31:
1637:, may be regarded as either the equation of a line or the equation of a point. In general, there is no difference either algebraically or logically between homogeneous coordinates of points and lines. So plane geometry with points as the fundamental elements and plane geometry with lines as the fundamental elements are equivalent except for interpretation. This leads to the concept of
2184:(or barycenter) of a system of three point masses placed at the vertices of a fixed triangle. Points within the triangle are represented by positive masses and points outside the triangle are represented by allowing negative masses. Multiplying the masses in the system by a scalar does not affect the center of mass, so this is a special case of a system of homogeneous coordinates.
141:. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point
836:, are the lines through the origin with the origin removed. The origin does not really play an essential part in the previous discussion so it can be added back in without changing the properties of the projective plane. This produces a variation on the definition, namely the projective plane is defined as the set of lines in
1641:
in projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective geometry and the result will also be a theorem. Analogously, the theory of points in projective 3-space is dual to the theory of planes in projective 3-space, and so on for higher
2359:. The point is then mapped to a plane by finding the point of intersection of that plane and the line. This produces an accurate representation of how a three-dimensional object appears to the eye. In the simplest situation, the center of projection is the origin and points are mapped to the plane
2332:
to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple and efficient processing. By contrast, using
Cartesian coordinates, translations and perspective projection cannot be
1869:
Just as the selection of axes in the
Cartesian coordinate system is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore, it is useful to know how the different systems are related to each other.
1656:
Assigning coordinates to lines in projective 3-space is more complicated since it would seem that a total of 8 coordinates, either the coordinates of two points which lie on the line or two planes whose intersection is the line, are required. A useful method, due to
2571:
2302:, resulting in a different system of homogeneous coordinates with the same triangle of reference. This is, in fact, the most general type of system of homogeneous coordinates for points in the plane if none of the lines is the line at infinity.
348:. As any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates.
475:, emphasizes that the coordinates are to be considered ratios. Square brackets, as in emphasize that multiple sets of coordinates are associated with a single point. Some authors use a combination of colons and square brackets, as in .
2088:
1976:
1175:
80:, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including
111:
then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the
213:. The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.
1011:
2573:
Matrices representing other geometric transformations can be combined with this and each other by matrix multiplication. As a result, any perspective projection of space can be represented as a single matrix.
116:
being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.
2481:
1619:
and may be interpreted as the equation of the point in line-coordinates. In the same way, planes in 3-space may be given sets of four homogeneous coordinates, and so on for higher dimensions.
1603:
are taken as constants then the equation becomes an equation of a set of lines in the space of all lines in the plane. Geometrically it represents the set of lines that pass through the point
177:
for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. In particular,
1324:
397:
Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same non-zero constant.
2250:
are determined exactly, not just up to proportionality. There is a linear relationship between them however, so these coordinates can be made homogeneous by allowing multiples of
2001:
1892:
1411:
1052:
447:
Some authors use different notations for homogeneous coordinates which help distinguish them from
Cartesian coordinates. The use of colons instead of commas, for example (
926:
2238:
with respect to the triangle whose vertices are the pairwise intersections of the lines. Strictly speaking these are not homogeneous, since the values of
2614:
34:
Rational Bézier curve – polynomial curve defined in homogeneous coordinates (blue) and its projection on plane – rational curve (red)
2995:
2869:
2175:
3139:
3095:
3045:
3014:
2938:
2907:
1449:
and it defines the same curve when restricted to the
Euclidean plane. For example, the homogeneous form of the equation of the line
856:
of a line are taken to be homogeneous coordinates of the line. These lines are now interpreted as points in the projective plane.
