1365:
8575:
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."
38:
8517:
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
9791:
4477:
1710:
9803:
3257:
1300:
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
3988:
9827:
9815:
8429:
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.
1254:
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
8526:
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.
1336:(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.
8525:
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
8472:
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
8428:
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
4975:
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
1253:
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
1690:
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
8574:
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
8516:
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
1299:
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
4450:
7798:
7188:
2305:
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
1691:
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
8473:
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.
1967:
5017:
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
1733:
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:
4363:
3390:
4346:
3937:
2894:
8804:"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."
2648:
7688:
7078:
4146:
8806:(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.)
1308:. 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.
7195:: 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
5843:
2759:
1291:, 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
7681:
7071:
5355:
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.
1840:
1279:
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.
1713:
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.
8715:
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.
5715:
2412:
6750:
6816:
3025:
2458:
4191:
7428:
6467:
7597:
7251:
6987:
6017:
3425:
4949:
2340:
5466:
4882:
8688:
6631:
5110:
3688:
3523:
6583:
6410:
6241:
3282:
2978:
2591:
2569:
2500:
3636:
6059:
3474:
3166:
3125:
3084:
7637:
7027:
4207:
3809:
7979:
7495:
7462:
6883:
6850:
5550:
5515:
5391:
5051:
4534:
2571:. 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
2252:
8071:
7305:
7278:
6660:
6299:
6175:
6143:
6091:
5939:
5907:
5875:
5637:
5582:
5015:
3727:
2527:
6267:
4815:
4724:
4653:
4584:
3582:
3556:
5414:
1364:
8091:
8039:
8019:
7999:
7941:
7921:
7901:
7881:
7858:
7838:
7537:
7517:
7365:
7345:
7325:
6925:
6905:
6680:
6535:
6515:
6495:
6362:
6342:
6319:
6195:
6111:
5959:
5735:
5605:
5350:
5330:
5310:
5290:
5270:
5250:
5230:
5210:
5190:
5170:
5150:
5130:
4969:
4902:
4838:
4787:
4764:
4744:
4696:
4673:
4627:
4604:
4558:
2547:
2478:
2775:
8521:
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
1667:
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
5132:
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
2596:
4046:
3279:
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
4445:{\displaystyle \mathbf {A} \cdot \mathbf {B} {\stackrel {\mathrm {def} }{=}}\left\|\mathbf {A} \right\|\left\|\mathbf {B} \right\|\cos \theta ,}
1320:
saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and
9251:
8329:, 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.
8933:
8773:
8791:
8589:
5192:
value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like
2657:
1166:
1295:
further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of
7641:
7031:
1678:
Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of
5352:
values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.
8974:
8885:
8830:
1508:
266:
9614:
6685:
6754:
7793:{\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}
7183:{\displaystyle \left(1/2,{\frac {+{\sqrt {3}}}{2}}\right)\;\;{\text{and}}\;\;\left(1/2,{\frac {-{\sqrt {3}}}{2}}\right).}
5740:
4155:
1604:
1503:
7372:
6414:
9759:
9244:
9097:
9035:
9017:
7542:
6932:
3392:
As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional
1837:). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae:
9831:
9069:
8911:
8708:
8657:
8563:
8505:
8461:
8421:
5845:. The intersection of these two circles is the collection of points which make both equations true. Does the point
5419:
1682:(Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes'
232:
6587:
2159: = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a
9295:
6542:
6369:
2927:
8344:
that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in
9709:
8664:
the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments.
8104:
Also for this may be used the common language use as a: normal (perpendicular) line, otherwise in engineering as
1528:
1287:
3437:
3129:
3088:
3047:
1328:, but the decisive step came later with Descartes. Omar Khayyam is credited with identifying the foundations of
9807:
9115:
7601:
6991:
5642:
2345:
1159:
1113:
719:
178:
4498:
Transformations are applied to a parent function to turn it into a new function with similar characteristics.
9237:
9181:
8818:
7466:
7432:
6854:
6820:
2222:
9819:
2984:
2417:
9734:
9290:
3268:
2030:
1749:
1743:
1456:
1441:
20:
9081:
9853:
9305:
9043:
7198:
5964:
3398:
1962:{\displaystyle x=r\,\cos \theta ,\,y=r\,\sin \theta ;\,r={\sqrt {x^{2}+y^{2}}},\,\theta =\arctan(y/x).}
1556:
1134:
744:
4907:
2316:
1618:
1348:
9719:
9691:
9328:
4843:
2164:
1571:
1426:
1255:
1152:
7820:
One type of intersection which is widely studied is the intersection of a geometric object with the
5056:
3648:
3483:
9764:
8930:
8770:
6200:
1533:
121:
8198:
Tangent is the linear approximation of a spherical or other curved or twisted line of a function.
2574:
2552:
2483:
1647:
Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences
9649:
9639:
9609:
9543:
9278:
8989:
While this discussion is limited to the xy-plane, it can easily be extended to higher dimensions.
8953:
5477:
3596:
1992:
1986:
1970:
1597:
1518:
547:
227:
84:
8788:
8700:
8692:
6022:
9747:
9644:
9624:
9619:
9548:
9273:
4977:
1394:
1247:
623:
334:
212:
97:
8587:
Cooper, Glen M. (2003). "Review: Omar Khayyam, the Mathmetician by R. Rashed, B. Vahabzadeh".
8555:
8497:
8489:
8453:
8445:
9774:
9704:
9581:
9505:
9444:
9429:
9424:
9401:
9283:
8918:
8413:
8405:
8345:
8337:
8143:
4012:
3763:
2160:
1974:
1513:
1431:
1239:
1223:
395:
356:
315:
310:
163:
7949:
5520:
5485:
5361:
5021:
4504:
4152:. Similarly, the angle that a line makes with the horizontal can be defined by the formula
2460:
be a nonzero vector. The plane determined by this point and vector consists of those points
1695:
who first applied the coordinate method in a systematic study of space curves and surfaces.
9754:
9634:
9629:
9553:
9454:
8044:
7283:
7256:
6638:
6272:
6148:
6116:
6064:
5912:
5880:
5848:
5610:
5555:
4986:
3775:
3700:
2505:
2286:
2215:
1675:
in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.
1538:
1063:
986:
834:
739:
261:
156:
70:
8:
9769:
9679:
9601:
9500:
9434:
9391:
9381:
9361:
9159:
9137:
8949:
8541:
8485:
8441:
8401:
8376:
8341:
6246:
4792:
4701:
4632:
4563:
4204:
In three dimensions, distance is given by the generalization of the Pythagorean theorem:
4149:
3958:
3767:
3561:
3535:
3272:
3261:
3041:
2168:
2089:
2071:
1651:
1561:
1451:
1282:
1068:
1012:
925:
779:
759:
684:
574:
445:
435:
298:
173:
168:
151:
126:
114:
66:
61:
42:
8240:
points on the curve. More precisely, a straight line is said to be a tangent of a curve
8236:
that "just touches" the curve at that point. Informally, it is a line through a pair of
5396:
2213:
equations. In two dimensions, the equation for non-vertical lines is often given in the
1645:, one of the three accompanying essays (appendices) published in 1637 together with his
9795:
9714:
9654:
9586:
9576:
9515:
9490:
9366:
9323:
9318:
9198:
8606:
8076:
8024:
8004:
7984:
7926:
7906:
7886:
7866:
7843:
7823:
7522:
7502:
7350:
7330:
7310:
6910:
6890:
6665:
6520:
6500:
6480:
6347:
6327:
6304:
6180:
6096:
5944:
5720:
5590:
5335:
5315:
5295:
5275:
5255:
5235:
5215:
5195:
5175:
5155:
5135:
5115:
4954:
4887:
4823:
4772:
4749:
4729:
4681:
4658:
4612:
4589:
4543:
4008:
3802:
3771:
3276:
3226:
2532:
2463:
2206:
2129:
1704:
1672:
1590:
1576:
1421:
1329:
1235:
1204:
1027:
754:
594:
222:
146:
136:
107:
92:
19:
This article is about coordinate geometry. For the geometry of analytic varieties, see
9223:
8868:
VujiÄiÄ, Milan; Sanderson, Jeffrey (2008), VujiÄiÄ, Milan; Sanderson, Jeffrey (eds.),
3385:{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0{\text{ with }}A,B,C{\text{ not all zero.}}}
1772:). This system can also be used for three-dimensional geometry, where every point in
9790:
9510:
9439:
9386:
9093:
9077:
9065:
9059:
9031:
9013:
8970:
8907:
8881:
8826:
8704:
8653:
8559:
8546:
8501:
8457:
8417:
8325:
As it passes through the point where the tangent line and the curve meet, called the
8151:
1807:
1801:
1640:
1523:
1466:
1321:
1243:
1200:
1098:
886:
864:
789:
648:
374:
303:
195:
102:
6471:
Traditional methods for finding intersections include substitution and elimination.
1624:
1356:
1088:
1017:
814:
724:
9724:
9699:
9571:
9419:
9356:
9190:
9179:
Coolidge, J. L. (1948), "The Beginnings of Analytic Geometry in Three Dimensions",
9168:
9146:
9135:
Boyer, Carl B. (1944), "Analytic Geometry: The Discovery of Fermat and Descartes",
9124:
9051:
8873:
8598:
8371:
8361:
8229:
8171:
3966:
3393:
2196:
1760:-coordinate representing its vertical position. These are typically written as an
1722:
1628:
1477:
1446:
1410:
1078:
819:
529:
407:
342:
200:
185:
50:
4341:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}
2120:
is said to be the equation for this line. In general, linear equations involving
9664:
9591:
9520:
9313:
8937:
8901:
8795:
8777:
8319:
8167:
3745:
2769:
2202:
2192:
2183:
is the equation for any circle centered at the origin (0, 0) with a radius of r.
2109:
1773:
1730:
1718:
1660:
1461:
1399:
1389:
501:
364:
207:
190:
131:
37:
3932:{\displaystyle \sum _{i,j=1}^{3}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{3}P_{i}x_{i}+R=0.}
1073:
1042:
976:
824:
769:
704:
9742:
9669:
9376:
9047:
8645:
1692:
1566:
1404:
1325:
1272:
1129:
1037:
981:
946:
854:
764:
734:
694:
599:
9113:
Bissell, Christopher C. (1987), "Cartesian geometry: The Dutch contribution",
8877:
4476:
3040:
be described by a single linear equation, so they are frequently described by
2035:
In spherical coordinates, every point in space is represented by its distance
1709:
1103:
714:
9847:
9530:
9462:
9414:
8366:
8349:
8237:
8233:
8187:
8159:
8155:
8120:
8101:
Axis in geometry is the perpendicular line to any line, object or a surface.
3251:
2307:
2133:
1969:
This system may be generalized to three-dimensional space through the use of
1829:-axis. Using this notation, points are typically written as an ordered pair (
1635:, the alternative term used for analytic geometry, is named after Descartes.
1227:
1108:
1093:
1022:
839:
799:
749:
524:
487:
454:
292:
288:
8769:(Toulouse, France: Jean Pech, 1679), "Ad locos planos et solidos isagoge,"
8635:
Cooper, G. (2003). Journal of the American Oriental Society,123(1), 248-249.
9472:
9467:
9371:
8381:
8128:
3430:
2085:
2067:
1761:
1471:
1379:
1317:
1047:
996:
809:
664:
579:
369:
9229:
3429:
The conic sections described by this equation can be classified using the
9674:
9338:
9261:
9172:
9150:
8311:
7903:-intercept of the object. The intersection of a geometric object and the
7815:
7811:
4349:
3954:
2651:
2276:
1726:
1484:
1384:
1231:
1215:
1180:
1083:
956:
774:
709:
637:
609:
584:
1638:
Descartes made significant progress with the methods in an essay titled
9659:
9538:
9333:
9202:
9128:
8652:(Second ed.). Springer Science + Business Media Inc. p. 105.
8610:
8303:
4007:. These definitions are designed to be consistent with the underlying
3991:
The distance formula on the plane follows from the Pythagorean theorem.
3950:
3256:
2889:{\displaystyle ax+by+cz+d=0,{\text{ where }}d=-(ax_{0}+by_{0}+cz_{0}).}
1416:
1276:
941:
920:
910:
900:
859:
804:
699:
689:
589:
440:
7802:
For conic sections, as many as 4 points might be in the intersection.
8945:
3942:
3730:
3691:
1296:
951:
669:
632:
496:
468:
9194:
8602:
8193:
2643:{\displaystyle \mathbf {n} \cdot (\mathbf {r} -\mathbf {r} _{0})=0.}
2136:, and more complicated equations describe more complicated figures.
