2245:
2242:
2239:
2199:
2196:
2145:conic sections
2074:
2071:
2040:Main article:
2037:
2034:
2021:-axis and the
1996:Main article:
1993:
1990:
1969:
1966:
1963:
1959:
1955:
1952:
1949:
1946:
1943:
1940:
1936:
1929:
1925:
1921:
1916:
1912:
1906:
1903:
1899:
1896:
1893:
1890:
1886:
1883:
1880:
1876:
1873:
1870:
1867:
1863:
1860:
1857:
1811:Main article:
1808:
1805:
1789:ordered triple
1753:Main article:
1750:
1747:
1736:has a pair of
1714:Main article:
1711:
1708:
1704:Leonhard Euler
1636:René Descartes
1624:
1623:
1621:
1620:
1613:
1606:
1598:
1595:
1594:
1591:
1590:
1585:
1580:
1578:Baruch Spinoza
1575:
1570:
1564:
1561:
1560:
1557:
1556:
1553:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1511:
1508:
1507:
1504:
1503:
1500:
1499:
1492:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1418:
1416:Dream argument
1413:
1408:
1403:
1398:
1393:
1387:
1384:
1383:
1380:
1379:
1371:
1370:
1368:René Descartes
1364:
1363:
1352:
1351:Western Europe
1349:
1324:
1321:
1286:mathematician
1279:
1278:Ancient Greece
1276:
1274:
1271:
1229:, and also in
1186:
1185:
1183:
1182:
1175:
1168:
1160:
1157:
1156:
1151:
1150:
1149:
1148:
1143:
1135:
1134:
1130:
1129:
1128:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1069:
1068:
1064:
1063:
1062:
1061:
1056:
1051:
1046:
1041:
1036:
1031:
1026:
1018:
1017:
1013:
1012:
1011:
1010:
1005:
1000:
995:
990:
985:
980:
975:
970:
965:
960:
955:
947:
946:
942:
941:
940:
939:
934:
929:
924:
919:
914:
909:
901:
900:
892:
888:
887:
886:
883:
882:
879:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
672:
668:
667:
666:
663:
662:
656:
655:
652:
651:
646:
640:
633:
632:
631:
628:
627:
624:
623:
618:
613:
611:Platonic Solid
608:
603:
598:
593:
588:
583:
582:
581:
570:
569:
563:
557:
556:
555:
552:
551:
546:
545:
544:
543:
538:
533:
525:
524:
518:
517:
516:
515:
510:
502:
501:
495:
494:
493:
492:
487:
482:
477:
469:
468:
462:
461:
460:
459:
454:
449:
441:
440:
434:
433:
432:
431:
426:
421:
411:
405:
404:
403:
400:
399:
396:
395:
390:
389:
388:
383:
372:
366:
365:
364:
361:
360:
357:
356:
350:
344:
343:
342:
339:
338:
335:
334:
329:
324:
318:
317:
312:
307:
297:
292:
287:
281:
280:
271:
267:
266:
263:
259:
258:
257:
256:
253:
252:
249:
248:
247:
246:
236:
231:
226:
221:
216:
215:
214:
204:
199:
194:
193:
192:
187:
182:
172:
171:
170:
165:
155:
150:
145:
140:
135:
130:
129:
128:
123:
122:
121:
106:
100:
94:
93:
92:
89:
88:
86:
85:
75:
69:
66:
65:
52:
44:
43:
26:
9:
6:
4:
3:
2:
9877:
9866:
9863:
9862:
9860:
9845:
9844:
9835:
9833:
9832:
9823:
9821:
9820:
9811:
9809:
9808:
9803:
9797:
9796:
9793:
9787:
9784:
9782:
9779:
9777:
9774:
9772:
9769:
9767:
9764:
9760:
9757:
9756:
9755:
9752:
9751:
9749:
9747:
9743:
9737:
9734:
9732:
9729:
9727:
9724:
9722:
9719:
9717:
9714:
9712:
9709:
9708:
9706:
9704:
9703:Computational
9700:
9692:
9689:
9687:
9684:
9682:
9679:
9678:
9677:
9674:
9672:
9669:
9667:
9664:
9662:
9659:
9657:
9654:
9652:
9649:
9647:
9644:
9642:
9639:
9637:
9634:
9632:
9629:
9627:
9624:
9622:
9619:
9618:
9616:
9614:
9610:
9604:
9601:
9599:
9596:
9594:
9591:
9589:
9586:
9584:
9581:
9580:
9578:
9576:
9572:
9566:
9563:
9561:
9558:
9556:
9553:
9551:
9548:
9547:
9545:
9543:
9542:Number theory
9539:
9533:
9530:
9528:
9525:
9523:
9520:
9518:
9515:
9513:
9510:
9508:
9505:
9503:
9500:
9499:
9497:
9495:
9491:
9485:
9482:
9480:
9477:
9475:
9474:Combinatorics
9472:
9471:
9469:
9467:
9463:
9457:
9454:
9452:
9449:
9447:
9444:
9442:
9439:
9437:
9434:
9432:
9429:
9427:
9426:Real analysis
9424:
9422:
9419:
9418:
9416:
9414:
9410:
9404:
9401:
9399:
9396:
9394:
9391:
9389:
9386:
9384:
9381:
9379:
9376:
9374:
9371:
9369:
9366:
9365:
9363:
9361:
9357:
9351:
9348:
9346:
9343:
9341:
9338:
9336:
9333:
9331:
9328:
9326:
9323:
9322:
9320:
9318:
9314:
9308:
9305:
9303:
9300:
9296:
9293:
9291:
9288:
9287:
9286:
9283:
9282:
9279:
9274:
9266:
9261:
9259:
9254:
9252:
9247:
9246:
9243:
9236:
9233:
9232:
9223:
9218:
9215:
9211:
9207:
9203:
9199:
9195:
9194:
9188:
9185:
9181:
9177:
9173:
9172:
9166:
9163:
9159:
9156:(3): 99â105,
9155:
9151:
9150:
9144:
9141:
9137:
9133:
9129:
9128:
9122:
9121:
9112:
9106:
9102:
9097:
9094:
9090:
9087:
9084:
9082:0-321-01618-1
9078:
9074:
9073:
9067:
9064:
9060:
9056:
9053:
9050:
9044:
9040:
9035:
9032:
9026:
9022:
9017:
9016:
8997:
8989:
8983:
8979:
8972:
8966:
8962:
8958:
8954:
8950:
8946:
8943:
8937:
8931:
8926:
8924:0-471-75715-2
8920:
8916:
8915:
8907:
8900:
8894:
8890:
8886:
8882:
8875:
8868:
8862:
8855:
8849:
8843:
8839:
8835:
8831:
8826:
8824:
8816:
8812:
8808:
8804:
8801:
8796:
8790:
8786:
8783:
8779:
8773:
8766:
8761:
8754:
8749:
8747:
8739:
8734:
8727:
8723:
8721:0-471-18082-3
8717:
8713:
8709:
8705:
8701:
8695:
8688:
8683:
8676:
8672:
8670:0-387-95336-1
8666:
8662:
8658:
8652:
8643:
8636:
8631:
8623:
8619:
8615:
8611:
8607:
8603:
8602:
8594:
8587:
8584:
8578:
8576:9780471543978
8572:
8568:
8564:
8559:
8554:
8548:
8546:
8538:
8535:
8531:
8526:
8520:
8518:0-471-54397-7
8514:
8510:
8506:
8502:
8498:
8492:
8485:
8482:
8476:
8474:0-471-54397-7
8470:
8466:
8462:
8458:
8454:
8448:
8441:
8436:
8434:0-471-54397-7
8430:
8426:
8422:
8418:
8414:
8408:
8404:
8394:
8391:
8389:
8386:
8384:
8381:
8379:
8378:Cross product
8376:
8374:
8371:
8370:
8364:
8362:
8361:Tangent space
8358:
8354:
8350:
8346:
8345:tangent plane
8341:
8339:
8334:
8332:
8329:-dimensional
8328:
8324:
8320:
8316:
8309:
8303:
8296:
8289:
8285:
8281:
8274:
8270:
8263:
8259:
8255:
8250:
8246:
8245:straight line
8242:
8238:
8235:) to a plane
8234:
8230:
8226:
8220:
8210:
8202:
8200:
8199:orthogonality
8196:
8192:
8191:normal vector
8188:
8184:
8180:
8176:
8172:
8171:tangent plane
8168:
8167:perpendicular
8164:
8160:
8156:
8152:
8148:
8143:
8141:
8137:
8133:
8132:perpendicular
8129:
8125:
8120:
8118:
8113:
8105:
8091:
8068:
8065:
8062:
8052:or the point
8039:
8019:
7999:
7979:
7976:
7973:
7970:
7967:
7964:
7957:For the line
7955:
7941:
7921:
7901:
7881:
7872:
7858:
7838:
7828:
7824:
7814:
7811:
7798:
7794:
7788:
7782:
7777:
7771:
7768:
7764:
7760:
7756:
7742:
7736:
7730:
7725:
7719:
7716:
7712:
7708:
7704:
7694:
7681:
7676:
7670:
7665:
7659:
7656:
7637:
7633:
7629:
7626:
7621:
7617:
7597:
7594:
7589:
7585:
7581:
7576:
7568:
7564:
7560:
7537:
7517:
7508:
7495:
7491:
7487:
7484:
7481:
7462:
7459:
7456:
7453:
7450:
7447:
7428:
7425:
7420:
7416:
7412:
7409:
7406:
7403:
7400:
7397:
7392:
7388:
7379:
7365:
7345:
7325:
7303:
7299:
7276:
7272:
7251:
7248:
7243:
7239:
7235:
7230:
7222:
7219:
7216:
7205:
7201:
7188:
7184:
7178:
7172:
7167:
7161:
7158:
7154:
7150:
7146:
7132:
7126:
7120:
7115:
7109:
7106:
7102:
7098:
7094:
7084:
7071:
7066:
7060:
7055:
7049:
7046:
7027:
7023:
7019:
7016:
7011:
7007:
6987:
6984:
6979:
6975:
6971:
6966:
6958:
6954:
6950:
6939:
6925:
6905:
6896:
6883:
6879:
6875:
6872:
6869:
6850:
6847:
6844:
6841:
6838:
6835:
6816:
6813:
6808:
6804:
6800:
6797:
6794:
6791:
6788:
6785:
6782:
6779:
6774:
6770:
6750:
6747:
6739:
6735:
6731:
6728:
6722:
6717:
6709:
6706:
6703:
6680:
6658:
6654:
6644:
6631:
6626:
6622:
6618:
6615:
6612:
6607:
6603:
6583:
6580:
6575:
6571:
6567:
6562:
6558:
6549:
6535:
6515:
6495:
6487:
6486:Substitution:
6483:
6480:
6467:
6464:
6459:
6455:
6451:
6446:
6438:
6435:
6432:
6410:
6407:
6402:
6398:
6394:
6389:
6385:
6376:
6362:
6342:
6333:
6319:
6296:
6293:
6290:
6267:
6264:
6261:
6241:
6238:
6233:
6229:
6225:
6220:
6216:
6195:
6172:
6169:
6166:
6140:
6137:
6134:
6111:
6088:
6085:
6082:
6059:
6056:
6051:
6043:
6040:
6017:
6014:
6009:
6005:
6001:
5996:
5988:
5985:
5982:
5959:
5936:
5933:
5930:
5904:
5901:
5898:
5872:
5869:
5866:
5840:
5837:
5832:
5828:
5824:
5819:
5811:
5808:
5805:
5791:
5788:
5785:
5776:
5773:
5770:
5764:
5761:
5758:
5735:
5712:
5709:
5704:
5700:
5696:
5691:
5687:
5675:
5672:
5669:
5660:
5657:
5634:
5631:
5628:
5605:
5598:For example,
5596:
5579:
5576:
5573:
5547:
5544:
5541:
5535:
5512:
5509:
5506:
5500:
5490:
5480:
5466:
5463:
5460:
5457:
5452:
5448:
5444:
5439:
5435:
5414:
5411:
5388:
5385:
5382:
5376:
5369:Suppose that
5367:
5364:
5350:
5330:
5310:
5290:
5270:
5250:
5230:
5210:
5190:
5170:
5150:
5130:
5110:
5107:
5098:
5095:
5092:
5086:
5080:
5077:
5074:
5071:
5048:
5042:
5039:
5036:
5015:
5011:
5007:
5004:
5001:
4992:
4990:
4969:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4902:
4895:and changing
4882:
4879:
4876:
4873:
4870:
4867:
4864:
4861:
4858:
4838:
4830:
4815:
4811:
4807:
4787:
4779:
4764:
4744:
4724:
4720:
4716:
4696:
4688:
4673:
4653:
4650:
4647:
4627:
4619:
4604:
4584:
4581:
4578:
4558:
4550:
4549:
4548:
4531:
4528:
4525:
4519:
4512:The graph of
4510:
4489:
4480:
4478:
4474:
4470:
4466:
4450:
4447:
4444:
4441:
4395:
4383:
4370:
4366:
4362:
4346:
4339:
4329:
4325:
4321:
4316:
4312:
4305:
4300:
4290:
4286:
4282:
4277:
4273:
4266:
4261:
4251:
4247:
4243:
4238:
4234:
4225:
4222:
4213:
4212:of the line.
4211:
4207:
4191:
4185:
4179:
4176:
4173:
4170:
4162:
4146:
4139:
4129:
4125:
4121:
4116:
4112:
4105:
4100:
4090:
4086:
4082:
4077:
4073:
4064:
4061:
4050:
4043:
4036:
4029:
4025:
4021:
4017:
4013:
4009:
4000:
3995:
3991:
3981:
3979:
3975:
3971:
3967:
3963:
3959:
3955:
3950:
3937:
3934:
3931:
3928:
3923:
3919:
3913:
3909:
3903:
3898:
3895:
3892:
3888:
3884:
3879:
3875:
3869:
3866:
3862:
3856:
3852:
3846:
3841:
3838:
3835:
3832:
3829:
3825:
3816:
3815:
3807:
3800:
3793:
3788:
3784:
3780:
3776:
3773:-dimensional
3772:
3768:
3764:
3758:
3743:
3727:
3724:
3721:
3718:
3715:
3707:
3706:
3704:
3688:
3685:
3682:
3679:
3676:
3673:
3668:
3664:
3655:
3652:
3636:
3633:
3630:
3627:
3624:
3621:
3616:
3612:
3603:
3598:
3582:
3579:
3576:
3556:
3553:
3550:
3542:
3541:
3539:
3523:
3520:
3517:
3514:
3511:
3508:
3503:
3499:
3490:
3489:
3488:
3474:
3471:
3468:
3465:
3462:
3457:
3453:
3444:
3443:
3438:
3425:
3420:
3406:
3385:
3382:
3379:
3376:
3373:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3333:
3329:
3325:
3322:
3319:
3316:
3313:
3310:
3305:
3301:
3297:
3289:
3285:
3281:
3274:
3269:
3264:
3263:Conic section
3251:
3247:
3243:
3239:
3235:
3231:
3227:
3224:
3218:
3211:
3204:
3200:
3197:
3193:
3189:
3185:
3182:
3181:
3180:
3166:
3163:
3160:
3155:
3151:
3147:
3144:
3125:
3122:
3119:
3114:
3110:
3106:
3103:
3084:
3081:
3078:
3073:
3069:
3065:
3062:
3054:
3050:
3045:
3043:
3022:
3019:
3016:
3013:
3010:
3004:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2934:
2930:
2926:
2922:
2918:
2914:
2910:
2894:
2886:
2882:
2878:
2875:
2870:
2866:
2862:
2859:
2854:
2850:
2846:
2840:
2837:
2834:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2782:
2776:
2773:which is the
2759:
2756:
2753:
2745:
2741:
2737:
2734:
2728:
2725:
2717:
2713:
2709:
2706:
2700:
2697:
2689:
2685:
2681:
2678:
2672:
2664:
2648:
2645:
2637:
2627:
2616:
2547:
2525:
2521:
2478:
2455:
2452:
2449:
2446:
2443:
2437:
2407:
2403:
2399:
2394:
2390:
2386:
2381:
2377:
2370:
2365:
2361:
2338:
2322:
2320:
2319:normal vector
2311:
2307:
2303:
2299:
2295:
2292:
2289:
2285:
2282:
2279:
2275:
2271:
2268:
2267:
2266:
2252:
2249:
2246:
2243:
2240:
2237:
2229:
2228:
2223:
2219:
2215:
2209:
2205:
2195:
2193:
2190: =
2189:
2186: +
2185:
2181:
2177:
2173:
2169:
2166: +
2165:
2161:
2158: =
2157:
2153:
2148:
2146:
2142:
2138:
2134:
2130:
2127: =
2126:
2122:
2118:
2114:
2110:
2107: =
2106:
2102:
2098:
2094:
2090:
2084:
2080:
2070:
2068:
2064:
2060:
2056:
2053:
2049:
2043:
2033:
2031:
2027:
2024:
2020:
2016:
2013:
2009:
2005:
1999:
1989:
1988:coordinates.