394:
The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
807:
defines an inclusion from the
Euclidean plane to the projective plane and the complement of the image is the set of points with
1238:
1857:
and can be regarded as the common points of intersection of all circles. This can be generalized to curves of higher order as
3120:
2355:
For example, in perspective projection, a position in space is associated with the line from it to a fixed point called the
483:
The discussion in the preceding section applies analogously to projective spaces other than the plane. So the points on the
1487:
3173:
1854:
1791:
1351:
913:
defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is
569:
101:
69:
51:
1535:
determines a line, the line determined is unchanged if it is multiplied by a non-zero scalar, and at least one of
2876:
uses homogeneous coordinates in its rendering pipeline. Page 2 indicates that OpenGL is a software interface to
554:(a skew field). However, in this case, care must be taken to account for the fact that multiplication may not be
93:
2316:
Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as
3178:
2619:
1858:
532:
875:
Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say
3168:
3149:
576:
536:
2566:{\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}}}
2329:
2321:
2317:
1651:
1202:
1837:
which has two solutions over the complex numbers, giving rise to the points with homogeneous coordinates
523:
gives homogeneous coordinates of points in the classical case of the real projective spaces, however any
2311:
2193:
97:
89:
61:
1771:
914:
859:
Again, this discussion applies analogously to other dimensions. So the projective space of dimension
692:
555:
432:
108:
2605:
524:
337:
as the homogeneous coordinates of the point at infinity corresponding to the direction of the line
240:
65:
2976:
2180:
Möbius's original formulation of homogeneous coordinates specified the position of a point as the
2864:
Shreiner, Dave; Woo, Mason; Neider, Jackie; Davis, Tom; "OpenGL Programming Guide", 4th
Edition,
2639:
2338:
2325:
894:, does not determine a function defined on points as with Cartesian coordinates. But a condition
562:
130:
77:
73:
2930:
2924:
2609:
3137:
2899:
2840:
3116:
3091:
3085:
3041:
3024:
3010:
2991:
2985:
2934:
2903:
2865:
1658:
588:
134:
81:
3035:
2083:{\displaystyle {\begin{pmatrix}X\\Y\\Z\end{pmatrix}}=A{\begin{pmatrix}x\\y\\z\end{pmatrix}}.}
1796:
The homogeneous form for the equation of a circle in the real or complex projective plane is
1774:
is the generalization of this to create homogeneous coordinates of elements of any dimension
3056:
2891:
2349:
1971:{\displaystyle A={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}},}
1564:
138:
126:
113:
3153:
3143:
484:
85:
3067:
2970:
2633:
2166:
are a new system of homogeneous coordinates for the same point of the projective plane.
2181:
1170:{\displaystyle f(x,y,z)=0\iff f(\lambda x,\lambda y,\lambda z)=\lambda ^{k}f(x,y,z)=0.}
550:
Homogeneous coordinates for projective spaces can also be created with elements from a
540:
528:
2959:
1563:
may be taken to be homogeneous coordinates of a line in the projective plane, that is
17:
3162:
3109:
2892:
2877:
2345:
2341:
551:
1819:. The intersection of this curve with the line at infinity can be found by setting
544:
729:
Lines in this space are defined to be sets of solutions of equations of the form
2654:
1979:
520:
39:
2366:, working for the moment in Cartesian coordinates. For a given point in space,
30:
1889:
be homogeneous coordinates of a point in the projective plane. A fixed matrix
1180:
309:
approaches infinity, in other words, as the point moves away from the origin,
2872:, published December 2004. Page 38 and Appendix F (pp. 697-702) Discuss how
2770:, pp. 107–108 (adapted to the plane according to the footnote on p. 108)
787:
so the equation defines a set of points in the projective plane. The mapping
2333:
expressed as matrix multiplications, though other operations can. Modern
1332:
is a polynomial, so it makes sense to extend its domain to triples where
587:
Another definition of the real projective plane can be given in terms of
840:
that pass through the origin and the coordinates of a non-zero element
1006:{\displaystyle f(\lambda x,\lambda y,\lambda z)=\lambda ^{k}f(x,y,z).}
695:
and the projective plane can be defined as the equivalence classes of
2873:
2334:
76:. They have the advantage that the coordinates of points, including
575:
can be defined with homogeneous factors acting on the left and the
107:
If homogeneous coordinates of a point are multiplied by a non-zero
29:
313:
approaches 0 and the homogeneous coordinates of the point become
1492:
The equation of a line in the projective plane may be given as
2653:
Igoe, Kevin; McGrew, David; Salter, Margaret (February 2011).
3058:
Projective
Differential Geometry of Curves and Ruled Surfaces
1853:
in the complex projective plane. These points are called the
539:
uses two homogeneous complex coordinates and is known as the
355:
Any point in the projective plane is represented by a triple
2210:
be three lines in the plane and define a set of coordinates
269:, so the coordinates of a point on the line may be written
189:
is such a system of homogeneous coordinates for the point
3136:
Jules
Bloomenthal and Jon Rokne, Homogeneous coordinates
2462:
and the point it maps to on the plane is represented by
2344:
take advantage of homogeneous coordinates to implement a
863:
can be defined as the set of lines through the origin in
2975:. trans. J.H. Boyd. Werner school book company. p.