1663:
tongue, and its philosophical principles, provided a foundation for
1234:. It is the foundation of most modern fields of geometry, including
9563:
9482:
9409:
8964:
8536:
8534:
8213:
8112:
3996:
3987:
3978:
3639:
2077:
1664:
1219:
1196:
1032:
991:
961:
849:
844:
794:
519:
478:
426:
320:
283:
29:
9348:
8940:
in "Geometry Formulas and Facts", excerpted from 30th Edition of
8207:
8123:
to a given object. For example, in the two-dimensional case, the
4004:
3526:
2143:
on the plane. This is not always the case: the trivial equation
1211:
966:
679:
473:
417:
217:
9048:
Analytic Geometry of the Point, Line, Circle, and Conic Sections
8531:
4141:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}},}
2050:-plane makes with respect to the horizontal axis, and the angle
5482:
For two geometric objects P and Q represented by the relations
3946:
3585:
2081:
2000:
915:
905:
784:
729:
604:
567:
555:
510:
463:
381:
46:
8767:
Varia Opera Mathematica d. Petri de Fermat, Senatoris Tolosani
8624:
Mathematical Masterpieces: Further Chronicles by the Explorers
7347:
has been eliminated. We then solve the remaining equation for
2058:-axis. The names of the angles are often reversed in physics.
1737:
9157:
Boyer, Carl B. (1965), "Johann Hudde and space coordinates",
8225:
8175:
8127:
to a curve at a given point is the line perpendicular to the
5471:
4457:
4198:
4000:
3982:
3962:
2266:
2262:
2140:
2040:
2011:
1815:
1777:
1668:
971:
895:
829:
674:
278:
273:
2100:
corresponds to the set of all the points on the plane whose
562:
412:
1810:, every point of the plane is represented by its distance
8570:
Omar Khayyam (ca. 1050â1123), the "tent-maker," wrote an
8737:
8735:
8496:(Second ed.). John Wiley & Sons, Inc. pp.
8452:(Second ed.). John Wiley & Sons, Inc. pp.
8412:(Second ed.). John Wiley & Sons, Inc. pp.
1756:-coordinate representing its horizontal position, and a
1980:
8732:
8079:
8047:
8027:
8007:
7987:
7952:
7929:
7909:
7889:
7869:
7846:
7826:
7691:
7644:
7604:
7545:
7525:
7505:
7469:
7435:
7375:
7353:
7333:
7313:
7286:
7259:
7201:
7081:
7034:
6994:
6935:
6913:
6893:
6857:
6823:
6757:
6688:
6668:
6641:
6590:
6545:
6523:
6503:
6483:
6417:
6372:
6350:
6330:
6307:
6275:
6249:
6203:
6183:
6151:
6119:
6099:
6067:
6025:
5967:
5947:
5915:
5883:
5851:
5743:
5723:
5645:
5613:
5593:
5558:
5523:
5488:
5422:
5399:
5364:
5338:
5318:
5298:
5278:
5258:
5238:
5218:
5198:
5178:
5158:
5138:
5118:
5059:
5024:
4989:
4957:
4910:
4890:
4846:
4826:
4795:
4775:
4752:
4732:
4704:
4684:
4661:
4635:
4615:
4592:
4566:
4546:
4507:
4366:
4210:
4158:
4049:
3812:
3703:
3651:
3599:
3564:
3538:
3486:
3440:
3401:
3285:
3132:
3091:
3050:
2987:
2930:
2778:
2660:
2599:
2577:
2555:
2535:
2508:
2486:
2466:
2420:
2348:
2319:
2225:
2024:
1843:
1825:
normally measured counterclockwise from the positive
1631:, although Descartes is sometimes given sole credit.
8166:. The word "normal" is also used as an adjective: a
6364:
can be found by solving the simultaneous equations:
4348:
while the angle between two vectors is given by the
3225:
are related to the slope of the line, such that the
2654:, not scalar multiplication.) Expanded this becomes
1995:, every point of space is represented by its height
9061:
A History of Mathematics: An Introduction (2nd Ed.)
5838:{\displaystyle (1,0):Q=\{(x,y)|(x-1)^{2}+y^{2}=1\}}
4536:is changed by standard transformations as follows:
2754:{\displaystyle a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0,}
8841:Percey Franklyn Smith, Arthur Sullivan Gale (1905)
8545:
8265:on the curve if the line passes through the point
8085:
8065:
8033:
8013:
7993:
7973:
7935:
7915:
7895:
7875:
7852:
7832:
7792:
7675:
7631:
7591:
7531:
7519:in either of the original equations and solve for
7511:
7489:
7456:
7422:
7359:
7339:
7319:
7299:
7272:
7245:
7182:
7065:
7021:
6981:
6919:
6907:in either of the original equations and solve for
6899:
6877:
6844:
6810:
6744:
6674:
6654:
6625:
6577:
6529:
6509:
6489:
6461:
6404:
6356:
6336:
6313:
6293:
6261:
6235:
6189:
6169:
6137:
6105:
6085:
6053:
6011:
5953:
5933:
5901:
5869:
5837:
5729:
5709:
5631:
5599:
5576:
5544:
5509:
5460:
5408:
5385:
5344:
5324:
5304:
5284:
5264:
5244:
5224:
5204:
5184:
5164:
5144:
5124:
5104:
5045:
5009:
4963:
4943:
4896:
4876:
4832:
4809:
4781:
4758:
4738:
4718:
4690:
4667:
4647:
4621:
4598:
4578:
4552:
4528:
4444:
4340:
4185:
4140:
3931:
3721:
3682:
3630:
3576:
3550:
3517:
3468:
3419:
3384:
3160:
3119:
3078:
3019:
2972:
2888:
2753:
2642:
2585:
2563:
2541:
2521:
2494:
2472:
2452:
2406:
2334:
2246:
2021:-plane makes with respect to the horizontal axis.
1961:
9082:Lectures in Geometry Semester I Analytic Geometry
8802:, 9 February 1665, pp. 69â72. From p. 70:
8194:Spherical and nonlinear planes and their tangents
7367:, in the same way as in the substitution method:
6662:into the other equation and proceed to solve for
5552:the intersection is the collection of all points
3029:This familiar equation for a plane is called the
2924:are not all zero, then the graph of the equation
1795:
1334:Treatise on Demonstrations of Problems of Algebra
9845:
8965:M.R. Spiegel; S. Lipschutz; D. Spellman (2009).
8867:
5468:is the relation that describes the unit circle.
4726:stretches the graph horizontally by a factor of
3995:In analytic geometry, geometric notions such as
1748:The most common coordinate system to use is the
1623:Analytic geometry was independently invented by
8001:specifies the point where the line crosses the
7863:The intersection of a geometric object and the
7676:{\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}
7066:{\displaystyle y={\frac {\pm {\sqrt {3}}}{2}}.}
4015:on the plane, the distance between two points (
2167:of two surfaces (see below), or as a system of
8119:is an object such as a line or vector that is
3183:are all functions of the independent variable
9245:
8942:CRC Standard Mathematical Tables and Formulas
7280:in the first equation is subtracted from the
5737:might be the circle with radius 1 and center
5607:might be the circle with radius 1 and center
2151:specifies the entire plane, and the equation
1721:is given a coordinate system, by which every
1598:
1160:
8590:The Journal of the American Oriental Society
5832:
5768:
5704:
5652:
2139:Usually, a single equation corresponds to a
2108:-coordinate are equal. These points form a
9259:
8958:
8903:Math refresher for scientists and engineers
8201:
5472:Finding intersections of geometric objects
4352:. The dot product of two Euclidean vectors
1738:Cartesian coordinates (in a plane or space)
1324:with his geometric solution of the general
21:Algebraic geometry § Analytic geometry
16:Study of geometry using a coordinate system
9252:
9238:
8814:
8812:
8697:The History of Mathematics: A Brief Course
8348:and has been extensively generalized; see
7743:
7742:
7736:
7735:
7133:
7132:
7126:
7125:
1605:
1591:
1167:
1153:
36:
8825:, 6th ed., Brooks Cole Cengage Learning.
8644:
5710:{\displaystyle P=\{(x,y)|x^{2}+y^{2}=1\}}
5152:values, the function is reflected in the
2407:{\displaystyle P_{0}=(x_{0},y_{0},z_{0})}
1926:
1889:
1876:
1866:
1853:
1686:. Clearly written and well received, the
9178:
8858:Courier Dover Publications, Jan 27, 2012
4475:
4148:which can be viewed as a version of the
3986:
3801:, the general quadric is defined by the
3588:, which is a special case of an ellipse;
3255:
1708:
9112:
9090:A Source Book in Mathematics, 1200-1800
8906:, John Wiley and Sons, pp. 44â45,
8809:
8021:axis. Depending on the context, either
6517:and then substitute the expression for
2896:Conversely, it is easily shown that if
2163:, and a curve must be specified as the
2061:
1619:René Descartes § Analytic geometry
1316:The 11th-century Persian mathematician
9846:
9087:
9025:
8899:
8586:
3766:in 3-dimensional space defined as the
3476:If the conic is non-degenerate, then:
2080:involving the coordinates specifies a
267:Straightedge and compass constructions
9233:
9156:
9134:
9007:
8856:Analytic Geometry of Three Dimensions
8835:
8726:
8687:
8675:
8629:
8540:
8484:
8440:
8400:
7805:
6745:{\displaystyle (x-1)^{2}+(1-x^{2})=1}
3972:
2342:be the position vector of some point
9814:
9057:
8923:
8753:
8741:
7685:So our intersection has two points:
7075:So our intersection has two points:
6811:{\displaystyle x^{2}-2x+1+1-x^{2}=1}
3020:{\displaystyle \mathbf {n} =(a,b,c)}
2453:{\displaystyle \mathbf {n} =(a,b,c)}
2209:, can be described algebraically by
1981:Cylindrical coordinates (in a space)
9826:
9064:, Reading: Addison Wesley Longman,
8967:Vector Analysis (Schaum's Outlines)
8870:Linear Algebra Thoroughly Explained
5112:. In the new transformed function,
4186:{\displaystyle \theta =\arctan(m),}
3739:
3187:which ranges over the real numbers.
2186:
1504:Rules for the Direction of the Mind
13:
8623:
8310:. A similar definition applies to
7423:{\displaystyle x^{2}-2x+1-x^{2}=0}
7307:in the second equation leaving no
6635:We then substitute this value for
6462:{\displaystyle (x-1)^{2}+y^{2}=1.}
6321:so it is not in the intersection.
6113:. On the other hand, still using
5877:make both equations true? Using
5053:, then it can be transformed into
4471:
4396:
4393:
4390:
2502:, such that the vector drawn from
2054:that it makes with respect to the
2025:Spherical coordinates (in a space)
1680:Ad locos planos et solidos isagoge
1671:and the addition of commentary by
14:
9865:
9217:
8843:Introduction to Analytic Geometry
8096:
7592:{\displaystyle (1/2)^{2}+y^{2}=1}
7246:{\displaystyle (x-1)^{2}-x^{2}=0}
6982:{\displaystyle (1/2)^{2}+y^{2}=1}
6012:{\displaystyle (0-1)^{2}+0^{2}=1}
4983:For example, the parent function
3420:{\displaystyle \mathbf {P} ^{5}.}
3245:
2310:) to indicate its "inclination".
1339:
1266:
233:Noncommutative algebraic geometry
9825:
9813:
9802:
9801:
9789:
9208:
8512:The method of Apollonius in the
8406:"The Age of Plato and Aristotle"
8134:In the three-dimensional case a
4944:{\displaystyle -x\sin A+y\cos A}
4422:
4409:
4376:
4368:
3404:
2989:
2766:form of the equation of a plane.
2621:
2612:
2601:
2579:
2557:
2488:
2422:
2335:{\displaystyle \mathbf {r} _{0}}
2322:
1363:
9710:Computational complexity theory
8983:
8893:
8861:
8848:
8823:Calculus: Early Transcendentals
8782:
8759:
8747:
8720:
8699:. Wiley-Interscience. pp.
8681:
5461:{\displaystyle x^{2}+y^{2}-1=0}
5272:values introduce translations,
5232:-axis when it is negative. The
5212:, reflects the function in the
4877:{\displaystyle x\cos A+y\sin A}
4817:stretches the graph vertically.
3036:In three dimensions, lines can
1529:Meditations on First Philosophy
9116:The Mathematical Intelligencer
8669:
8638:
8617:
8580:
8478:
8434:
8394:
8060:
8048:
7561:
7546:
7215:
7202:
6951:
6936:
6733:
6714:
6702:
6689:
6626:{\displaystyle y^{2}=1-x^{2}.}
6431:
6418:
6288:
6276:
6164:
6152:
6132:
6120:
6080:
6068:
6036:
6026:
5981:
5968:
5928:
5916:
5896:
5884:
5864:
5852:
5804:
5791:
5787:
5783:
5771:
5756:
5744:
5671:
5667:
5655:
5626:
5614:
5571:
5559:
5539:
5527:
5504:
5492:
5380:
5368:
5105:{\displaystyle y=af(b(x-k))+h}
5093:
5090:
5078:
5072:
5040:
5034:
4951:rotates the graph by an angle
4523:
4511:
4491:
4487:
4483:
4426:
4418:
4413:
4405:
4324:
4297:
4285:
4258:
4246:
4219:
4177:
4171:
4124:
4097:
4085:
4058:
3683:{\displaystyle B^{2}-4AC>0}
3518:{\displaystyle B^{2}-4AC<0}
3033:of the equation of the plane.
