5817:
5825:
4952:
4852:
4729:
5223:
5370:
5075:
5430:
31:
6474:
2702:
1918:
6005:
2531:
3402:
If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a transformation as a rotation by a certain angle with respect to a certain axis
117:
line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.
2298:
Ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coordinate "1" to every vector, one essentially considers the space to be
5217:
1923:
1922:
1919:
46:. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the dark blue leaf and any of the light blue leaves by a combination of reflection, rotation, scaling, and translation.
5447:
in a given direction, with respect to a line in another direction (not necessarily perpendicular), combined with translation that is not purely in the direction of scaling; taking "scaling" in a generalized sense it includes the cases that the scale factor is zero
1924:
1310:
2087:
2526:
5867:
4617:, the affine transformations are analogous to printing on a sheet of rubber and stretching the sheet's edges parallel to the plane. This transform relocates pixels requiring intensity interpolation to approximate the value of moved pixels, bicubic
4846:
5364:
5069:
1940:. The technique requires that all vectors be augmented with a "1" at the end, and all matrices be augmented with an extra row of zeros at the bottom, an extra column—the translation vector—to the right, and a "1" in the lower right corner. If
1507:
1194:
4547:
1921:
4946:
2349:. A translation within the original space by means of a linear transformation of the higher-dimensional space is then possible (specifically, a shear transformation). The coordinates in the higher-dimensional space are an example of
4723:
2697:{\displaystyle M={\begin{bmatrix}\mathbf {y} _{1}&\cdots &\mathbf {y} _{n+1}\\1&\cdots &1\end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{1}&\cdots &\mathbf {x} _{n+1}\\1&\cdots &1\end{bmatrix}}^{-1}.}
3620:
4103:
2445:, where these new points can lie in a space with any number of dimensions. (Furthermore, the new points need not be distinct from each other and need not form a non-degenerate simplex.) The unique augmented matrix
1907:
1725:
3902:
2299:
mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the additional coordinate is 1. Thus the origin of the original space can be found at
5091:
149:. Unlike a purely linear transformation, an affine transformation need not preserve the origin of the affine space. Thus, every linear transformation is affine, but not every affine transformation is linear.
5714:
1615:
1928:
Affine transformations on the 2D plane can be performed by linear transformations in three dimensions. Translation is done by shearing along over the z axis, and rotation is performed around the z axis.
5402:
4281:
3981:
3449:
2137:
6010:
Transforming the three corner points of the original triangle (in red) gives three new points which form the new triangle (in blue). This transformation skews and translates the original triangle.
4159:
1055:
3068:
3004:
2452:
5409:
4350:
627:
3403:
may give a clearer idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation. However, this depends on application and context.
2940:
2894:
4426:
4621:
is the standard for image transformations in image processing applications. Affine transformations scale, rotate, translate, mirror and shear images as shown in the following examples:
1201:
3941:
2285:
905:
3691:
1966:
3787:
3537:
6000:{\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}\mapsto {\begin{bmatrix}0&1\\2&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}+{\begin{bmatrix}-100\\-100\end{bmatrix}}}
3728:
540:
1920:
2347:
484:
5859:
4012:
804:
4745:
866:
5263:
4374:
4241:
4217:
2388:
Suppose you have three points that define a non-degenerate triangle in a plane, or four points that define a non-degenerate tetrahedron in 3-dimensional space, or generally
5776:
4968:
1843:
1801:
3816:
1371:
3500:
1073:
5802:
4438:
3227:
2848:
2821:
2794:
2767:
2247:
2197:
3284:
5754:
5734:
4868:
4571:
4182:
3748:
3478:
3389:
3339:
3311:
3258:
3110:
1958:
1821:
1779:
4648:
5828:
Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix.
6013:
In fact, all triangles are related to one another by affine transformations. This is also true for all parallelograms, but not for all quadrilaterals.
1745:
This representation of affine transformations is often taken as the definition of an affine transformation (with the choice of origin being implicit).
3545:
1761:
to represent translations. Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by an invertible matrix
5394:
The affine transform preserves parallel lines. However, the stretching and shearing transformations warp shapes, as the following example shows:
1851:
1646:
4020:
3824:
2723:
between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation.