499:, not both zero. In this case, the point at infinity is
1661:, creates a set of six coordinates as the determinants
2984:
Cox, David A.; Little, John B.; O'Shea, Donal (2007).
2490:
2382:, the point where the line and the plane intersect is
2049:
2010:
1907:
722:
then these are taken to be homogeneous coordinates of
2484:
2478:, so projection can be represented in matrix form as
2004:
1895:
1354:
1241:
1055:
1015:
If a set of coordinates represents the same point as
929:
153:
on the
Euclidean plane, for any non-zero real number
2655:"Fundamental Elliptic Curve Cryptography Algorithms"
2738:
2594:, Verlag von Johann Ambrosius Barth, Leipzig, 1827.
3108:
2565:
2082:
1970:
1434:can then be thought of as the homogeneous form of
1405:
1318:
1169:
1005:
821:is an equation of a line in the projective plane (
423:is 0 the point represented is a point at infinity.
2972:Elements of Analytical Geometry of Two Dimensions
822:
431:is omitted and does not represent any point. The
205:can be represented in homogeneous coordinates as
823:see definition of a line in the projective plane
718:is one of the elements of the equivalence class
2841:"Viewports and Clipping (Direct3D 9) (Windows)"
1706:from the homogeneous coordinates of two points
137:, and are considered to lie on a new line, the
133:with additional points added, which are called
2894:Mathematics for Computer Graphics Applications
2969:Briot, Charles; Bouquet, Jean Claude (1896).
2266:to represent the same point. More generally,
2106:by a scalar results in the multiplication of
27:Coordinate system used in projective geometry
8:
2306:Use in computer graphics and computer vision
404:is not 0 the point represented is the point
55:
3148:Ching-Kuang Shene, Homogeneous coordinates
3111:Mathematical elements for computer graphics
2230:to these three lines. These are called the
487:may be represented by pairs of coordinates
2791:
1090:
1086:
289:. In homogeneous coordinates this becomes
216:The equation of a line through the origin
2929:. Jones & Bartlett Learning. p.
2485:
2483:
2044:
2005:
2003:
1902:
1894:
1353:
1339:. The process can be reversed by setting
1319:{\displaystyle f(x,y,z)=z^{k}g(x/z,y/z).}
1302:
1288:
1273:
1240:
1131:
1054:
970:
928:
771:depends only on the equivalence class of
435:of the Euclidean plane is represented by
2426:. In homogeneous coordinates, the point
756:are zero. Satisfaction of the condition
3026:An Introduction to Algebraical Geometry
2926:Computer Graphics: Theory into Practice
2751:
2702:
2615:MacTutor History of Mathematics Archive
2583:
1567:as opposed to point coordinates. If in
2803:
2767:
2725:
2713:
2691:
1982:, defines a new system of coordinates
3037:Algebraic Curves and Riemann Surfaces
2827:
2815:
2779:
2755:
2680:
2668:
2176:Barycentric coordinates (mathematics)
503:. Similarly the points in projective
7:
100:. They are also used in fundamental
1778:in a projective space of dimension
1047:for some non-zero value of λ. Then
917:. Specifically, suppose there is a
201:. For example, the Cartesian point
3055:Wilczynski, Ernest Julius (1906).
25:
3007:An Outline of Projective Geometry
2987:Ideals, Varieties, and Algorithms
2898:. Industrial Press Inc. p.
1547:must be non-zero. So the triple
1406:{\displaystyle g(x,y)=f(x,y,1).}
527:may be used, in particular, the
2638:. J. Wiley & Sons. p.
2402:. Dropping the now superfluous
3023:Jones, Alfred Clement (1912).
2961:Introduction to Higher Algebra
2923:McConnell, Jeffrey J. (2006).
2890:Mortenson, Michael E. (1999).
1397:
1379:
1370:
1358:
1310:
1282:
1263:
1245:
1158:
1140:
1121:
1094:
1087:
1077:
1059:
997:
979:
960:
933:
175:set of homogeneous coordinates
96:to be easily represented by a
1:
3072:. Ginn and Co. pp. 27ff.