3014:
2996:
2880:
2832:
2739:
2720:
2711:
2692:
2683:
2664:
2631:
2608:
2447:
2429:
2401:
2362:
1953:
1939:
1796:Polar coordinates (in a plane)
1698:
626:- / other-dimensional
1:
9182:American Mathematical Monthly
8996:
8969:(2nd ed.). McGraw Hill.
8789:"Eloge de Monsieur de Fermat"
8648:(2004). "Analytic Geometry".
6887:Next, we place this value of
6578:{\displaystyle x^{2}+y^{2}=1}
6477:Solve the first equation for
6405:{\displaystyle x^{2}+y^{2}=1}
6236:{\displaystyle 0^{2}+0^{2}=1}
5584:which are in both relations.
4586:moves the graph to the right
3525:, the equation represents an
2981:is a plane having the vector
2973:{\displaystyle ax+by+cz+d=0,}
2092:. For example, the equation
1210:Analytic geometry is used in
9211:Newton and analytic geometry
9092:, Harvard University Press,
9010:History of Analytic Geometry
8174:, the normal component of a
7499:We then place this value of
4043:) is defined by the formula
3729:, the equation represents a
3690:, the equation represents a
3638:, the equation represents a
3584:, the equation represents a
2586:{\displaystyle \mathbf {r} }
2564:{\displaystyle \mathbf {n} }
2495:{\displaystyle \mathbf {r} }
7:
9226:with interactive animations
9106:
8798:(Eulogy of Mr. de Fermat),
8650:Mathematics and its History
8355:
8281:on the curve and has slope
8131:to the curve at the point.
3631:{\displaystyle B^{2}-4AC=0}
3269:Cartesian coordinate system
3211:) is any point on the line.
2031:Spherical coordinate system
1750:Cartesian coordinate system
1744:Cartesian coordinate system
10:
9870:
9760:Films about mathematicians
9224:Coordinate Geometry topics
8205:
7943:-intercept of the object.
7809:
6537:into the second equation:
6054:{\displaystyle (-1)^{2}=1}
5475:
4003:measure are defined using
3976:
3743:
3469:{\displaystyle B^{2}-4AC.}
3249:
3241:) is parallel to the line.
3161:{\displaystyle z=z_{0}+ct}
3120:{\displaystyle y=y_{0}+bt}
3079:{\displaystyle x=x_{0}+at}
2190:
2076:In analytic geometry, any
2065:
2028:
1984:
1799:
1752:, where each point has an
1741:
1717:In analytic geometry, the
1702:
1649:, commonly referred to as
1616:
1557:Christina, Queen of Sweden
1261:
18:
9783:
9733:
9690:
9600:
9562:
9529:
9481:
9453:
9400:
9347:
9329:Philosophy of mathematics
9304:
9269:
8878:10.1007/978-3-540-74639-3
7632:{\displaystyle y^{2}=3/4}
7022:{\displaystyle y^{2}=3/4}
3941:Quadric surfaces include
2084:of the plane, namely the
1572:Gottfried Wilhelm Leibniz
1427:Causal adequacy principle
1311:
9765:Recreational mathematics
9058:Katz, Victor J. (1998),
9028:A History of Mathematics
9026:Cajori, Florian (1999),
9008:Boyer, Carl B. (2004) ,
9001:
8900:Fanchi, John R. (2006),
8872:, Springer, p. 27,
8552:A History of Mathematics
8494:A History of Mathematics
8468:The Apollonian treatise
8450:A History of Mathematics
8410:A History of Mathematics
8387:
8340:at a given point is the
8202:Tangent lines and planes
2205:, or more generally, in
1814:from the origin and its
1729:coordinates. Similarly,
1659:, written in his native
1534:Principles of Philosophy
122:Non-Archimedean geometry
9650:Mathematical statistics
9640:Mathematical psychology
9610:Engineering mathematics
9544:Algebraic number theory
8954:University of Minnesota
5478:Intersection (geometry)
2480:, with position vector
1993:cylindrical coordinates
1987:Cylindrical coordinates
1519:Discourse on the Method
228:Noncommutative geometry
9796:Mathematics portal
9645:Mathematical sociology
9625:Mathematical economics
9620:Mathematical chemistry
9549:Analytic number theory
9430:Differential equations
9088:Struik, D. J. (1969),
9012:, Dover Publications,
8800:Le Journal des Scavans
8470:On Determinate Section
8182:, etc. The concept of
8087:
8067:
8035:
8015:
7995:
7975:
7974:{\displaystyle y=mx+b}
7937:
7917:
7897:
7877:
7854:
7834:
7794:
7677:
7633:
7593:
7533:
7513:
7491:
7490:{\displaystyle x=1/2.}
7458:
7457:{\displaystyle -2x=-1}
7424:
7361:
7341:
7321:
7301:
7274:
7247:
7184:
7067:
7023:
6983:
6921:
6901:
6879:
6878:{\displaystyle x=1/2.}
6846:
6845:{\displaystyle -2x=-1}
6812:
6746:
6676:
6656:
6627:
6579:
6531:
6511:
6491:
6463:
6406:
6358:
6338:
6315:
6295:
6263:
6237:
6191:
6171:
6139:
6107:
6087:
6055:
6013:
5955:
5935:
5903:
5871:
5839:
5731:
5711:
5633:
5601:
5578:
5546:
5545:{\displaystyle Q(x,y)}
5511:
5510:{\displaystyle P(x,y)}
5462:
5410:
5387:
5386:{\displaystyle R(x,y)}
5346:
5326:
5306:
5286:
5266:
5246:
5226:
5206:
5186:
5166:
5146:
5126:
5106:
5047:
5046:{\displaystyle y=f(x)}
5011:
4978:affine transformations
4965:
4945:
4898:
4878:
4834:
4811:
4783:
4760:
4740:
4720:
4692:
4669:
4649:
4623:
4600:
4580:
4554:
4530:
4529:{\displaystyle R(x,y)}
4495:
4446:
4342:
4187:
4142:
4011:. For example, using
3992:
3933:
3896:
3839:
3723:
3684:
3632:
3578:
3552:
3519:
3470:
3421:
3386:
3264:
3162:
3121:
3080:
3021:
2974:
2890:
2755:
2650:(The dot here means a
2644:
2587:
2565:
2543:
2523:
2496:
2474:
2454:
2408:
2336:
2248:
2247:{\displaystyle y=mx+b}
2046:its projection on the
2017:its projection on the
1963:
1714:
1288:On Determinate Section
1248:computational geometry
1203:. This contrasts with
196:Discrete/Combinatorial
9775:Mathematics education
9705:Theory of computation
9425:Hypercomplex analysis
8547:"The Arabic Hegemony"
8490:"Apollonius of Perga"
8446:"Apollonius of Perga"
8346:differential geometry
8088:
8068:
8066:{\displaystyle (0,b)}
8036:
8016:
7996:
7976:
7938:
7918:
7898:
7878:
7855:
7835:
7795:
7678:
7634:
7594:
7534:
7514:
7492:
7459:
7425:
7362:
7342:
7322:
7302:
7300:{\displaystyle y^{2}}
7275:
7273:{\displaystyle y^{2}}
7248:
7185:
7068:
7024:
6984:
6922:
6902:
6880:
6847:
6813:
6747:
6677:
6657:
6655:{\displaystyle y^{2}}
6628:
6580:
6532:
6512:
6492:
6464:
6407:
6359:
6339:
6316:
6296:
6294:{\displaystyle (0,0)}
6264:
6238:
6192:
6172:
6170:{\displaystyle (x,y)}
6140:
6138:{\displaystyle (0,0)}
6108:
6088:
6086:{\displaystyle (0,0)}
6056:
6014:
5956:
5936:
5934:{\displaystyle (x,y)}
5904:
5902:{\displaystyle (0,0)}
5872:
5870:{\displaystyle (0,0)}
5840:
5732:
5712:
5634:
5632:{\displaystyle (0,0)}
5602:
5579:
5577:{\displaystyle (x,y)}
5547:
5512:
5463:
5416:plane. For example,
5411:
5393:is a relation in the
5388:
5347:
5327:
5312:horizontal. Positive
5307:
5287:
5267:
5247:
5227:
5207:
5187:
5167:
5147:
5127:
5107:
5048:
5012:
5010:{\displaystyle y=1/x}
4966:
4946:
4899:
4879:
4835:
4812:
4784:
4761:
4741:
4721:
4693:
4670:
4650:
4624:
4601:
4581:
4555:
4531:
4479:
4447:
4343:
4188:
4143:
4013:Cartesian coordinates
3990:
3934:
3876:
3813:
3731:rectangular hyperbola
3724:
3722:{\displaystyle A+C=0}
3685:
3633:
3579:
3553:
3520:
3471:
3422:
3387:
3259:
3163:
3122:
3081:
3022:
2975:
2891:
2756:
2645:
2588:
2566:
2544:
2524:
2522:{\displaystyle P_{0}}
2497:
2475:
2455:
2409:
2337:
2249:
2088:for the equation, or
2039:from the origin, the
1964:
1776:is represented by an
1712:
1256:CantorâDedekind axiom
179:Discrete differential
9755:Informal mathematics
9635:Mathematical physics
9630:Mathematical finance
9615:Mathematical biology
9554:Diophantine geometry
9173:10.5951/MT.58.1.0033
9151:10.5951/MT.37.3.0099
9084:via Internet Archive
8919:Section 3.2, page 45
8077:
8045:
8025:
8005:
7985:
7950:
7927:
7923:-axis is called the
7907:
7887:
7883:-axis is called the
7867:
7844:
7824:
7689:
7642:
7602:
7543:
7523:
7503:
7467:
7433:
7373:
7351:
7331:
7311:
7284:
7257:
7199:
7079:
7032:
6992:
6933:
6911:
6891:
6855:
6821:
6755:
6686:
6666:
6639:
6588:
6543:
6521:
6501:
6481:
6415:
6370:
6348:
6328:
6324:The intersection of
6305:
6273:
6247:
6201:
6181:
6149:
6117:
6097:
6065:
6023:
5965:
5945:
5913:
5881:
5849:
5741:
5721:
5643:
5611:
5591:
5556:
5521:
5486:
5420:
5397:
5362:
5336:
5316:
5296:
5276:
5256:
5236:
5216:
5196:
5176:
5156:
5136:
5116:
5057:
5022:
4987:
4955:
4908:
4888:
4844:
4824:
4793:
4773:
4750:
4730:
4702:
4682:
4659:
4633:
4613:
4590:
4564:
4544:
4505:
4364:
4208:
4156:
4047:
3810:
3776:quadratic polynomial
3701:
3649:
3597:
3562:
3536:
3484:
3438:
3399:
3283:
3260:A hyperbola and its
3130:
3089:
3048:
3042:parametric equations
2985:
2928:
2776:
2658:
2597:
2575:
2553:
2549:is perpendicular to
2533:
2506:
2484:
2464:
2418:
2346:
2317:
2287:independent variable
2223:
2216:slope-intercept form
2169:parametric equations
2062:Equations and curves
1841:
1539:Passions of the Soul
1509:The Search for Truth
9770:Mathematics and art
9680:Operations research
9435:Functional analysis
9160:Mathematics Teacher
9138:Mathematics Teacher
8950:The Geometry Center
8854:William H. McCrea,
8377:Translation of axes
8162:to that surface at
7327:term. The variable
6262:{\displaystyle 0=1}
6093:is in the relation
5941:, the equation for
4810:{\displaystyle y/a}
4719:{\displaystyle x/b}
4655:moves the graph up
4648:{\displaystyle y-k}
4579:{\displaystyle x-h}
4150:Pythagorean theorem
3577:{\displaystyle B=0}
3551:{\displaystyle A=C}
3379: not all zero.