1987:
1983:
1967:
1961:
1957:
1953:
1947:
1944:
1941:
1938:
1934:
1927:
1923:
1919:
1914:
1910:
1904:
1901:
1897:
1894:
1891:
1888:
1884:
1881:
1878:
1874:
1871:
1868:
1865:
1861:
1858:
1855:
1847:
1843:
1839:
1835:
1831:
1828:
1824:
1820:
1814:
1804:
1802:
1798:
1794:
1790:
1786:
1782:
1778:
1774:
1770:
1766:
1762:
1756:
1746:
1743:
1739:
1735:
1731:
1722:
1717:
1707:
1705:
1700:
1696:
1692:
1687:
1685:
1681:
1677:
1673:
1669:
1665:
1664:
1659:
1655:
1653:
1647:
1645:
1641:
1637:
1631:
1619:
1614:
1612:
1607:
1605:
1600:
1599:
1597:
1596:
1589:
1586:
1584:
1581:
1579:
1576:
1574:
1571:
1569:
1566:
1565:
1559:
1558:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1512:
1506:
1505:
1498:
1497:
1493:
1491:
1490:
1486:
1484:
1481:
1479:
1476:
1474:
1471:
1469:
1468:Rule of signs
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1423:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1388:
1382:
1381:
1377:
1373:
1372:
1369:
1366:
1365:
1361:
1357:
1356:
1348:
1346:
1342:
1338:
1334:
1330:
1320:
1318:
1314:
1309:
1305:
1301:
1300:
1295:
1291:
1289:
1285:
1270:
1268:
1262:
1260:
1256:
1252:
1248:
1244:
1240:
1239:space science
1236:
1232:
1228:
1224:
1219:
1217:
1213:
1209:
1205:
1201:
1197:
1193:
1181:
1176:
1174:
1169:
1167:
1162:
1161:
1159:
1158:
1147:
1144:
1142:
1139:
1138:
1137:
1136:
1132:
1131:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1072:
1071:
1070:
1066:
1065:
1060:
1057:
1055:
1052:
1050:
1047:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1021:
1020:
1019:
1015:
1014:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
950:
949:
948:
944:
943:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
904:
903:
902:
899:
896:
895:
885:
884:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
673:
665:
664:
661:
658:
657:
650:
647:
645:
642:
641:
636:
630:
629:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
584:
580:
577:
576:
575:
572:
571:
568:
565:
564:
560:
554:
553:
542:
539:
537:
536:Circumference
534:
532:
529:
528:
527:
526:
523:
520:
519:
514:
511:
509:
506:
505:
504:
503:
500:
499:Quadrilateral
497:
496:
491:
488:
486:
483:
481:
478:
476:
473:
472:
471:
470:
467:
466:Parallelogram
464:
463:
458:
455:
453:
450:
448:
445:
444:
443:
442:
439:
436:
435:
430:
427:
425:
422:
420:
417:
416:
415:
414:
408:
402:
401:
394:
391:
387:
384:
382:
379:
378:
377:
374:
373:
369:
363:
362:
355:
352:
351:
347:
341:
340:
333:
330:
328:
325:
323:
320:
319:
316:
313:
311:
308:
305:
304:Perpendicular
301:
300:Orthogonality
298:
296:
293:
291:
288:
286:
283:
282:
279:
276:
275:
274:
264:
261:
260:
255:
254:
245:
242:
241:
240:
237:
235:
232:
230:
227:
225:
224:Computational
222:
220:
217:
213:
210:
209:
208:
205:
203:
200:
198:
195:
191:
188:
186:
183:
181:
178:
177:
176:
173:
169:
166:
164:
161:
160:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
134:
131:
127:
124:
120:
117:
116:
115:
112:
111:
110:
109:Non-Euclidean
107:
105:
102:
101:
97:
91:
90:
83:
79:
76:
74:
71:
70:
68:
67:
63:
59:
55:
50:
46:
45:
42:
39:
38:
33:
19:
9841:
9829:
9817:
9798:
9731:Optimization
9593:Differential
9517:Differential
9506:
9484:Order theory
9479:Graph theory
9383:Group theory
9221:
9200:(2): 76â86,
9197:
9191:
9178:(1): 33â36,
9175:
9169:
9153:
9147:
9134:(4): 38â44,
9131:
9125:
9100:
9071:
9061:, link from
9038:
9020:
8996:
8977:
8971:
8952:
8940:Silvio Levy
8936:
8913:
8906:
8880:
8874:
8866:
8861:
8853:
8848:
8833:
8814:
8810:
8795:
8777:
8772:
8760:
8740:, p. 82
8733:
8725:
8707:
8700:Cooke, Roger
8694:
8689:, p. 74
8682:
8674:
8660:
8651:
8642:
8630:
8605:
8599:
8593:
8582:
8580:
8562:
8533:
8530:a posteriori
8529:
8524:
8522:
8504:
8491:
8480:
8478:
8460:
8447:
8438:
8420:
8407:
8393:Vector space
8344:
8342:
8337:
8335:
8326:
8323:space curves
8318:
8307:
8301:
8294:
8287:
8283:
8279:
8272:
8268:
8261:
8257:
8253:
8232:
8229:tangent line
8228:
8222:
8208:
8194:
8190:
8181:normal to a
8174:
8158:
8150:
8149:, or simply
8146:
8144:
8140:tangent line
8135:
8127:
8121:
8116:
8114:
8111:
8104:-intercept.
7956:
7873:
7830:
7812:
7695:
7509:
7380:
7203:
7202:
7085:
6940:
6897:
6645:
6550:
6508:in terms of
6485:
6484:
6481:
6377:
6334:
5597:
5492:
5368:
5365:
4993:
4985:
4511:
4508:
4476:
4472:
4464:
4368:
4364:
4214:
4205:
4048:
4041:
4034:
4027:
4005:
3966:hyperboloids
3951:
3817:
3805:
3798:
3791:
3770:
3766:
3762:
3760:
3445:
3442:discriminant
3439:
3277:
3249:
3245:
3241:
3233:
3229:
3225:
3216:
3209:
3202:
3195:
3191:
3187:
3183:
3048:
3046:
3042:general form
3041:
3038:as a normal.