5212:{\displaystyle {\begin{bmatrix}\cos(\theta )&-\sin(\theta )&0\\\sin(\theta )&\cos(\theta )&0\\0&0&1\end{bmatrix}}}
6566:
1628:. Translations are affine transformations and the composition of affine transformations is an affine transformation. For this choice of
5652:
1569:
5378:
The affine transforms are applicable to the registration process where two or more images are aligned (registered). An example of
1753:
As shown above, an affine map is the composition of two functions: a translation and a linear map. Ordinary vector algebra uses
6478:
5417:
This is an example of image warping. However, the affine transformations do not facilitate projection onto a curved surface or
4246:
3946:
3414:
3135:
2160:
2098:
2006:
6955:
6928:
6439:
6371:
6346:
6254:
988:
6971:
6575:
3290:
2368:
any number of affine transformations into one by multiplying the respective matrices. This property is used extensively in
161:
3009:
2945:
737:
These two conditions are satisfied by affine transformations, and express what is precisely meant by the expression that "
4590:
4112:
6997:
6695:
4290:
113:(meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of
6457:
6420:
6402:
6328:
6297:
2899:
2853:
2144:
1305:{\displaystyle rx=m_{c}^{-1}\left(rm_{c}(x)\right),{\text{ for all }}r{\text{ in }}k{\text{ and }}x{\text{ in }}X.}
4382:
6559:
17:
2082:{\displaystyle {\begin{bmatrix}\mathbf {y} \\1\end{bmatrix}}=\left{\begin{bmatrix}\mathbf {x} \\1\end{bmatrix}}}
6818:
6619:
6533:
5433:
A central dilation. The triangles A1B1Z, A1C1Z, and B1C1Z get mapped to A2B2Z, A2C2Z, and B2C2Z, respectively.
2252:
871:
6943:
6933:
6823:
6642:
6495:
3631:
3753:
2521:{\displaystyle {\begin{bmatrix}\mathbf {y} \\1\end{bmatrix}}=M{\begin{bmatrix}\mathbf {x} \\1\end{bmatrix}}}
1936:
and an augmented vector, it is possible to represent both the translation and the linear map using a single
545:
6938:
6803:
6743:
6485:
3506:
3911:
6490:
5449:
3703:
1329:
an origin has been specified. This identification permits points to be viewed as vectors and vice versa.
4841:{\displaystyle {\begin{bmatrix}1&0&v_{x}>0\\0&1&v_{y}=0\\0&0&1\end{bmatrix}}}
6731:
6552:
5359:{\displaystyle {\begin{bmatrix}1&c_{x}=0.5&0\\c_{y}=0&1&0\\0&0&1\end{bmatrix}}}
2302:
421:
192:
5835:
4598:
775:
5467:
5064:{\displaystyle {\begin{bmatrix}c_{x}=2&0&0\\0&c_{y}=1&0\\0&0&1\end{bmatrix}}}
2204:
827:
744:
These conditions are not independent as the second follows from the first. Furthermore, if the field
157:
4355:
4222:
4198:
6992:
5453:
4860:
4614:
1502:{\displaystyle L(c,\lambda )(x)=m_{c}^{-1}\left(\lambda (m_{c}(x))\right)=c+\lambda ({\vec {cx}}).}
165:
130:
67:
5759:
1826:
1784:
5600:
Affine transformations do not respect lengths or angles; they multiply area by a constant factor
3986:
2350:
1189:{\displaystyle x+y=m_{c}^{-1}\left(m_{c}(x)+m_{c}(y)\right),{\text{ for all }}x,y{\text{ in }}X,}
196:
6539:
4542:{\displaystyle f\left(\sum _{i\in I}\lambda _{i}a_{i}\right)=\sum _{i\in I}\lambda _{i}f(a_{i})}
3792:
6950:
6900:
6865:
6843:
6838:
4737:
1325:, but common practice is to denote it by the same symbol and mention that it is a vector space
497:
184:
169:
142:
98:
75:
3485:
2742:
ratios of lengths of parallel line segments: for distinct parallel segments defined by points
6772:
6716:
6244:
5781:
4585:
The word "affine" as a mathematical term is defined in connection with tangents to curves in
3456:
2358:
1937:
1754:
1333:
134:
3126:
6676:
6662:
6606:
6203:
4602:
3392:
3237:
2826:
2799:
2772:
2745:
2365:
2225:
2151:
6513:
6271:
5816:
4941:{\displaystyle {\begin{bmatrix}-1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}
8:
6808:
6736:
6624:
5418:
5401:
4718:{\displaystyle {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}
3263:
3086:
2729:: two or more lines which are parallel, continue to be parallel after the transformation.