3262:conjugate hyperbola
2130:quadratic equations
2072:Locus (mathematics)
1652:Discourse on Method
1562:Nicolas Malebranche
1432:Mindâbody dichotomy
1400:Doubt and certainty
1283:Apollonius of Perga
1189:coordinate geometry
446:Pythagorean theorem
9715:Numerical analysis
9324:Mathematical logic
9319:Information theory
9129:10.1007/BF03023730
8936:2018-07-18 at the
8794:2015-08-04 at the
8776:2015-08-04 at the
8765:Pierre de Fermat,
8083:
8063:
8031:
8011:
7991:
7971:
7933:
7913:
7893:
7873:
7850:
7830:
7806:Finding intercepts
7790:
7673:
7629:
7589:
7529:
7509:
7487:
7454:
7420:
7357:
7337:
7317:
7297:
7270:
7243:
7180:
7063:
7019:
6979:
6917:
6897:
6875:
6842:
6808:
6742:
6672:
6652:
6623:
6575:
6527:
6507:
6487:
6459:
6402:
6354:
6334:
6311:
6291:
6259:
6233:
6187:
6167:
6135:
6103:
6083:
6061:which is true, so
6051:
6009:
5951:
5931:
5899:
5867:
5835:
5727:
5707:
5629:
5597:
5574:
5542:
5507:
5458:
5409:{\displaystyle xy}
5406:
5383:
5342:
5322:
5302:
5282:
5262:
5242:
5222:
5202:
5182:
5162:
5142:
5122:
5102:
5043:
5007:
4961:
4941:
4894:
4874:
4830:
4807:
4779:
4756:
4736:
4716:
4688:
4665:
4645:
4619:
4596:
4576:
4550:
4526:
4496:
4481:a) y = f(x) = |x|
4442:
4338:
4183:
4138:
4009:Euclidean geometry
3993:
3973:Distance and angle
3929:
3803:algebraic equation
3719:
3680:
3628:
3574:
3548:
3515:
3466:
3417:
3382:
3277:quadratic equation
3265:
3158:
3117:
3076:
3017:
2970:
2912:are constants and
2886:
2751:
2640:
2583:
2561:
2539:
2519:
2492:
2470:
2450:
2404:
2332:
2313:Specifically, let
2244:
2207:affine coordinates
1959:
1715:
1705:Coordinate systems
1633:Cartesian geometry
1577:Francine Descartes
1422:Trademark argument
1330:algebraic geometry
1205:synthetic geometry
1195:, is the study of
1193:Cartesian geometry
9854:Analytic geometry
9841:
9840:
9440:Harmonic analysis
9078:Mikhail Postnikov
8976:978-0-07-161545-7
8887:978-3-540-74637-9
8845:, Athaeneum Press
8831:978-0-495-01166-8
8327:point of tangency
8086:{\displaystyle y}
8034:{\displaystyle b}
8014:{\displaystyle y}
7994:{\displaystyle b}
7936:{\displaystyle x}
7916:{\displaystyle x}
7896:{\displaystyle y}
7876:{\displaystyle y}
7860:coordinate axes.
7853:{\displaystyle y}
7833:{\displaystyle x}
7780:
7774:
7740:
7728:
7722:
7668:
7662:
7532:{\displaystyle y}
7512:{\displaystyle x}
7360:{\displaystyle x}
7340:{\displaystyle y}
7320:{\displaystyle y}
7170:
7164:
7130:
7118:
7112:
7058:
7052:
6920:{\displaystyle y}
6900:{\displaystyle x}
6675:{\displaystyle x}
6530:{\displaystyle y}
6510:{\displaystyle x}
6490:{\displaystyle y}
6357:{\displaystyle Q}
6337:{\displaystyle P}
6314:{\displaystyle P}
6269:which is false.
6190:{\displaystyle P}
6177:the equation for
6106:{\displaystyle Q}
5954:{\displaystyle Q}
5730:{\displaystyle Q}
5600:{\displaystyle P}
5345:{\displaystyle k}
5325:{\displaystyle h}
5305:{\displaystyle k}
5285:{\displaystyle h}
5265:{\displaystyle h}
5245:{\displaystyle k}
5225:{\displaystyle y}
5205:{\displaystyle a}
5185:{\displaystyle b}
5165:{\displaystyle x}
5145:{\displaystyle a}
5125:{\displaystyle a}
4964:{\displaystyle A}
4897:{\displaystyle y}
4833:{\displaystyle x}
4782:{\displaystyle y}
4766:as being dilated)
4759:{\displaystyle x}
4746:. (think of the
4739:{\displaystyle b}
4691:{\displaystyle x}
4668:{\displaystyle k}
4622:{\displaystyle y}
4599:{\displaystyle h}
4553:{\displaystyle x}
4401:
4333:
4133:
3778:. In coordinates
3380:
3360:
2821:
2820: where
2542:{\displaystyle P}
2473:{\displaystyle P}
1921:
1808:polar coordinates
1802:Polar coordinates
1615:
1614:
1467:Balloonist theory
1442:Coordinate system
1437:Analytic geometry
1322:geometric algebra
1201:coordinate system
1185:analytic geometry
1177:
1176:
1142:
1141:
865:List of geometers
548:Three-dimensional
537:
536:
9861:
9829:
9828:
9817:
9816:
9805:
9804:
9794:
9793:
9725:Computer algebra
9700:Computer science
9420:Complex analysis
9254:
9247:
9240:
9231:
9230:
9213:
9205:
9175:
9153:
9131:
9102:
9074:
9052:Internet Archive
9040:
9022:
8990:
8987:
8981:
8980:
8962:
8956:
8927:
8921:
8916:
8897:
8891:
8890:
8865:
8859:
8852:
8846:
8839:
8833:
8816:
8807:
8786:
8780:
8771:pp. 91â103.
8763:
8757:
8751:
8745:
8739:
8730:
8724:
8718:
8717:
8685:
8679:
8673:
8667:
8666:
8642:
8636:
8633:
8627:
8621:
8615:
8614:
8584:
8578:
8577:
8549:
8538:
8529:
8528:
8482:
8476:
8475:
8438:
8432:
8431:
8398:
8372:Rotation of axes
8362:Applied geometry
8301:
8294:
8288:
8280:
8264:
8254:
8238:infinitely close
8092:
8090:
8089:
8084:
8072:
8070:
8069:
8064:
8040:
8038:
8037:
8032:
8020:
8018:
8017:
8012:
8000:
7998:
7997:
7992:
7981:, the parameter
7980:
7978:
7977:
7972:
7942:
7940:
7939:
7934:
7922:
7920:
7919:
7914:
7902:
7900:
7899:
7894:
7882:
7880:
7879:
7874:
7859:
7857:
7856:
7851:
7839:
7837:
7836:
7831:
7799:
7797:
7796:
7791:
7786:
7782:
7781:
7776:
7775:
7770:
7764:
7756:
7741:
7738:
7734:
7730:
7729:
7724:
7723:
7718:
7712:
7704:
7682:
7680:
7679:
7674:
7669:
7664:
7663:
7658:
7652:
7638:
7636:
7635:
7630:
7625:
7614:
7613:
7598:
7596:
7595:
7590:
7582:
7581:
7569:
7568:
7556:
7538:
7536:
7535:
7530:
7518:
7516:
7515:
7510:
7496:
7494:
7493:
7488:
7483:
7463:
7461:
7460:
7455:
7429:
7427:
7426:
7421:
7413:
7412:
7385:
7384:
7366:
7364:
7363:
7358:
7346:
7344:
7343:
7338:
7326:
7324:
7323:
7318:
7306:
7304:
7303:
7298:
7296:
7295:
7279:
7277:
7276:
7271:
7269:
7268:
7252:
7250:
7249:
7244:
7236:
7235:
7223:
7222:
7189:
7187:
7186:
7181:
7176:
7172:
7171:
7166:
7165:
7160:
7154:
7146:
7131:
7128:
7124:
7120:
7119:
7114:
7113:
7108:
7102:
7094:
7072:
7070:
7069:
7064:
7059:
7054:
7053:
7048:
7042:
7028:
7026:
7025:
7020:
7015:
7004:
7003:
6988:
6986:
6985:
6980:
6972:
6971:
6959:
6958:
6946:
6926:
6924:
6923:
6918:
6906:
6904:
6903:
6898:
6884:
6882:
6881:
6876:
6871:
6851:
6849:
6848:
6843:
6817:
6815:
6814:
6809:
6801:
6800:
6767:
6766:
6751:
6749:
6748:
6743:
6732:
6731:
6710:
6709:
6681:
6679:
6678:
6673:
6661:
6659:
6658:
6653:
6651:
6650:
6632:
6630:
6629:
6624:
6619:
6618:
6600:
6599:
6584:
6582:
6581:
6576:
6568:
6567:
6555:
6554:
6536:
6534:
6533:
6528:
6516:
6514:
6513:
6508:
6496:
6494:
6493:
6488:
6468:
6466:
6465:
6460:
6452:
6451:
6439:
6438:
6411:
6409:
6408:
6403:
6395:
6394:
6382:
6381:
6363:
6361:
6360:
6355:
6343:
6341:
6340:
6335:
6320:
6318:
6317:
6312:
6300:
6298:
6297:
6292:
6268:
6266:
6265:
6260:
6242:
6240:
6239:
6234:
6226:
6225:
6213:
6212:
6196:
6194:
6193:
6188:
6176:
6174:
6173:
6168:
6144:
6142:
6141:
6136:
6112:
6110:
6109:
6104:
6092:
6090:
6089:
6084:
6060:
6058:
6057:
6052:
6044:
6043:
6018:
6016:
6015:
6010:
6002:
6001:
5989:
5988:
5960:
5958:
5957:
5952:
5940:
5938:
5937:
5932:
5908:
5906:
5905:
5900:
5876:
5874:
5873:
5868:
5844:
5842:
5841:
5836:
5825:
5824:
5812:
5811:
5790:
5736:
5734:
5733:
5728:
5716:
5714:
5713:
5708:
5697:
5696:
5684:
5683:
5674:
5638:
5636:
5635:
5630:
5606:
5604:
5603:
5598:
5583:
5581:
5580:
5575:
5551:
5549:
5548:
5543:
5516:
5514:
5513:
5508:
5467:
5465:
5464:
5459:
5445:
5444:
5432:
5431:
5415:
5413:
5412:
5407:
5392:
5390:
5389:
5384:
5351:
5349:
5348:
5343:
5331:
5329:
5328:
5323:
5311:
5309:
5308:
5303:
5292:, vertical, and
5291:
5289:
5288:
5283:
5271:
5269:
5268:
5263:
5251:
5249:
5248:
5243:
5231:
5229:
5228:
5223:
5211:
5209:
5208:
5203:
5191:
5189:
5188:
5183:
5171:
5169:
5168:
5163:
5151:
5149:
5148:
5143:
5131:
5129:
5128:
5123:
5111:
5109:
5108:
5103:
5052:
5050:
5049:
5044:
5016:
5014:
5013:
5008:
5003:
4970:
4968:
4967:
4962:
4950:
4948:
4947:
4942:
4903:
4901:
4900:
4895:
4883:
4881:
4880:
4875:
4839:
4837:
4836:
4831:
4816:
4814:
4813:
4808:
4803:
4788:
4786:
4785:
4780:
4765:
4763:
4762:
4757:
4745:
4743:
4742:
4737:
4725:
4723:
4722:
4717:
4712:
4697:
4695:
4694:
4689:
4674:
4672:
4671:
4666:
4654:
4652:
4651:
4646:
4628:
4626:
4625:
4620:
4605:
4603:
4602:
4597:
4585:
4583:
4582:
4577:
4559:
4557:
4556:
4551:
4535:
4533:
4532:
4527:
4492:
4488:
4484:
4451:
4449:
4448:
4443:
4429:
4425:
4416:
4412:
4403:
4402:
4400:
4399:
4387:
4382:
4379:
4371:
4347:
4345:
4344:
4339:
4334:
4332:
4331:
4322:
4321:
4309:
4308:
4293:
4292:
4283:
4282:
4270:
4269:
4254:
4253:
4244:
4243:
4231:
4230:
4218:
4192:
4190:
4189:
4184:
4147:
4145:
4144:
4139:
4134:
4132:
4131:
4122:
4121:
4109:
4108:
4093:
4092:
4083:
4082:
4070:
4069:
4057:
3938:
3936:
3935:
3930:
3916:
3915:
3906:
3905:
3895:
3890:
3872:
3871:
3862:
3861:
3849:
3848:
3838:
3833:
3800:
3740:Quadric surfaces
3728:
3726:
3725:
3720:
3697:if we also have
3689:
3687:
3686:
3681:
3661:
3660:
3637:
3635:
3634:
3629:
3609:
3608:
3583:
3581:
3580:
3575:
3557:
3555:
3554:
3549:
3524:
3522:
3521:
3516:
3496:
3495:
3475:
3473:
3472:
3467:
3450:
3449:
3426:
3424:
3423:
3418:
3413:
3412:
3407:
3394:projective space
3391:
3389:
3388:
3383:
3381:
3378:
3361:
3359: with
3358:
3326:
3325:
3298:
3297:
3167:
3165:
3164:
3159:
3148:
3147:
3126:
3124:
3123:
3118:
3107:
3106:
3085:
3083:
3082:
3077:
3066:
3065:
3028:
3026:
3024:
3023:
3018:
2992:
2979:
2977:
2976:
2971:
2895:
2893:
2892:
2887:
2879:
2878:
2863:
2862:
2847:
2846:
2822:
2819:
2767:
2760:
2758:
2757:
2752:
2738:
2737:
2710:
2709:
2682:
2681:
2649:
2647:
2646:
2641:
2630:
2629:
2624:
2615:
2604:
2592:
2590:
2589:
2584:
2582:
2570:
2568:
2567:
2562:
2560:
2548:
2546:
2545:
2540:
2528:
2526:
2525:
2520:
2518:
2517:
2501:
2499:
2498:
2493:
2491:
2479:
2477:
2476:
2471:
2459:
2457:
2456:
2451:
2425:
2413:
2411:
2410:
2405:
2400:
2399:
2387:
2386:
2374:
2373:
2358:
2357:
2341:
2339:
2338:
2333:
2331:
2330:
2325:
2289:of the function
2253:
2251:
2250:
2245:
2197:Plane (geometry)
2187:Lines and planes
2104:-coordinate and
1968:
1966:
1965:
1960:
1949:
1922:
1920:
1919:
1907:
1906:
1897:
1780:of coordinates (
1629:Pierre de Fermat
1607:
1600:
1593:
1447:Cartesian circle
1411:Cogito, ergo sum
1367:
1344:
1343:
1187:, also known as
1169:
1162:
1155:
883:
882:
402:
401:
335:Zero-dimensional
40:
26:
25:
9869:
9868:
9864:
9863:
9862:
9860:
9859:
9858:
9844:
9843:
9842:
9837:
9788:
9779:
9729:
9686:
9665:Systems science
9596:
9592:Homotopy theory
9558:
9525:
9477:
9449:
9396:
9343:
9314:Category theory
9300:
9265:
9258:
9220:
9195:10.2307/2305740
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9004:
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8646:Stillwell, John
8643:
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8603:10.2307/3217882
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8395:
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8332:Similarly, the
8320:Euclidean space
8299:
8286:
8282:
8266:
8256:
8241:
8210:
8204:
8196:
8186:generalizes to
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4493:d) y = 1/2 f(x)
4490:
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4472:Transformations
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3977:Main articles:
3975:
3945:(including the
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3756:quadric surface
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3746:Quadric surface
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2770:linear equation
2768:This is just a
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2203:Cartesian plane
2199:
2193:Line (geometry)
2191:Main articles:
2189:
2171:. The equation
2128:specify lines,
2074:
2066:Main articles:
2064:
2033:
2027:
1989:
1983:
1945:
1915:
1911:
1902:
1898:
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1838:
1804:
1798:
1774:Euclidean space
1746:
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1731:Euclidean space
1707:
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1621:
1611:
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1544:
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1490:
1462:Cartesian diver
1390:Foundationalism
1375:
1342:
1332:, and his book
1326:cubic equations
1314:
1269:
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1143:
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396:Two-dimensional
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9781:
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9743:Mathematicians
9739:
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9735:Related topics
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9670:Control theory
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9445:Measure theory
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9218:External links
9216:
9215:
9214:
9206:
9176:
9154:
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9099:978-0674823556
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9037:978-0821821022
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8819:Stewart, James
8808:
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8731:
8719:
8709:
8693:"The Calculus"
8680:
8668:
8658:
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8628:
8616:
8597:(1): 248â249.