2932:
2928:
2924:
2920:
2916:
2912:
2908:
2775:point-normal
2774:
2323:
2315:
2309:
2305:
2301:
2293:
2290:of the line.
2283:
2280:of the line.
2269:
2225:
2221:
2211:
2191:
2187:
2183:
2176:intersection
2167:
2163:
2159:
2155:
2149:
2136:
2132:
2128:
2124:
2116:
2112:
2108:
2104:
2097:solution set
2086:
2079:Solution set
2066:
2062:
2058:
2054:
2047:
2045:
2029:
2025:
2018:
2014:
2007:
2001:
1845:
1841:
1837:
1833:
1829:
1822:
1816:
1800:
1796:
1792:
1780:
1776:
1773:ordered pair
1768:
1764:
1758:
1727:
1699:Introduction
1698:
1694:
1690:
1688:
1684:van Schooten
1668:La Geometrie
1667:
1661:
1657:
1652:La Géométrie
1650:
1648:
1643:
1633:
1535:La Géométrie
1494:
1489:Res cogitans
1487:
1483:Wax argument
1447:
1420:
1391:Cartesianism
1344:
1329:Omar Khayyam
1326:
1316:
1313:a posteriori
1312:
1303:
1297:
1292:
1281:
1263:
1251:differential
1220:
1203:
1199:
1195:
1189:
1008:Parameshvara
821:Parameshvara
591:Dodecahedron
175:Differential
152:
9843:WikiProject
9686:Game theory
9666:Probability
9403:Homological
9393:Multilinear
9373:Commutative
9350:Type theory
9317:Foundations
9273:mathematics
8565:. pp.
8266:at a point
8239:at a given
8231:(or simply
8157:at a point
8136:normal line
7827:y-intercept
7823:x-intercept
7204:Elimination
5183:-axis. The
4361:dot product
3962:paraboloids
2663:dot product
2288:y-intercept
2212:Lines in a
1982:cylindrical
1738:real number
1710:Coordinates
1496:Res extensa
1396:Rationalism
1315:instead of
1243:spaceflight
1227:engineering
1192:mathematics
1133:Present day
1080:Lobachevsky
1067:1700sâ1900s
1024:JyeáčŁáčhadeva
1016:1400sâ1700s
968:Brahmagupta
791:Lobachevsky
771:JyeáčŁáčhadeva
721:Brahmagupta
649:Hypersphere
621:Tetrahedron
596:Icosahedron
168:Diophantine
9671:Statistics
9550:Arithmetic
9512:Arithmetic
9378:Elementary
9345:Set theory
9220:Pecl, J.,
9055:John Casey
9008:References
8738:Boyer 2004
8687:Boyer 2004
8315:derivative
8117:axial line
6312:is not in
3954:ellipsoids
2604:such that
2425:, and let
1654:(Geometry)
1628:See also:
1428:Evil demon
1385:Philosophy
1288:Menaechmus
993:al-Yasamin
937:Apollonius
932:Archimedes
922:Pythagoras
912:Baudhayana
866:al-Yasamin
816:Pythagoras
711:Baudhayana
701:Archimedes
696:Apollonius
601:Octahedron
452:Hypotenuse
327:Similarity
322:Congruence
234:Incidence
185:Symplectic
180:Riemannian
163:Arithmetic
138:Projective
126:Hyperbolic
54:Projecting
9598:Geometric
9588:Algebraic
9527:Euclidean
9502:Algebraic
9398:Universal
8957:CRC Press
8767:, pg. 436
8765:Katz 1998
8755:, pg. 442
8753:Katz 1998
8195:normality
7778:−
7666:±
7460:−
7448:−
7413:−
7398:−
7236:−
7220:−
7168:−
7056:±
6848:−
6836:−
6801:−
6780:−
6732:−
6707:−
6619:−
6436:−
6041:−
5986:−
5809:−
5458:−
5096:−
4947:
4932:
4923:−
4880:
4865:
4831:Changing
4780:Changing
4689:Changing
4651:−
4620:Changing
4582:−
4551:Changing
4448:θ
4445:
4384:⋅
4322:−
4283:−
4244:−
4180:
4171:θ
4122:−
4083:−
3970:cylinders
3889:∑
3826:∑
3703:hyperbola
3674:−
3622:−
3509:−
3463:−
2841:−
2738:−
2710:−
2682:−
2628:−
2617:⋅
2017:from the
1986:spherical
1948:
1939:θ
1895:θ
1892:
1872:θ
1869:
1695:Discourse
1525:The World
1406:Mechanism
1308:Descartes
1247:algebraic
1110:Minkowski
1029:Descartes
963:Aryabhata
958:KÄtyÄyana
889:by period
801:Minkowski
776:KÄtyÄyana
736:Descartes
681:Aryabhata
660:Geometers
644:Tesseract
508:Trapezoid
480:Rectangle
273:Dimension
158:Algebraic
148:Synthetic
119:Spherical
104:Euclidean
9859:Category
9819:Category
9575:Topology
9522:Discrete
9507:Analytic
9494:Geometry
9466:Discrete
9421:Calculus
9413:Analysis
9368:Abstract
9307:Glossary
9290:Timeline
9118:Articles
8945:Archived
8942:Quadrics
8832:(2008).
8803:Archived
8785:Archived
8702:(1997).
8555:(1991).
8534:a priori
8499:(1991).
8455:(1991).
8415:(1991).