3040:. AMS Bookstore. p. 13.
2739:Cox, Little & O'Shea 2007
2635:History of Modern Mathematics
2226:as the signed distances from
1826:. This produces the equation
1488:Duality (projective geometry)
1233:, in other words by defining
3090:. Springer. pp. 134ff.
3066:Woods, Frederick S. (1922).
2632:Smith, David Eugene (1906).
2278:can be defined as constants
1865:Change of coordinate systems
1482:Line coordinates and duality
641:to mean there is a non-zero
3087:Mathematics and its History
2964:. Macmillan. pp. 11ff.
1855:circular points at infinity
1792:Circular points at infinity
1591:are taken as variables and
1579: = 0 the letters
1519:are constants. Each triple
591:. For non-zero elements of
507:-space are represented by (
102:elliptic curve cryptography
3195:
2352:with 4-element registers.
2309:
2191:
2173:
1789:
1649:
1485:
543:. Other fields, including
94:projective transformations
3107:Rogers, David F. (1976).
2990:. Springer. p. 357.
2610:"August Ferdinand Möbius"
2592:Der barycentrische Calcul
2590:August Ferdinand Möbius:
2406:coordinate, this becomes
1859:circular algebraic curves
832:The equivalence classes,
243:form this can be written
129:can be thought of as the
57:Der barycentrische Calcul
3084:Stillwell, John (2002).
3005:Garner, Lynn E. (1981),
2620:University of St Andrews
2122:by the same scalar, and
533:complex projective space
511: + 1)-tuples.
2958:Bôcher, Maxime (1907).
2290:times the distances to
2170:Barycentric coordinates
2134:cannot be all 0 unless
1328:The resulting function
1031:then it can be written
577:projective linear group
537:complex projective line
515:Other projective spaces
416:in the Euclidean plane.
373:homogeneous coordinates
52:August Ferdinand Möbius
44:homogeneous coordinates
3034:Miranda, Rick (1995).
2567:
2330:perspective projection
2084:
1972:
1407:
1320:
1203:homogeneous polynomial
1171:
1007:
583:Alternative definition
377:projective coordinates
90:affine transformations
56:
48:projective coordinates
35:
18:Homogeneous coordinate
2568:
2312:Transformation matrix
2232:trilinear coordinates
2194:Trilinear coordinates
2188:Trilinear coordinates
2085:
1973:
1408:
1321:
1201:can be turned into a
1172:
1008:
825:), and is called the
579:acting on the right.
570:projective line over
127:real projective plane
70:Cartesian coordinates
62:system of coordinates
33:
2606:Robertson, Edmund F.
2482:
2357:center of projection
2002:
1893:
1352:
1239:
1053:
927:
693:equivalence relation
379:of the point, where
3174:Projective geometry
2604:O'Connor, John J.;
2150:is nonsingular. So
2146:are all zero since
1652:Plücker coordinates
1646:Plücker coordinates
1622:The same relation,
1229:and multiplying by
589:equivalence classes
535:. For example, the
305:. In the limit, as
239:are not both 0. In
88:, where they allow
66:projective geometry
3142:2021-02-26 at the
2845:msdn.microsoft.com
2741:, pp. 360–362
2671:, pp. 120–122
2563:
2557:
2442:is represented by
2348:efficiently using
2090:Multiplication of
2080:
2071:
2032:
1968:
1959:
1403:
1316:
1167:
1003:
135:points at infinity
78:points at infinity
74:Euclidean geometry
36:
3154:Wolfram MathWorld
3009:, North Holland,
2997:978-0-387-35650-1
2878:graphics hardware
2870:978-0-321-17348-5
2830:, pp. 452 ff
2754:, p. 14 and
2750:For the section:
2737:For the section:
2667:For the section:
2350:vector processors
1772:Plücker embedding
1770:on the line. The
744:where not all of
325:. Thus we define
92:and, in general,
82:computer graphics
54:in his 1827 work
16:(Redirected from
3186:
3126:
3114:
3101:
3073:
3062:
3051:
3030:
3019:
3001:
2980:
2965:
2945:
2944:
2920:
2914:
2913:
2897:
2887:
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2862:
2856:
2855:
2853:
2851:
2837:
2831:
2825:
2819:
2813:
2807:
2801:
2795:
2789:
2783:
2782:, pp. 2, 40
2777:
2771:
2765:
2759:
2748:
2742:
2735:
2729:
2728:, pp. 32–33
2723:
2717:
2716:, pp. 13–14
2711:
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2165:
2121:
2105:
2089:
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2081:
2076:
2075:
2037:
2036:
1998:by the equation
1997:
1977:
1975:
1974:
1969:
1964:
1963:
1888:
1852:
1844:
1836:
1825:
1818:
1769:
1737:
1705:
1636:
1618:
1565:line coordinates
1562:
1534:
1506:
1478:
1463:
1448:
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975:
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912:
893:
855:
827:line at infinity
820:
813:
806:
786:
770:
743:
717:
701:
690:
640:
561:For the general
531:may be used for
502:
498:
479:Other dimensions
474:
438:
430:
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370:
347:
336:
324:
304:
288:
260:
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219:
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139:line at infinity
114:projective space
59:
50:, introduced by
21:
3194:
3193:
3189:
3188:
3187:
3185:
3184:
3183:
3179:1827 in science
3159:
3158:
3144:Wayback Machine
3133:
3123:
3115:. McGraw Hill.