8579:
8564:
8530:
8506:
8486:Boyer, Carl B.
8477:
8462:
8442:Boyer, Carl B.
8433:
8422:
8402:Boyer, Carl B.
8392:
8391:
8389:
8386:
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8374:
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8357:
8354:
8314:and curves in
8206:Main article:
8203:
8200:
8195:
8192:
8136:surface normal
8098:
8097:Geometric axis
8095:
8082:
8073:is called the
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5476:Main article:
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4569:
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4513:
4510:
4489:c) y = f(x)-3
4485:b) y = f(x+3)
4480:
4473:
4470:
4441:
4438:
4435:
4432:
4428:
4424:
4420:
4415:
4411:
4407:
4398:
4395:
4392:
4386:
4378:
4374:
4370:
4360:is defined by
4337:
4330:
4326:
4320:
4316:
4312:
4307:
4303:
4299:
4296:
4291:
4287:
4281:
4277:
4273:
4268:
4264:
4260:
4257:
4252:
4248:
4242:
4238:
4234:
4229:
4225:
4221:
4216:
4213:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4137:
4130:
4126:
4120:
4116:
4112:
4107:
4103:
4099:
4096:
4091:
4087:
4081:
4077:
4073:
4068:
4064:
4060:
4055:
4052:
4040:
4033:
4026:
4019:
3974:
3971:
3928:
3925:
3922:
3919:
3914:
3910:
3904:
3900:
3894:
3889:
3886:
3883:
3879:
3875:
3870:
3866:
3860:
3857:
3853:
3847:
3843:
3837:
3832:
3829:
3826:
3823:
3820:
3816:
3797:
3790:
3783:
3744:Main article:
3741:
3738:
3737:
3736:
3735:
3734:
3718:
3715:
3712:
3709:
3706:
3679:
3676:
3673:
3670:
3667:
3664:
3659:
3655:
3643:
3627:
3624:
3621:
3618:
3615:
3612:
3607:
3603:
3591:
3590:
3589:
3573:
3570:
3567:
3547:
3544:
3541:
3514:
3511:
3508:
3505:
3502:
3499:
3494:
3490:
3465:
3462:
3459:
3456:
3453:
3448:
3444:
3416:
3411:
3406:
3376:
3373:
3370:
3367:
3364:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3324:
3320:
3316:
3313:
3310:
3307:
3304:
3301:
3296:
3292:
3288:
3250:Main article:
3247:
3246:Conic sections
3244:
3243:
3242:
3212:
3208:
3201:
3194:
3188:
3157:
3154:
3151:
3146:
3142:
3138:
3135:
3116:
3113:
3110:
3105:
3101:
3097:
3094:
3075:
3072:
3069:
3064:
3060:
3056:
3053:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2991:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2885:
2882:
2877:
2873:
2869:
2866:
2861:
2857:
2853:
2850:
2845:
2841:
2837:
2834:
2831:
2828:
2825:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2750:
2747:
2744:
2741:
2736:
2732:
2728:
2725:
2722:
2719:
2716:
2713:
2708:
2704:
2700:
2697:
2694:
2691:
2688:
2685:
2680:
2676:
2672:
2669:
2666:
2663:
2639:
2636:
2633:
2628:
2623:
2618:
2614:
2610:
2607:
2603:
2581:
2559:
2538:
2516:
2512:
2490:
2469:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2424:
2403:
2398:
2394:
2390:
2385:
2381:
2377:
2372:
2368:
2364:
2361:
2356:
2352:
2329:
2324:
2303:
2302:
2280:
2270:
2243:
2240:
2237:
2234:
2231:
2228:
2188:
2185:
2134:conic sections
2063:
2060:
2029:Main article:
2026:
2023:
2010:-axis and the
1985:Main article:
1982:
1979:
1958:
1955:
1952:
1948:
1944:
1941:
1938:
1935:
1932:
1929:
1925:
1918:
1914:
1910:
1905:
1901:
1895:
1892:
1888:
1885:
1882:
1879:
1875:
1872:
1869:
1865:
1862:
1859:
1856:
1852:
1849:
1846:
1800:Main article:
1797:
1794:
1778:ordered triple
1742:Main article:
1739:
1736:
1725:has a pair of
1703:Main article:
1700:
1697:
1693:Leonhard Euler
1625:René Descartes
1613:
1612:
1610:
1609:
1602:
1595:
1587:
1584:
1583:
1580:
1579:
1574:
1569:
1567:Baruch Spinoza
1564:
1559:
1553:
1550:
1549:
1546:
1545:
1542:
1541:
1536:
1531:
1526:
1521:
1516:
1511:
1506:
1500:
1497:
1496:
1493:
1492:
1489:
1488:
1481:
1474:
1469:
1464:
1459:
1454:
1449:
1444:
1439:
1434:
1429:
1424:
1419:
1414:
1407:
1405:Dream argument
1402:
1397:
1392:
1387:
1382:
1376:
1373:
1372:
1369:
1368:
1360:
1359:
1357:René Descartes
1353:
1352:
1341:
1340:Western Europe
1338:
1313:
1310:
1275:mathematician
1268:
1267:Ancient Greece
1265:
1263:
1260:
1218:, and also in
1175:
1174:
1172:
1171:
1164:
1157:
1149:
1146:
1145:
1140:
1139:
1138:
1137:
1132:
1124:
1123:
1119:
1118:
1117:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1066:
1058:
1057:
1053:
1052:
1051:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1007:
1006:
1002:
1001:
1000:
999:
994:
989:
984:
979:
974:
969:
964:
959:
954:
949:
944:
936:
935:
931:
930:
929:
928:
923:
918:
913:
908:
903:
898:
890:
889:
881:
877:
876:
875:
872:
871:
868:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
661:
657:
656:
655:
652:
651:
645:
644:
641:
640:
635:
629:
622:
621:
620:
617:
616:
613:
612:
607:
602:
600:Platonic Solid
597:
592:
587:
582:
577:
572:
571:
570:
559:
558:
552:
546:
545:
544:
541:
540:
535:
534:
533:
532:
527:
522:
514:
513:
507:
506:
505:
504:
499:
491:
490:
484:
483:
482:
481:
476:
471:
466:
458:
457:
451:
450:
449:
448:
443:
438:
430:
429:
423:
422:
421:
420:
415:
410:
400:
394:
393:
392:
389:
388:
385:
384:
379:
378:
377:
372:
361:
355:
354:
353:
350:
349:
346:
345:
339:
333:
332:
331:
328:
327:
324:
323:
318:
313:
307:
306:
301:
296:
286:
281:
276:
270:
269:
260:
256:
255:
252:
248:
247:
246:
245:
242:
241:
238:
237:
236:
235:
225:
220:
215:
210:
205:
204:
203:
193:
188:
183:
182:
181:
176:
171:
161:
160:
159:
154:
144:
139:
134:
129:
124:
119:
118:
117:
112:
111:
110:
95:
89:
83:
82:
81:
78:
77:
75:
74:
64:
58:
55:
54:
41:
33:
32:
15:
9:
6:
4:
3:
2:
9866:
9855:
9852:
9851:
9849:
9834:
9833:
9824:
9822:
9821:
9812:
9810:
9809:
9800:
9798:
9797:
9792:
9786:
9785:
9782:
9776:
9773:
9771:
9768:
9766:
9763:
9761:
9758:
9756:
9753:
9749:
9746:
9745:
9744:
9741:
9740:
9738:
9736:
9732:
9726:
9723:
9721:
9718:
9716:
9713:
9711:
9708:
9706:
9703:
9701:
9698:
9697:
9695:
9693:
9692:Computational
9689:
9681:
9678:
9676:
9673:
9671:
9668:
9667:
9666:
9663:
9661:
9658:
9656:
9653:
9651:
9648:
9646:
9643:
9641:
9638:
9636:
9633:
9631:
9628:
9626:
9623:
9621:
9618:
9616:
9613:
9611:
9608:
9607:
9605:
9603:
9599:
9593:
9590:
9588:
9585:
9583:
9580:
9578:
9575:
9573:
9570:
9569:
9567:
9565:
9561:
9555:
9552:
9550:
9547:
9545:
9542:
9540:
9537:
9536:
9534:
9532:
9531:Number theory
9528:
9522:
9519:
9517:
9514:
9512:
9509:
9507:
9504:
9502:
9499:
9497:
9494:
9492:
9489:
9488:
9486:
9484:
9480:
9474:
9471:
9469:
9466:
9464:
9463:Combinatorics
9461:
9460:
9458:
9456:
9452:
9446:
9443:
9441:
9438:
9436:
9433:
9431:
9428:
9426:
9423:
9421:
9418:
9416:
9415:Real analysis
9413:
9411:
9408:
9407:
9405:
9403:
9399:
9393:
9390:
9388:
9385:
9383:
9380:
9378:
9375:
9373:
9370:
9368:
9365:
9363:
9360:
9358:
9355:
9354:
9352:
9350:
9346:
9340:
9337:
9335:
9332:
9330:
9327:
9325:
9322:
9320:
9317:
9315:
9312:
9311:
9309:
9307:
9303:
9297:
9294:
9292:
9289:
9285:
9282:
9280:
9277:
9276:
9275:
9272:
9271:
9268:
9263:
9255:
9250:
9248:
9243:
9241:
9236:
9235:
9232:
9225:
9222:
9221:
9212:
9207:
9204:
9200:
9196:
9192:
9188:
9184:
9183:
9177:
9174:
9170:
9166:
9162:
9161:
9155:
9152:
9148:
9145:(3): 99â105,
9144:
9140:
9139:
9133:
9130:
9126:
9122:
9118:
9117:
9111:
9110:
9101:
9095:
9091:
9086:
9083:
9079:
9076:
9073:
9071:0-321-01618-1
9067:
9063:
9062:
9056:
9053:
9049:
9045:
9042:
9039:
9033:
9029:
9024:
9021:
9015:
9011:
9006:
9005:
8986:
8978:
8972:
8968:
8961:
8955:
8951:
8947:
8943:
8939:
8935:
8932:
8926:
8920:
8915:
8913:0-471-75715-2
8909:
8905:
8904:
8896:
8889:
8883:
8879:
8875:
8871:
8864:
8857:
8851:
8844:
8838:
8832:
8828:
8824:
8820:
8815:
8813:
8805:
8801:
8797:
8793:
8790:
8785:
8779:
8775:
8772:
8768:
8762:
8755:
8750:
8743:
8738:
8736:
8728:
8723:
8716:
8712:
8710:0-471-18082-3
8706:
8702:
8698:
8694:
8690:
8684:
8677:
8672:
8665:
8661:
8659:0-387-95336-1
8655:
8651:
8647:
8641:
8632:
8625:
8620:
8612:
8608:
8604:
8600:
8596:
8592:
8591:
8583:
8576:
8573:
8567:
8565:9780471543978
8561:
8557:
8553:
8548:
8543:
8537:
8535:
8527:
8524:
8520:
8515:
8509:
8507:0-471-54397-7
8503:
8499:
8495:
8491:
8487:
8481:
8474:
8471:
8465:
8463:0-471-54397-7
8459:
8455:
8451:
8447:
8443:
8437:
8430:
8425:
8423:0-471-54397-7
8419:
8415:
8411:
8407:
8403:
8397:
8393:
8383:
8380:
8378:
8375:
8373:
8370:
8368:
8367:Cross product
8365:
8363:
8360:
8359:
8353:
8351:
8350:Tangent space
8347:
8343:
8339:
8335:
8334:tangent plane
8330:
8328:
8323:
8321:
8318:-dimensional
8317:
8313:
8309:
8305:
8298:
8292:
8285:
8278:
8274:
8270:
8263:
8259:
8252:
8248:
8244:
8239:
8235:
8234:straight line
8231:
8227:
8224:) to a plane
8223:
8219:
8215:
8209:
8199:
8191:
8189:
8188:orthogonality
8185:
8181:
8180:normal vector
8177:
8173:
8169:
8165:
8161:
8160:tangent plane
8157:
8156:perpendicular
8153:
8149:
8145:
8141:
8137:
8132:
8130:
8126:
8122:
8121:perpendicular
8118:
8114:
8109:
8107:
8102:
8094:
8080:
8057:
8054:
8051:
8041:or the point
8028:
8008:
7988:
7968:
7965:
7962:
7959:
7956:
7953:
7946:For the line
7944:
7930:
7910:
7890:
7870:
7861:
7847:
7827:
7817:
7813:
7803:
7800:
7787:
7783:
7777:
7771:
7766:
7760:
7757:
7753:
7749:
7745:
7731:
7725:
7719:
7714:
7708:
7705:
7701:
7697:
7693:
7683:
7670:
7665:
7659:
7654:
7648:
7645:
7626:
7622:
7618:
7615:
7610:
7606:
7586:
7583:
7578:
7574:
7570:
7565:
7557:
7553:
7549:
7526:
7506:
7497:
7484:
7480:
7476:
7473:
7470:
7451:
7448:
7445:
7442:
7439:
7436:
7417:
7414:
7409:
7405:
7401:
7398:
7395:
7392:
7389:
7386:
7381:
7377:
7368:
7354:
7334:
7314:
7292:
7288:
7265:
7261:
7240:
7237:
7232:
7228:
7224:
7219:
7211:
7208:
7205:
7194:
7190:
7177:
7173:
7167:
7161:
7156:
7150:
7147:
7143:
7139:
7135:
7121:
7115:
7109:
7104:
7098:
7095:
7091:
7087:
7083:
7073:
7060:
7055:
7049:
7044:
7038:
7035:
7016:
7012:
7008:
7005:
7000:
6996:
6976:
6973:
6968:
6964:
6960:
6955:
6947:
6943:
6939:
6928:
6914:
6894:
6885:
6872:
6868:
6864:
6861:
6858:
6839:
6836:
6833:
6830:
6827:
6824:
6805:
6802:
6797:
6793:
6789:
6786:
6783:
6780:
6777:
6774:
6771:
6768:
6763:
6759:
6739:
6736:
6728:
6724:
6720:
6717:
6711:
6706:
6698:
6695:
6692:
6669:
6647:
6643:
6633:
6620:
6615:
6611:
6607:
6604:
6601:
6596:
6592:
6572:
6569:
6564:
6560:
6556:
6551:
6547:
6538:
6524:
6504:
6484:
6476:
6475:Substitution:
6472:
6469:
6456:
6453:
6448:
6444:
6440:
6435:
6427:
6424:
6421:
6399:
6396:
6391:
6387:
6383:
6378:
6374:
6365:
6351:
6331:
6322:
6308:
6285:
6282:
6279:
6256:
6253:
6250:
6230:
6227:
6222:
6218:
6214:
6209:
6205:
6184:
6161:
6158:
6155:
6129:
6126:
6123:
6100:
6077:
6074:
6071:
6048:
6045:
6040:
6032:
6029:
6006:
6003:
5998:
5994:
5990:
5985:
5977:
5974:
5971:
5948:
5925:
5922:
5919:
5893:
5890:
5887:
5861:
5858:
5855:
5829:
5826:
5821:
5817:
5813:
5808:
5800:
5797:
5794:
5780:
5777:
5774:
5765:
5762:
5759:
5753:
5750:
5747:
5724:
5701:
5698:
5693:
5689:
5685:
5680:
5676:
5664:
5661:
5658:
5649:
5646:
5623:
5620:
5617:
5594:
5587:For example,
5585:
5568:
5565:
5562:
5536:
5533:
5530:
5524:
5501:
5498:
5495:
5489:
5479:
5469:
5455:
5452:
5449:
5446:
5441:
5437:
5433:
5428:
5424:
5403:
5400:
5377:
5374:
5371:
5365:
5358:Suppose that
5356:
5353:
5339:
5319:
5299:
5279:
5259:
5239:
5219:
5199:
5179:
5159:
5139:
5119:
5099:
5096:
5087:
5084:
5081:
5075:
5069:
5066:
5063:
5060:
5037:
5031:
5028:
5025:
5004:
5000:
4996:
4993:
4990:
4981:
4979:
4958:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4891:
4884:and changing
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4850:
4847:
4827:
4819:
4804:
4800:
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4768:
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4733:
4713:
4709:
4705:
4685:
4677:
4662:
4642:
4639:
4636:
4616:
4608:
4593:
4573:
4570:
4567:
4547:
4539:
4538:
4537:
4520:
4517:
4514:
4508:
4501:The graph of
4499:
4478:
4469:
4467:
4463:
4459:
4455:
4439:
4436:
4433:
4430:
4384:
4372:
4359:
4355:
4351:
4335:
4328:
4318:
4314:
4310:
4305:
4301:
4294:
4289:
4279:
4275:
4271:
4266:
4262:
4255:
4250:
4240:
4236:
4232:
4227:
4223:
4214:
4211:
4202:
4201:of the line.
4200:
4196:
4180:
4174:
4168:
4165:
4162:
4159:
4151:
4135:
4128:
4118:
4114:
4110:
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4094:
4089:
4079:
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4025:
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4010:
4006:
4002:
3998:
3989:
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3970:
3968:
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3956:
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3944:
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3926:
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3821:
3818:
3814:
3805:
3804:
3796:
3789:
3782:
3777:
3773:
3769:
3765:
3762:-dimensional
3761:
3757:
3753:
3747:
3732:
3716:
3713:
3710:
3707:
3704:
3696:
3695:
3693:
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3625:
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3601:
3592:
3587:
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3506:
3503:
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3497:
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3463:
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3457:
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3414:
3409:
3395:
3374:
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3333:
3330:
3327:
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3318:
3314:
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3308:
3305:
3302:
3299:
3294:
3290:
3286:
3278:
3274:
3270:
3263:
3258:
3253:
3252:Conic section
3240:
3236:
3232:
3228:
3224:
3220:
3216:
3213:
3207:
3200:
3193:
3189:
3186:
3182:
3178:
3174:
3171:
3170:
3169:
3155:
3152:
3149:
3144:
3140:
3136:
3133:
3114:
3111:
3108:
3103:
3099:
3095:
3092:
3073:
3070:
3067:
3062:
3058:
3054:
3051:
3043:
3039:
3034:
3032:
3011:
3008:
3005:
3002:
2999:
2993:
2967:
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2937:
2934:
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2923:
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2915:
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2907:
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2883:
2875:
2871:
2867:
2864:
2859:
2855:
2851:
2848:
2843:
2839:
2835:
2829:
2826:
2823:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2771:
2765:
2762:which is the
2748:
2745:
2742:
2734:
2730:
2726:
2723:
2717:
2714:
2706:
2702:
2698:
2695:
2689:
2686:
2678:
2674:
2670:
2667:
2661:
2653:
2637:
2634:
2626:
2616:
2605:
2536:
2514:
2510:
2467:
2444:
2441:
2438:
2435:
2432:
2426:
2396:
2392:
2388:
2383:
2379:
2375:
2370:
2366:
2359:
2354:
2350:
2327:
2311:
2309:
2308:normal vector
2300:
2296:
2292:
2288:
2284:
2281:
2278:
2274:
2271:
2268:
2264:
2260:
2257:
2256:
2255:
2241:
2238:
2235:
2232:
2229:
2226:
2218:
2217:
2212:
2208:
2204:
2198:
2194:
2184:
2182:
2179: =
2178:
2175: +
2174:
2170:
2166:
2162:
2158:
2155: +
2154:
2150:
2147: =
2146:
2142:
2137:
2135:
2131:
2127:
2123:
2119:
2116: =
2115:
2111:
2107:
2103:
2099:
2096: =
2095:
2091:
2087:
2083:
2079:
2073:
2069:
2059:
2057:
2053:
2049:
2045:
2042:
2038:
2032:
2022:
2020:
2016:
2013:
2009:
2005:
2002:
1998:
1994:
1988:
1978:
1977:coordinates.
1976:
1972:
1956:
1950:
1946:
1942:
1936:
1933:
1930:
1927:
1923:
1916:
1912:
1908:
1903:
1899:
1893:
1890:
1886:
1883:
1880:
1877:
1873:
1870:
1867:
1863:
1860:
1857:
1854:
1850:
1847:
1844:
1836:
1832:
1828:
1824:
1820:
1817:
1813:
1809:
1803:
1793:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1745:
1735:
1732:
1728:
1724:
1720:
1711:
1706:
1696:
1694:
1689:
1685:
1681:
1676:
1674:
1670:
1666:
1662:
1658:
1654:
1653:
1648:
1644:
1642:
1636:
1634:
1630:
1626:
1620:
1608:
1603:
1601:
1596:
1594:
1589:
1588:
1586:
1585:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1558:
1555:
1554:
1548:
1547:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1501:
1495:
1494:
1487:
1486:
1482:
1480:
1479:
1475:
1473:
1470:
1468:
1465:
1463:
1460:
1458:
1457:Rule of signs
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1412:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1377:
1371:
1370:
1366:
1362:
1361:
1358:
1355:
1354:
1350:
1346:
1345:
1337:
1335:
1331:
1327:
1323:
1319:
1309:
1307:
1303:
1298:
1294:
1290:
1289:
1284:
1280:
1278:
1274:
1259:
1257:
1251:
1249:
1245:
1241:
1237:
1233:
1229:
1228:space science
1225:
1221:
1217:
1213:
1208:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1170:
1165:
1163:
1158:
1156:
1151:
1150:
1148:
1147:
1136:
1133:
1131:
1128:
1127:
1126:
1125:
1121:
1120:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1061:
1060:
1059:
1055:
1054:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1010:
1009:
1008:
1004:
1003:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
948:
945:
943:
940:
939:
938:
937:
933:
932:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
893:
892:
891:
888:
885:
884:
874:
873:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
662:
654:
653:
650:
647:
646:
639:
636:
634:
631:
630:
625:
619:
618:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
569:
566:
565:
564:
561:
560:
557:
554:
553:
549:
543:
542:
531:
528:
526:
525:Circumference
523:
521:
518:
517:
516:
515:
512:
509:
508:
503:
500:
498:
495:
494:
493:
492:
489:
488:Quadrilateral
486:
485:
480:
477:
475:
472:
470:
467:
465:
462:
461:
460:
459:
456:
455:Parallelogram
453:
452:
447:
444:
442:
439:
437:
434:
433:
432:
431:
428:
425:
424:
419:
416:
414:
411:
409:
406:
405:
404:
403:
397:
391:
390:
383:
380:
376:
373:
371:
368:
367:
366:
363:
362:
358:
352:
351:
344:
341:
340:
336:
330:
329:
322:
319:
317:
314:
312:
309:
308:
305:
302:
300:
297:
294:
293:Perpendicular
290:
289:Orthogonality
287:
285:
282:
280:
277:
275:
272:
271:
268:
265:
264:
263:
253:
250:
249:
244:
243:
234:
231:
230:
229:
226:
224:
221:
219:
216:
214:
213:Computational
211:
209:
206:
202:
199:
198:
197:
194:
192:
189:
187:
184:
180:
177:
175:
172:
170:
167:
166:
165:
162:
158:
155:
153:
150:
149:
148:
145:
143:
140:
138:
135:
133:
130:
128:
125:
123:
120:
116:
113:
109:
106:
105:
104:
101:
100:
99:
98:Non-Euclidean
96:
94:
91:
90:
86:
80:
79:
72:
68:
65:
63:
60:
59:
57:
56:
52:
48:
44:
39:
35:
34:
31:
28:
27:
22:
9830:
9818:
9806:
9787:
9720:Optimization
9582:Differential
9506:Differential
9495:
9473:Order theory
9468:Graph theory
9372:Group theory
9210:
9189:(2): 76â86,
9186:
9180:
9167:(1): 33â36,
9164:
9158:
9142:
9136:
9123:(4): 38â44,
9120:
9114:
9089:
9060:
9050:, link from
9027:
9009:
8985:
8966:
8960:
8941:
8929:Silvio Levy
8925:
8902:
8895:
8869:
8863:
8855:
8850:
8842:
8837:
8822:
8803:
8799:
8784:
8766:
8761:
8749:
8729:, p. 82
8722:
8714:
8696:
8689:Cooke, Roger
8683:
8678:, p. 74
8671:
8663:
8649:
8640:
8631:
8619:
8594:
8588:
8582:
8571:
8569:
8551:
8522:
8519:a posteriori
8518:
8513:
8511:
8493:
8480:
8469:
8467:
8449:
8436:
8427:
8409:
8396:
8382:Vector space
8333:
8331:
8326:
8324:
8315:
8312:space curves
8307:
8296:
8290:
8283:
8276:
8272:
8268:
8261:
8257:
8250:
8246:
8242:
8221:
8218:tangent line
8217:
8211:
8197:
8183:
8179:
8170:normal to a
8163:
8147:
8139:
8138:, or simply
8135:
8133:
8129:tangent line
8124:
8116:
8110:
8105:
8103:
8100:
8093:-intercept.