8367:See also
8225:geometry
8165:that is
8124:geometry
6208:becomes
5972:becomes
4471:between
4438:‖
4430:‖
4425:‖
4417:‖
4016:formulas
4008:distance
3990:Distance
3651:parabola
2278:gradient
2143:specify
2089:equation
1676:calculus
1360:a series
1358:Part of
1317:a priori
1255:discrete
1235:rocketry
1231:aviation
1210:using a
1208:geometry
1100:Poincaré
1044:Minggatu
1003:Yang Hui
973:Virasena
861:Yang Hui
856:Virasena
826:Poincaré
806:Minggatu
586:Cylinder
531:Diameter
490:Rhomboid
447:Altitude
438:Triangle
332:Symmetry
310:Parallel
295:Diagonal
265:Features
262:Concepts
153:Analytic
114:Elliptic
96:Branches
82:Timeline
41:Geometry
9831:Commons
9613:Applied
9583:General
9360:Algebra
9285:History
9214:2305740
9091:(1982)
9057:(1885)
9041:, AMS,
8959:, from
8637:, p. 92
8622:3217882
8583:Algebra
8567:241â242
8349:surface
8313:is the
8243:is the
8233:tangent
8219:Tangent
8169:to the
8155:surface
8153:, to a
4467:is the
4208:is the
4047:,
4040:) and (
4033:,
3775:surface
3769:, is a
3763:quadric
3538:ellipse
3278:In the
3179:where:
2296:is the
2286:is the
2272:is the
2265:where:
2172:surface
1832:, with
1799:,
1795:,
1779:,
1273:History
1223:physics
1125:Coxeter
1105:Hilbert
1090:Riemann
1039:Huygens
998:al-Tusi
988:KhayyĂĄm
978:Alhazen
945:1â1400s
846:al-Tusi
831:Riemann
781:KhayyĂĄm
766:Huygens
761:Hilbert
731:Coxeter
691:Alhazen
669:by name
606:Pyramid
485:Rhombus
429:Polygon
381:segment
229:Fractal
212:Digital
197:Complex
78:History
73:Outline
9532:Finite
9388:Linear
9295:Future
9271:Major
9212:
9107:
9079:
9045:
9027:
8984:
8921:
8895:
8840:
8718:
8667:
8620:
8573:
8525:Conics
8515:
8471:
8431:
8306:where
8227:, the
8189:, the
8163:vector
8151:normal
8128:normal
7264:. The
4686:units.
4617:units.
4463:where
4204:where
4177:arctan
3978:planes
3976:, and
3958:sphere
3597:circle
3282:, the
3238:vector
3232:, and
3190:, and
2931:, and
2222:linear
2123:, and
2093:subset
2012:radius
2010:, its
1945:arctan
1672:French
1562:People
1463:Folium
1323:Persia
1304:Conics
1241:, and
1146:Gromov
1141:Atiyah
1120:Veblen
1115:Cartan
1085:Bolyai
1054:Sakabe
1034:Pascal
927:Euclid
917:Manava
851:Veblen
836:Sakabe
811:Pascal
796:Manava
756:Gromov
741:Euclid
726:Cartan
716:Bolyai
706:Atiyah
616:Sphere
579:cuboid
567:Volume
522:Circle
475:Square
393:Length
315:Vertex
219:Convex
202:Finite
143:Affine
58:sphere
9759:lists
9302:Lists
9275:areas
9210:JSTOR
9013:Books
8618:JSTOR
8553:Boyer
8425:94â95
8399:Notes
8353:plane
8347:to a
8311:'
8298:'
8241:point
8237:curve
8187:force
8183:plane
8161:is a
4469:angle
4210:slope
4012:angle
3994:Angle
3974:cones
3785:of a
3783:zeros
3779:locus
3765:, or
3286:of a
3284:graph
2274:slope
2152:curve
2101:locus
2052:angle
2023:angle
1827:angle
1734:point
1730:plane
1680:Latin
1509:Works
1296:, in
1284:Greek
1095:Klein
1075:Gauss
1049:Euler
983:Sijzi
953:Zhang
907:Ahmes
871:Zhang
841:Sijzi
786:Klein
751:Gauss
746:Euler
686:Ahmes
419:Plane
354:Point
290:Curve
285:Angle
62:plane
60:to a
9105:ISBN
9077:ISBN
9043:ISBN
9025:ISBN
8982:ISBN
8919:ISBN
8893:ISBN
8838:ISBN
8716:ISBN
8665:ISBN
8571:ISBN
8513:ISBN
8469:ISBN
8429:ISBN
8179:line
8126:, a
7851:and
7825:and
6355:and
6156:for
5920:for
5728:and
5528:and
5343:and
5263:and
4475:and
4367:and
4010:and
3992:and
3686:>
3569:and
3521:<
2919:and
2206:and
2135:and
2121:line
2081:and
1638:and
1282:The
1257:and
1225:and
1059:Aida
676:Aida
635:Four
574:Cube
541:Area
513:Kite
424:Area
376:Line
9202:doi
9180:doi
9158:doi
9136:doi
8963:at
8885:doi
8712:326
8610:doi
8606:123
8509:156
8465:142
8317:of
8223:In
8122:In
7750:and
7140:and
6254:or
6030:or
5650::
4987:on
4944:cos
4929:sin
4915:to
4877:sin
4862:cos
4851:to
4800:to
4709:to
4640:to
4571:to
4442:cos
3960:),
3781:of
3656:if
3604:if
3543:if
3491:if
3049:not
2540:to
2276:or
2002:In
1984:or
1889:sin
1866:cos
1817:In
1803:).
1202:or
1190:In
898:BCE
386:ray
9861::
9208:,
9198:55
9196:,
9176:58
9174:,
9154:37
9152:,
9130:,
8955:,
8928:,
8891:,
8822:^
8745:^
8724:.
8714:.
8706:.
8673:.
8616:.
8604:.
8579:.
8569:.
8561:.
8544:^
8521:.
8511:.
8503:.
8477:.
8467:.
8459:.
8437:.
8427:.
8419:.
8363:.
8333:.