3106:
3098:
3083:
3080:
3078:Further reading
3069:Higher Geometry
3065:
3061:. B.G. Teubner.
3054:
3048:
3033:
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3017:
3004:
2998:
2983:
2968:
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2954:
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2798:
2792:Wilczynski 1906
2790:
2786:
2778:
2774:
2766:
2762:
2749:
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2479:
2463:
2443:
2427:
2407:
2383:
2367:
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2314:
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2196:
2190:
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2151:
2107:
2091:
2070:
2069:
2063:
2062:
2056:
2055:
2045:
2031:
2030:
2024:
2023:
2017:
2016:
2006:
2000:
1999:
1983:
1958:
1957:
1952:
1947:
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1891:
1890:
1874:
1867:
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1838:
1827:
1820:
1797:
1794:
1788:
1786:Circular points
1767:
1760:
1753:
1746:
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1735:
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1127:
1051:
1050:
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1016:
966:
925:
924:
895:
876:
873:
841:
815:
814:. The equation
808:
788:
772:
757:
730:
703:
696:
691:. Then ~ is an
688:
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674:
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646:
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624:
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547:, can be used.
529:complex numbers
517:
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485:projective line
481:
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220:may be written
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131:Euclidean plane
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86:computer vision
28:
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22:
15:
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11:
5:
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3169:Linear algebra
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3132:
3131:External links
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2192:Main article:
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2182:center of mass
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1659:Julius Plücker
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541:Riemann sphere
516:
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425:
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398:
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392:
391:are not all 0.
351:To summarize:
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4:
3:
2:
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3097:0-387-95336-1
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3047:0-8218-0268-2
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2818:, p. 204
2817:
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2809:
2806:, p. 110
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2800:
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2761:
2758:, p. 120
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1995:
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1978:with nonzero
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1415:The equation
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1205:by replacing
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552:division ring
548:
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545:finite fields
542:
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530:
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459:) instead of
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176:
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157:, the triple
156:
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136:
132:
128:
120:
118:
115:
110:
105:
103:
99:
95:
91:
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83:
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71:
67:
63:
58:
53:
49:
45:
41:
32:
19:
3110:
3086:
3068:
3057:
3036:
3029:. Clarendon.
3025:
3006:
2986:
2971:
2960:
2925:
2918:
2893:
2885:
2860:
2848:. Retrieved
2844:
2835:
2823:
2811:
2799:
2794:, p. 50
2787:
2775:
2763:
2752:Miranda 1995
2746:
2733:
2721:
2709:
2703:Miranda 1995
2698:
2687:
2676:
2663:
2648:
2634:
2627:
2613:
2599:
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2437:
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2417:
2413:
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1701:
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1680:
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1671:
1667:
1663:
1655:
1642:dimensions.