7945:
7862:
7819:
7801:
7684:
7498:
7369:
7192:
7191:
7074:
6929:
6886:
6634:
6539:
6497:in terms of
6474:
6473:
6470:
6366:
6323:
5586:
5481:
5357:
5354:
4982:
4974:
4500:
4497:
4465:
4461:
4453:
4357:
4353:
4203:
4194:
4037:
4030:
4023:
4016:
3994:
3955:hyperboloids
3940:
3806:
3794:
3787:
3780:
3759:
3755:
3751:
3749:
3434:
3431:discriminant
3428:
3266:
3238:
3234:
3230:
3222:
3218:
3214:
3205:
3198:
3191:
3184:
3180:
3176:
3172:
3037:
3035:
3031:general form
3030:
3027:as a normal.
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2764:point-normal
2763:
2312:
2304:
2298:
2294:
2290:
2282:
2279:of the line.
2272:
2269:of the line.
2258:
2214:
2210:
2200:
2180:
2176:
2172:
2165:intersection
2156:
2152:
2148:
2144:
2138:
2125:
2121:
2117:
2113:
2105:
2101:
2097:
2093:
2086:solution set
2075:
2068:Solution set
2055:
2051:
2047:
2043:
2036:
2034:
2018:
2014:
2007:
2003:
1996:
1990:
1834:
1830:
1826:
1822:
1818:
1811:
1805:
1789:
1785:
1781:
1769:
1765:
1762:ordered pair
1757:
1753:
1747:
1716:
1688:Introduction
1687:
1683:
1679:
1677:
1673:van Schooten
1657:La Geometrie
1656:
1650:
1646:
1641:La Géométrie
1639:
1637:
1632:
1622:
1524:La Géométrie
1483:
1478:Res cogitans
1476:
1472:Wax argument
1436:
1409:
1380:Cartesianism
1333:
1318:Omar Khayyam
1315:
1305:
1302:a posteriori
1301:
1292:
1286:
1281:
1270:
1252:
1240:differential
1209:
1192:
1188:
1184:
1178:
997:Parameshvara
810:Parameshvara
580:Dodecahedron
164:Differential
141:
9832:WikiProject
9675:Game theory
9655:Probability
9392:Homological
9382:Multilinear
9362:Commutative
9339:Type theory
9306:Foundations
9262:mathematics
8554:. pp.
8255:at a point
8228:at a given
8220:(or simply
8146:at a point
8125:normal line
7816:y-intercept
7812:x-intercept
7193:Elimination
5172:-axis. The
4350:dot product
3951:paraboloids
2652:dot product
2277:y-intercept
2201:Lines in a
1971:cylindrical
1727:real number
1699:Coordinates
1485:Res extensa
1385:Rationalism
1304:instead of
1232:spaceflight
1216:engineering
1181:mathematics
1122:Present day
1069:Lobachevsky
1056:1700sâ1900s
1013:JyeáčŁáčhadeva
1005:1400sâ1700s
957:Brahmagupta
780:Lobachevsky
760:JyeáčŁáčhadeva
710:Brahmagupta
638:Hypersphere
610:Tetrahedron
585:Icosahedron
157:Diophantine
9660:Statistics
9539:Arithmetic
9501:Arithmetic
9367:Elementary
9334:Set theory
9209:Pecl, J.,
9044:John Casey
8997:References
8727:Boyer 2004
8676:Boyer 2004
8304:derivative
8106:axial line
6301:is not in
3943:ellipsoids
2593:such that
2414:, and let
1643:(Geometry)
1617:See also:
1417:Evil demon
1374:Philosophy
1277:Menaechmus
982:al-Yasamin
926:Apollonius
921:Archimedes
911:Pythagoras
901:Baudhayana
855:al-Yasamin
805:Pythagoras
700:Baudhayana
690:Archimedes
685:Apollonius
590:Octahedron
441:Hypotenuse
316:Similarity
311:Congruence
223:Incidence
174:Symplectic
169:Riemannian
152:Arithmetic
127:Projective
115:Hyperbolic
43:Projecting
9587:Geometric
9577:Algebraic
9516:Euclidean
9491:Algebraic
9387:Universal
8946:CRC Press
8756:, pg. 436
8754:Katz 1998
8744:, pg. 442
8742:Katz 1998
8184:normality
7767:−
7655:±
7449:−
7437:−
7402:−
7387:−
7225:−
7209:−
7157:−
7045:±
6837:−
6825:−
6790:−
6769:−
6721:−
6696:−
6608:−
6425:−
6030:−
5975:−
5798:−
5447:−
5085:−
4936:
4921:
4912:−
4869:
4854:
4820:Changing
4769:Changing
4678:Changing
4640:−
4609:Changing
4571:−
4540:Changing
4437:θ
4434:
4373:⋅
4311:−
4272:−
4233:−
4169:
4160:θ
4111:−
4072:−
3959:cylinders
3878:∑
3815:∑
3692:hyperbola
3663:−
3611:−
3498:−
3452:−
2830:−
2727:−
2699:−
2671:−
2617:−
2606:⋅
2006:from the
1975:spherical
1937:
1928:θ
1884:θ
1881:
1861:θ
1858:
1684:Discourse
1514:The World
1395:Mechanism
1297:Descartes
1236:algebraic
1099:Minkowski
1018:Descartes
952:Aryabhata
947:KÄtyÄyana
878:by period
790:Minkowski
765:KÄtyÄyana
725:Descartes
670:Aryabhata
649:Geometers
633:Tesseract
497:Trapezoid
469:Rectangle
262:Dimension
147:Algebraic
137:Synthetic
108:Spherical
93:Euclidean
9848:Category
9808:Category
9564:Topology
9511:Discrete
9496:Analytic
9483:Geometry
9455:Discrete
9410:Calculus
9402:Analysis
9357:Abstract
9296:Glossary
9279:Timeline
9107:Articles
8934:Archived
8931:Quadrics
8821:(2008).
8792:Archived
8774:Archived
8691:(1997).
8544:(1991).
8523:a priori
8488:(1991).
8444:(1991).
8404:(1991).
8356:See also
8214:geometry
8154:that is
8113:geometry
6197:becomes
5961:becomes
4460:between
4427:‖
4419:‖
4414:‖
4406:‖
4005:formulas
3997:distance
3979:Distance
3640:parabola
2267:gradient
2132:specify
2078:equation
1665:calculus
1349:a series
1347:Part of
1306:a priori
1244:discrete
1224:rocketry
1220:aviation
1199:using a
1197:geometry
1089:Poincaré
1033:Minggatu
992:Yang Hui
962:Virasena
850:Yang Hui
845:Virasena
815:Poincaré
795:Minggatu
575:Cylinder
520:Diameter
479:Rhomboid
436:Altitude
427:Triangle
321:Symmetry
299:Parallel
284:Diagonal
254:Features
251:Concepts
142:Analytic
103:Elliptic
85:Branches
71:Timeline
30:Geometry
9820:Commons
9602:Applied
9572:General
9349:Algebra
9274:History
9203:2305740
9080:(1982)
9046:(1885)
9030:, AMS,
8948:, from
8626:, p. 92
8611:3217882
8572:Algebra
8556:241â242
8338:surface
8302:is the
8232:is the
8222:tangent
8208:Tangent
8158:to the
8144:surface
8142:, to a
4456:is the
4197:is the
4036:,
4029:) and (
4022:,
3764:surface
3758:, is a
3752:quadric
3527:ellipse
3267:In the
3168:where:
2285:is the
2275:is the
2261:is the
2254:where:
2161:surface
1821:, with
1788:,
1784:,
1768:,
1262:History
1212:physics
1114:Coxeter
1094:Hilbert
1079:Riemann
1028:Huygens
987:al-Tusi
977:KhayyĂĄm
967:Alhazen
934:1â1400s
835:al-Tusi
820:Riemann
770:KhayyĂĄm
755:Huygens
750:Hilbert
720:Coxeter
680:Alhazen
658:by name
595:Pyramid
474:Rhombus
418:Polygon
370:segment
218:Fractal
201:Digital
186:Complex
67:History
62:Outline
9521:Finite
9377:Linear
9284:Future
9260:Major
9201:
9096:
9068:
9034:
9016:
8973:
8910:
8884:
8829:
8707:
8656:
8609:
8562:
8514:Conics
8504:
8460:
8420:
8295:where
8216:, the
8178:, the
8152:vector
8140:normal
8117:normal
7253:. The
4675:units.
4606:units.
4452:where
4193:where
4166:arctan
3967:planes
3965:, and
3947:sphere
3586:circle
3271:, the
3227:vector
3221:, and
3179:, and
2920:, and
2211:linear
2112:, and
2082:subset
2001:radius
1999:, its
1934:arctan
1661:French
1551:People
1452:Folium
1312:Persia
1293:Conics
1230:, and
1135:Gromov
1130:Atiyah
1109:Veblen
1104:Cartan
1074:Bolyai
1043:Sakabe
1023:Pascal
916:Euclid
906:Manava
840:Veblen
825:Sakabe
800:Pascal
785:Manava
745:Gromov
730:Euclid
715:Cartan
705:Bolyai
695:Atiyah
605:Sphere
568:cuboid
556:Volume
511:Circle
464:Square
382:Length
304:Vertex
208:Convex
191:Finite
132:Affine
47:sphere
9748:lists
9291:Lists
9264:areas
9199:JSTOR
9002:Books
8607:JSTOR
8542:Boyer
8414:94â95
8388:Notes
8342:plane
8336:to a
8300:'
8287:'
8230:point
8226:curve
8176:force
8172:plane
8150:is a
4458:angle
4199:slope
4001:angle
3983:Angle
3963:cones
3774:of a
3772:zeros
3768:locus
3754:, or
3275:of a
3273:graph
2263:slope
2141:curve
2090:locus
2041:angle
2012:angle
1816:angle
1723:point
1719:plane
1669:Latin
1498:Works
1285:, in
1273:Greek
1084:Klein
1064:Gauss
1038:Euler
972:Sijzi
942:Zhang
896:Ahmes
860:Zhang
830:Sijzi
775:Klein
740:Gauss
735:Euler
675:Ahmes
408:Plane
343:Point
279:Curve
274:Angle
51:plane
49:to a
9094:ISBN
9066:ISBN
9032:ISBN
9014:ISBN
8971:ISBN
8908:ISBN
8882:ISBN
8827:ISBN
8705:ISBN
8654:ISBN
8560:ISBN
8502:ISBN
8458:ISBN
8418:ISBN
8168:line
8115:, a
7840:and
7814:and
6344:and
6145:for
5909:for
5717:and
5517:and
5332:and
5252:and
4464:and
4356:and
3999:and
3981:and
3675:>
3558:and
3510:<
2908:and
2195:and
2124:and
2110:line
2070:and
1627:and
1271:The
1246:and
1214:and
1048:Aida
665:Aida
624:Four
563:Cube
530:Area
502:Kite
413:Area
365:Line
9191:doi
9169:doi
9147:doi
9125:doi
8952:at
8874:doi
8701:326
8599:doi
8595:123
8498:156
8454:142
8306:of
8212:In
8111:In
7739:and
7129:and
6243:or
6019:or
5639::
4976:on
4933:cos
4918:sin
4904:to
4866:sin
4851:cos
4840:to
4789:to
4698:to
4629:to
4560:to
4431:cos
3949:),
3770:of
3645:if
3593:if
3532:if
3480:if
3038:not
2529:to
2265:or
1991:In
1973:or
1878:sin
1855:cos
1806:In
1792:).