8290:))
8282:,
8271:=
8256:=
8201:.
8119:.
7550::
7496:2.
6938::
6884:2.
6693::
6468:1.
4991:.
4479:.
3980:.
3972:,
3968:,
3964:,
3938:0.
3797:,
3761:A
3705:;
3540:;
3248:,
3244:,
3228:,
3215:,
3208:,
3186:,
3055::
2927:,
2915:,
2911:,
2783::
2649:0.
2312:).
2304:=
2230::
2059:xy
2030:xy
1844:,
1666:.
1362:on
1269:.
1261:.
1253:,
1249:,
1237:,
1233:,
1218:.
1194:,
56:a
9264:e
9257:t
9250:v
9204::
9182::
9160::
9138::
9132:9
9065:.
8990:.
8887::
8624:.
8612::
8327:n
8319:f
8308:f
8304:)
8302:c
8300:(
8295:f
8288:c
8286:(
8284:f
8280:c
8278:(
8273:c
8269:x
8264:)
8262:x
8260:(
8258:f
8254:y
8175:P
8159:P
8092:y
8072:)
8069:b
8066:,
8063:0
8060:(
8040:b
8020:y
8000:b
7980:b
7977:+
7974:x
7971:m
7968:=
7965:y
7942:x
7922:x
7902:y
7882:y
7859:y
7839:x
7799:.
7795:)
7789:2
7783:3
7772:,
7769:2
7765:/
7761:1
7757:(
7743:)
7737:2
7731:3
7726:+
7720:,
7717:2
7713:/
7709:1
7705:(
7682:.
7677:2
7671:3
7660:=
7657:y
7638:4
7634:/
7630:3
7627:=
7622:2
7618:y
7598:1
7595:=
7590:2
7586:y
7582:+
7577:2
7573:)
7569:2
7565:/
7561:1
7558:(
7538:y
7518:x
7492:/
7488:1
7485:=
7482:x
7463:1
7457:=
7454:x
7451:2
7429:0
7426:=
7421:2
7417:x
7410:1
7407:+
7404:x
7401:2
7393:2
7389:x
7366:x
7346:y
7326:y
7304:2
7300:y
7277:2
7273:y
7252:0
7249:=
7244:2
7240:x
7231:2
7227:)
7223:1
7217:x
7214:(
7189:.
7185:)
7179:2
7173:3
7162:,
7159:2
7155:/
7151:1
7147:(
7133:)
7127:2
7121:3
7116:+
7110:,
7107:2
7103:/
7099:1
7095:(
7072:.
7067:2
7061:3
7050:=
7047:y
7028:4
7024:/
7020:3
7017:=
7012:2
7008:y
6988:1
6985:=
6980:2
6976:y
6972:+
6967:2
6963:)
6959:2
6955:/
6951:1
6948:(
6926:y
6906:x
6880:/
6876:1
6873:=
6870:x
6851:1
6845:=
6842:x
6839:2
6817:1
6814:=
6809:2
6805:x
6798:1
6795:+
6792:1
6789:+
6786:x
6783:2
6775:2
6771:x
6751:1
6748:=
6745:)
6740:2
6736:x
6729:1
6726:(
6723:+
6718:2
6714:)
6710:1
6704:x
6701:(
6681:x
6659:2
6655:y
6632:.
6627:2
6623:x
6616:1
6613:=
6608:2
6604:y
6584:1
6581:=
6576:2
6572:y
6568:+
6563:2
6559:x
6536:y
6516:x
6496:y
6465:=
6460:2
6456:y
6452:+
6447:2
6443:)
6439:1
6433:x
6430:(
6411:1
6408:=
6403:2
6399:y
6395:+
6390:2
6386:x
6363:Q
6343:P
6320:P
6300:)
6297:0
6294:,
6291:0
6288:(
6268:1
6265:=
6262:0
6242:1
6239:=
6234:2
6230:0
6226:+
6221:2
6217:0
6196:P
6176:)
6173:y
6170:,
6167:x
6164:(
6144:)
6141:0
6138:,
6135:0
6132:(
6112:Q
6092:)
6089:0
6086:,
6083:0
6080:(
6060:1
6057:=
6052:2
6048:)
6044:1
6038:(
6018:1
6015:=
6010:2
6006:0
6002:+
5997:2
5993:)
5989:1
5983:0
5980:(
5960:Q
5940:)
5937:y
5934:,
5931:x
5928:(
5908:)
5905:0
5902:,
5899:0
5896:(
5876:)
5873:0
5870:,
5867:0
5864:(
5844:}
5841:1
5838:=
5833:2
5829:y
5825:+
5820:2
5816:)
5812:1
5806:x
5803:(
5799:|
5795:)
5792:y
5789:,
5786:x
5783:(
5780:{
5777:=
5774:Q
5771::
5768:)
5765:0
5762:,
5759:1
5756:(
5736:Q
5716:}
5713:1
5710:=
5705:2
5701:y
5697:+
5692:2
5688:x
5683:|
5679:)
5676:y
5673:,
5670:x
5667:(
5664:{
5661:=
5658:P
5638:)
5635:0
5632:,
5629:0
5626:(
5606:P
5583:)
5580:y
5577:,
5574:x
5571:(
5551:)
5548:y
5545:,
5542:x
5539:(
5536:Q
5516:)
5513:y
5510:,
5507:x
5504:(
5501:P
5467:0
5464:=
5461:1
5453:2
5449:y
5445:+
5440:2
5436:x
5415:y
5412:x
5392:)
5389:y
5386:,
5383:x
5380:(
5377:R
5351:k
5331:h
5311:k
5291:h
5271:h
5251:k
5231:y
5211:a
5191:b
5171:x
5151:a
5131:a
5111:h
5108:+
5105:)
5102:)
5099:k
5093:x
5090:(
5087:b
5084:(
5081:f
5078:a
5075:=
5072:y
5052:)
5049:x
5046:(
5043:f
5040:=
5037:y
5016:x
5012:/
5008:1
5005:=
5002:y
4982:.