1638:
1632:
1628:
1624:
1621:
1614:
1610:
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1600:
1596:
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1588:
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1230:
1226:
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1218:
1214:
1210:
1206:
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1192:
1188:
1184:
1178:
1049:
1042:
1038:
1034:
1026:
1022:
1018:
1014:
923:
918:
908:
904:
900:
896:
889:
885:
881:
877:
874:
864:
860:
858:
851:
847:
843:
837:
833:
831:
826:
816:
809:
802:
798:
794:
790:
782:
778:
774:
766:
762:
758:
753:
749:
745:
739:
735:
731:
728:
723:
719:
713:
709:
705:
697:
683:
676:
669:
662:
655:
648:
642:
633:
626:
619:
612:
605:
598:
592:
586:
571:
565:
560:
549:
521:real numbers
518:
508:
504:
494:
490:
482:
470:
466:
462:
456:
452:
448:
446:
426:
420:
411:
407:
401:
388:
384:
380:
376:
372:
366:
362:
358:
350:
343:
339:
332:
328:
320:
316:
310:
306:
300:
296:
292:
284:
280:
276:
272:
266:
262:
257:
253:
249:
245:
236:
232:
226:
222:
215:
196:
192:
184:
180:
174:
173:is called a
168:
164:
160:
154:
148:
144:
124:
121:Introduction
106:
104:algorithms.
72:are used in
47:
43:
37:
2804:Bôcher 1907
2768:Bôcher 1907
2726:Garner 1981
2714:Bôcher 1907
2692:Garner 1981
2318:translation
2222:of a point
1980:determinant
915:homogeneous
871:Homogeneity
556:commutative
519:The use of
427:The triple
40:mathematics
3163:Categories
3122:0070535272
2952:References
2828:Jones 1912
2816:Jones 1912
2780:Woods 1922
2756:Jones 1912
2681:Woods 1922
2669:Jones 1912
2310:See also:
1197:of degree
1181:polynomial
921:such that
241:parametric
68:, just as
1129:λ
1116:λ
1107:λ
1098:λ
1088:⟺
968:λ
955:λ
946:λ
937:λ
595:, define
437:(0, 0, 1)
429:(0, 0, 0)
371:, called
211:(2, 4, 2)
207:(1, 2, 1)
3140:Archived
2850:10 April
2339:Direct3D
2322:rotation
645:so that
443:Notation
64:used in
60:, are a
2326:scaling
1639:duality
84:and 3D
3119:
3094:
3044:
3013:
2994:
2937:
2906:
2874:OpenGL
2868:
2335:OpenGL
1507:where
700:∖ {0}.
501:(1, 0)
433:origin
261:. Let
231:where
218:(0, 0)
203:(1, 2)
109:scalar
98:matrix
2578:Notes
1847:(1, −
1700:<
1696:(1 ≤
1447:) = 0
1432:) = 0
1346:, or
1221:with
1209:with
911:) = 0
797:) → (
668:) = (
618:) ~ (
525:field
419:When
400:When
3117:ISBN
3092:ISBN
3042:ISBN
3011:ISBN
2992:ISBN
2935:ISBN
2904:ISBN
2866:ISBN
2852:2018
2400:, 1)
2337:and
2328:and
2298:and
2286:and
2274:and
2246:and
2218:and
2198:Let
2142:and
2130:and
1873:Let
1851:, 0)
1845:and
1843:, 0)
1839:(1,
1738:and
1704:≤ 4)
1599:and
1587:and
1543:and
1515:and
1477:= 0.
805:, 1)
752:and
568:, a
563:ring
387:and
335:, 0)
323:, 0)
295:, −
279:, −
265:= 1/
235:and
187:, 1)
125:The
2977:380
2931:120
2900:318
2364:= 1
2234:of
1835:= 0
1824:= 0
1817:= 0
1813:+ c
1811:byz
1809:+ 2
1807:axz
1805:+ 2
1635:= 0
1505:= 0
1464:is
1462:= 0
1344:= 1
1337:= 0
1041:, λ
1037:, λ
819:= 0
812:= 0
769:= 0
742:= 0
702:If
412:Y/Z
408:X/Z
375:or
346:= 0
331:, −
319:, −
256:= −
229:= 0
209:or
46:or
38:In
3165::
2933:.
2902:.
2843:.
2640:53
2618:,
2612:,
2608:,
2474:zw
2472:,
2470:yw
2468:,
2466:xw
2456:,
2454:zw
2452:,
2450:yw
2448:,
2446:xw
2436:,
2432:,
2416:,
2392:,
2376:,
2372:,
2324:,
2320:,
2294:,
2282:,
2270:,
2260:,
2256:,
2242:,
2214:,
2206:,
2202:,
2160:,
2156:,
2138:,
2126:,
2116:,
2112:,
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