1191:or
1179:In
887:BCE
375:ray
9850::
9197:,
9187:55
9185:,
9165:58
9163:,
9143:37
9141:,
9119:,
8944:,
8917:,
8880:,
8811:^
8734:^
8713:.
8703:.
8695:.
8662:.
8605:.
8593:.
8568:.
8558:.
8550:.
8533:^
8510:.
8500:.
8492:.
8466:.
8456:.
8448:.
8426:.
8416:.
8408:.
8352:.
8322:.
8279:))
8271:,
8260:=
8245:=
8190:.
8108:.
7539::
7485:2.
6927::
6873:2.
6682::
6457:1.
4980:.
4468:.
3969:.
3961:,
3957:,
3953:,
3927:0.
3786:,
3750:A
3694:;
3529:;
3237:,
3233:,
3217:,
3204:,
3197:,
3175:,
3044::
2916:,
2904:,
2900:,
2772::
2638:0.
2301:).
2293:=
2219::
2048:xy
2019:xy
1833:,
1655:.
1351:on
1258:.
1250:.
1242:,
1238:,
1226:,
1222:,
1207:.
1183:,
45:a
9253:e
9246:t
9239:v
9193::
9171::
9149::
9127::
9121:9
9054:.
8979:.
8876::
8613:.
8601::
8316:n
8308:f
8297:f
8293:)
8291:c
8289:(
8284:f
8277:c
8275:(
8273:f
8269:c
8267:(
8262:c
8258:x
8253:)
8251:x
8249:(
8247:f
8243:y
8164:P
8148:P
8081:y
8061:)
8058:b
8055:,
8052:0
8049:(
8029:b
8009:y
7989:b
7969:b
7966:+
7963:x
7960:m
7957:=
7954:y
7931:x
7911:x
7891:y
7871:y
7848:y
7828:x
7788:.
7784:)
7778:2
7772:3
7761:,
7758:2
7754:/
7750:1
7746:(
7732:)
7726:2
7720:3
7715:+
7709:,
7706:2
7702:/
7698:1
7694:(
7671:.
7666:2
7660:3
7649:=
7646:y
7627:4
7623:/
7619:3
7616:=
7611:2
7607:y
7587:1
7584:=
7579:2
7575:y
7571:+
7566:2
7562:)
7558:2
7554:/
7550:1
7547:(
7527:y
7507:x
7481:/
7477:1
7474:=
7471:x
7452:1
7446:=
7443:x
7440:2
7418:0
7415:=
7410:2
7406:x
7399:1
7396:+
7393:x
7390:2
7382:2
7378:x
7355:x
7335:y
7315:y
7293:2
7289:y
7266:2
7262:y
7241:0
7238:=
7233:2
7229:x
7220:2
7216:)
7212:1
7206:x
7203:(
7178:.
7174:)
7168:2
7162:3
7151:,
7148:2
7144:/
7140:1
7136:(
7122:)
7116:2
7110:3
7105:+
7099:,
7096:2
7092:/
7088:1
7084:(
7061:.
7056:2
7050:3
7039:=
7036:y
7017:4
7013:/
7009:3
7006:=
7001:2
6997:y
6977:1
6974:=
6969:2
6965:y
6961:+
6956:2
6952:)
6948:2
6944:/
6940:1
6937:(
6915:y
6895:x
6869:/
6865:1
6862:=
6859:x
6840:1
6834:=
6831:x
6828:2
6806:1
6803:=
6798:2
6794:x
6787:1
6784:+
6781:1
6778:+
6775:x
6772:2
6764:2
6760:x
6740:1
6737:=
6734:)
6729:2
6725:x
6718:1
6715:(
6712:+
6707:2
6703:)
6699:1
6693:x
6690:(
6670:x
6648:2
6644:y
6621:.
6616:2
6612:x
6605:1
6602:=
6597:2
6593:y
6573:1
6570:=
6565:2
6561:y
6557:+
6552:2
6548:x
6525:y
6505:x
6485:y
6454:=
6449:2
6445:y
6441:+
6436:2
6432:)
6428:1
6422:x
6419:(
6400:1
6397:=
6392:2
6388:y
6384:+
6379:2
6375:x
6352:Q
6332:P
6309:P
6289:)
6286:0
6283:,
6280:0
6277:(
6257:1
6254:=
6251:0
6231:1
6228:=
6223:2
6219:0
6215:+
6210:2
6206:0
6185:P
6165:)
6162:y
6159:,
6156:x
6153:(
6133:)
6130:0
6127:,
6124:0
6121:(
6101:Q
6081:)
6078:0
6075:,
6072:0
6069:(
6049:1
6046:=
6041:2
6037:)
6033:1
6027:(
6007:1
6004:=
5999:2
5995:0
5991:+
5986:2
5982:)
5978:1
5972:0
5969:(
5949:Q
5929:)
5926:y
5923:,
5920:x
5917:(
5897:)
5894:0
5891:,
5888:0
5885:(
5865:)
5862:0
5859:,
5856:0
5853:(
5833:}
5830:1
5827:=
5822:2
5818:y
5814:+
5809:2
5805:)
5801:1
5795:x
5792:(
5788:|
5784:)
5781:y
5778:,
5775:x
5772:(
5769:{
5766:=
5763:Q
5760::
5757:)
5754:0
5751:,
5748:1
5745:(
5725:Q
5705:}
5702:1
5699:=
5694:2
5690:y
5686:+
5681:2
5677:x
5672:|
5668:)
5665:y
5662:,
5659:x
5656:(
5653:{
5650:=
5647:P
5627:)
5624:0
5621:,
5618:0
5615:(
5595:P
5572:)
5569:y
5566:,
5563:x
5560:(
5540:)
5537:y
5534:,
5531:x
5528:(
5525:Q
5505:)
5502:y
5499:,
5496:x
5493:(
5490:P
5456:0
5453:=
5450:1
5442:2
5438:y
5434:+
5429:2
5425:x
5404:y
5401:x
5381:)
5378:y
5375:,
5372:x
5369:(
5366:R
5340:k
5320:h
5300:k
5280:h
5260:h
5240:k
5220:y
5200:a
5180:b
5160:x
5140:a
5120:a
5100:h
5097:+
5094:)
5091:)
5088:k
5082:x
5079:(
5076:b
5073:(
5070:f
5067:a
5064:=
5061:y
5041:)
5038:x
5035:(
5032:f
5029:=
5026:y
5005:x
5001:/
4997:1
4994:=
4991:y
4971:.
4959:A
4939:A
4930:y
4927:+
4924:A
4915:x
4892:y
4872:A
4863:y
4860:+
4857:A
4848:x
4828:x
4805:a
4801:/
4797:y
4777:y
4754:x
4734:b
4714:b
4710:/
4706:x
4686:x
4663:k
4643:k
4637:y
4617:y
4594:h
4574:h
4568:x
4548:x
4524:)
4521:y
4518:,
4515:x
4512:(
4509:R
4466:B
4462:A
4454:Ξ
4440:,
4423:B
4410:A
4397:f
4394:e
4391:d
4385:=
4377:B
4369:A
4358:B
4354:A
4336:,
4329:2
4325:)
4319:1
4315:z
4306:2
4302:z
4298:(
4295:+
4290:2
4286:)
4280:1
4276:y
4267:2
4263:y
4259:(
4256:+
4251:2
4247:)
4241:1
4237:x
4228:2
4224:x
4220:(
4215:=
4212:d
4195:m
4181:,
4178:)
4175:m
4172:(
4163:=
4136:,
4129:2
4125:)
4119:1
4115:y
4106:2
4102:y
4098:(
4095:+
4090:2
4086:)
4080:1
4076:x
4067:2
4063:x
4059:(
4054:=
4051:d
4041:2
4038:y
4034:2
4031:x
4027:1
4024:y
4020:1
4017:x
3924:=
3921:R
3918:+
3913:i
3909:x
3903:i
3899:P
3893:3
3888:1
3885:=
3882:i
3874:+
3869:j
3865:x
3859:j
3856:i
3852:Q
3846:i
3842:x
3836:3
3831:1
3828:=
3825:j
3822:,
3819:i
3798:3
3795:x
3793:,
3791:2
3788:x
3784:1
3781:x
3760:2
3733:.
3717:0
3714:=
3711:C
3708:+
3705:A
3678:0
3672:C
3669:A
3666:4
3658:2
3654:B
3642:;
3626:0
3623:=
3620:C
3617:A
3614:4
3606:2
3602:B
3572:0
3569:=
3566:B
3546:C
3543:=
3540:A
3513:0
3507:C
3504:A
3501:4
3493:2
3489:B
3464:.
3461:C
3458:A
3455:4
3447:2
3443:B
3415:.
3410:5
3405:P
3375:C
3372:,
3369:B
3366:,
3363:A
3355:0
3352:=
3349:F
3346:+
3343:y
3340:E
3337:+
3334:x
3331:D
3328:+
3323:2
3319:y
3315:C
3312:+
3309:y
3306:x
3303:B
3300:+
3295:2
3291:x
3287:A
3239:c
3235:b
3231:a
3229:(
3223:c
3219:b
3215:a
3209:0
3206:z
3202:0
3199:y
3195:0
3192:x
3190:(
3185:t
3181:z
3177:y
3173:x
3156:t
3153:c
3150:+
3145:0
3141:z
3137:=
3134:z
3115:t
3112:b
3109:+
3104:0
3100:y
3096:=
3093:y
3074:t
3071:a
3068:+
3063:0
3059:x
3055:=
3052:x
3015:)
3012:c
3009:,
3006:b
3003:,
3000:a
2997:(
2994:=
2990:n
2968:,
2965:0
2962:=
2959:d
2956:+
2953:z
2950:c
2947:+
2944:y
2941:b
2938:+
2935:x
2932:a
2922:c
2918:b
2914:a
2910:d
2906:c
2902:b
2898:a
2884:.
2881:)
2876:0
2872:z
2868:c
2865:+
2860:0
2856:y
2852:b
2849:+
2844:0
2840:x
2836:a
2833:(
2827:=
2824:d
2816:,
2813:0
2810:=
2807:d
2804:+
2801:z
2798:c
2795:+
2792:y
2789:b
2786:+
2783:x
2780:a
2749:,
2746:0
2743:=
2740:)
2735:0
2731:z
2724:z
2721:(
2718:c
2715:+
2712:)
2707:0
2703:y
2696:y
2693:(
2690:b
2687:+
2684:)
2679:0
2675:x
2668:x
2665:(
2662:a
2635:=
2632:)
2627:0
2622:r
2613:r
2609:(
2602:n
2580:r
2558:n
2537:P
2515:0
2511:P
2489:r
2468:P
2448:)
2445:c
2442:,
2439:b
2436:,
2433:a
2430:(
2427:=
2423:n
2402:)
2397:0
2393:z
2389:,
2384:0
2380:y
2376:,
2371:0
2367:x
2363:(
2360:=
2355:0
2351:P
2328:0
2323:r
2299:x
2297:(
2295:f
2291:y
2283:x
2273:b
2259:m
2242:b
2239:+
2236:x
2233:m
2230:=
2227:y
2181:r
2177:y
2173:x
2157:y
2153:x
2149:x
2145:x
2126:y
2122:x
2118:x
2114:y
2106:y
2102:x
2098:x
2094:y
2056:z
2052:Ï
2044:Ξ
2037:Ï
2015:Ξ
2008:z
2004:r
1997:z
1957:.
1954:)
1951:x
1947:/
1943:y
1940:(
1931:=
1924:,
1917:2
1913:y
1909:+
1904:2
1900:x
1894:=
1891:r
1887:;
1874:r
1871:=
1868:y
1864:,
1851:r
1848:=
1845:x
1835:Ξ
1831:r
1827:x
1823:Ξ
1819:Ξ
1812:r
1790:z
1786:y
1782:x
1770:y
1766:x
1764:(
1758:y
1754:x
1606:e
1599:t
1592:v
1168:e
1161:t
1154:v
295:)
291:(
73:)
69:(
23:.
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