4970:A
4950:A
4941:y
4938:+
4935:A
4926:x
4903:y
4883:A
4874:y
4871:+
4868:A
4859:x
4839:x
4816:a
4812:/
4808:y
4788:y
4765:x
4745:b
4725:b
4721:/
4717:x
4697:x
4674:k
4654:k
4648:y
4628:y
4605:h
4585:h
4579:x
4559:x
4535:)
4532:y
4529:,
4526:x
4523:(
4520:R
4477:B
4473:A
4465:Ξ
4451:,
4434:B
4421:A
4408:f
4405:e
4402:d
4396:=
4388:B
4380:A
4369:B
4365:A
4347:,
4340:2
4336:)
4330:1
4326:z
4317:2
4313:z
4309:(
4306:+
4301:2
4297:)
4291:1
4287:y
4278:2
4274:y
4270:(
4267:+
4262:2
4258:)
4252:1
4248:x
4239:2
4235:x
4231:(
4226:=
4223:d
4206:m
4192:,
4189:)
4186:m
4183:(
4174:=
4147:,
4140:2
4136:)
4130:1
4126:y
4117:2
4113:y
4109:(
4106:+
4101:2
4097:)
4091:1
4087:x
4078:2
4074:x
4070:(
4065:=
4062:d
4052:2
4049:y
4045:2
4042:x
4038:1
4035:y
4031:1
4028:x
3935:=
3932:R
3929:+
3924:i
3920:x
3914:i
3910:P
3904:3
3899:1
3896:=
3893:i
3885:+
3880:j
3876:x
3870:j
3867:i
3863:Q
3857:i
3853:x
3847:3
3842:1
3839:=
3836:j
3833:,
3830:i
3809:3
3806:x
3804:,
3802:2
3799:x
3795:1
3792:x
3771:2
3744:.
3728:0
3725:=
3722:C
3719:+
3716:A
3689:0
3683:C
3680:A
3677:4
3669:2
3665:B
3653:;
3637:0
3634:=
3631:C
3628:A
3625:4
3617:2
3613:B
3583:0
3580:=
3577:B
3557:C
3554:=
3551:A
3524:0
3518:C
3515:A
3512:4
3504:2
3500:B
3475:.
3472:C
3469:A
3466:4
3458:2
3454:B
3426:.
3421:5
3416:P
3386:C
3383:,
3380:B
3377:,
3374:A
3366:0
3363:=
3360:F
3357:+
3354:y
3351:E
3348:+
3345:x
3342:D
3339:+
3334:2
3330:y
3326:C
3323:+
3320:y
3317:x
3314:B
3311:+
3306:2
3302:x
3298:A
3250:c
3246:b
3242:a
3240:(
3234:c
3230:b
3226:a
3220:0
3217:z
3213:0
3210:y
3206:0
3203:x
3201:(
3196:t
3192:z
3188:y
3184:x
3167:t
3164:c
3161:+
3156:0
3152:z
3148:=
3145:z
3126:t
3123:b
3120:+
3115:0
3111:y
3107:=
3104:y
3085:t
3082:a
3079:+
3074:0
3070:x
3066:=
3063:x
3026:)
3023:c
3020:,
3017:b
3014:,
3011:a
3008:(
3005:=
3001:n
2979:,
2976:0
2973:=
2970:d
2967:+
2964:z
2961:c
2958:+
2955:y
2952:b
2949:+
2946:x
2943:a
2933:c
2929:b
2925:a
2921:d
2917:c
2913:b
2909:a
2895:.
2892:)
2887:0
2883:z
2879:c
2876:+
2871:0
2867:y
2863:b
2860:+
2855:0
2851:x
2847:a
2844:(
2838:=
2835:d
2827:,
2824:0
2821:=
2818:d
2815:+
2812:z
2809:c
2806:+
2803:y
2800:b
2797:+
2794:x
2791:a
2760:,
2757:0
2754:=
2751:)
2746:0
2742:z
2735:z
2732:(
2729:c
2726:+
2723:)
2718:0
2714:y
2707:y
2704:(
2701:b
2698:+
2695:)
2690:0
2686:x
2679:x
2676:(
2673:a
2646:=
2643:)
2638:0
2633:r
2624:r
2620:(
2613:n
2591:r
2569:n
2548:P
2526:0
2522:P
2500:r
2479:P
2459:)
2456:c
2453:,
2450:b
2447:,
2444:a
2441:(
2438:=
2434:n
2413:)
2408:0
2404:z
2400:,
2395:0
2391:y
2387:,
2382:0
2378:x
2374:(
2371:=
2366:0
2362:P
2339:0
2334:r
2310:x
2308:(
2306:f
2302:y
2294:x
2284:b
2270:m
2253:b
2250:+
2247:x
2244:m
2241:=
2238:y
2192:r
2188:y
2184:x
2168:y
2164:x
2160:x
2156:x
2137:y
2133:x
2129:x
2125:y
2117:y
2113:x
2109:x
2105:y
2067:z
2063:Ï
2055:Ξ
2048:Ï
2026:Ξ
2019:z
2015:r
2008:z
1968:.
1965:)
1962:x
1958:/
1954:y
1951:(
1942:=
1935:,
1928:2
1924:y
1920:+
1915:2
1911:x
1905:=
1902:r
1898:;
1885:r
1882:=
1879:y
1875:,
1862:r
1859:=
1856:x
1846:Ξ
1842:r
1838:x
1834:Ξ
1830:Ξ
1823:r
1801:z
1797:y
1793:x
1781:y
1777:x
1775:(
1769:y
1765:x
1617:e
1610:t
1603:v
1179:e
1172:t
1165:v
306:)
302:(
84:)
80:(
34:.
20:)
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