2726:
2288:
374:
219:
173:
114:
6870:
6858:
6705:
6700:
6652:
6391:
6360:
5739:
5719:
5444:
5379:
4960:
4639:
4556:
4167:
3733:
3463:
3374:
3354:
3324:
3296:
3243:
3095:
2219:
1943:
1806:
1764:
153:
79:
51:
400:
6920:
6813:
6798:
6614:
6528:
6510:
6502:
6453:
6435:
6416:
6398:
6367:
6342:
6324:
6293:
6250:
3396:
3314:
3117:
2735:
of sets: a convex set continues to be convex after the transformation. Moreover, the
2369:
2211:
102:
6905:
6895:
6777:
6765:
5457:
2215:
1933:
188:
5824:
5408:
6726:
6591:
6583:
6043:
6028:
5487:
5479:
5383:
2373:
2354:
1758:
665:
110:
71:
43:
3232:
The invertible affine transformations (of an affine space onto itself) form the
6875:
6355:
4586:
4284:
3459:
on the vectors (that is, the vectors between points of the space). In symbols,
203:
5382:
is the generation of panoramic images that are the product of multiple images
3615:{\displaystyle {\overrightarrow {f(P)~f(Q)}}=\varphi ({\overrightarrow {PQ}})}
6986:
6880:
6754:
6684:
6316:
6226:
6048:
6033:
5633:
5491:
5473:
4618:
3362:
3342:
3090:
2736:
177:
39:
5513:, and each vertex similarly. Supposing we exclude the degenerate case where
2739:
of the original set are mapped to the extreme points of the transformed set.
6885:
6848:
6833:
6828:
6748:
6689:
6022:
5861:, the transformation shown at left is accomplished using the map given by:
5255:
4193:
3452:
3260:
as subgroup and is itself a subgroup of the general linear group of degree
3233:
2720:
2292:
753:
491:
274:
94:
90:
27:
Geometric transformation that preserves lines but not angles nor the origin
6413:
Differential
Geometry: Cartan's Generalization of Klein's Erlangen Program
4951:
6853:
6721:
6657:
4594:
3318:
3317:. For example, if the affine transformation acts on the plane and if the
487:
125:
is the point set of an affine space, then every affine transformation on
30:
4851:
4728:
2148:. In the general case, when the last row vector is not restricted to be
1902:{\displaystyle \mathbf {y} =f(\mathbf {x} )=A\mathbf {x} +\mathbf {b} .}
1720:{\displaystyle \sigma (x)=T_{\mathbf {w}}\left(L(c,\lambda )(x)\right).}
6038:
4574:
4098:{\displaystyle g\colon (O+{\vec {x}})\mapsto (O'+\varphi ({\vec {x}}))}
3073:
2732:
2424:-dimensional space. Suppose you have corresponding destination points
748:
has at least three elements, the first condition can be simplified to:
404:
296:
5222:
3897:{\displaystyle f\colon (O+{\vec {x}})\mapsto (B+\varphi ({\vec {x}}))}
6518:
5805:
5369:
5074:
647:
389:
106:
6544:
6242:
5501:. Whatever the choices of points, there is an affine transformation
6711:
5628:(reverse orientation), and this may be determined by its effect on
5463:
5429:
3943:
is also chosen, this can be decomposed as an affine transformation
3350:
3349:. A transformation that is both equi-affine and a similarity is an
2377:
3697:
We can interpret this definition in a few other ways, as follows.
3120:. The matrix representation of the inverse transformation is thus
218:) between two (potentially different) affine spaces over the same
6760:
2417:
35:
5709:{\displaystyle f\colon \mathbb {R} \to \mathbb {R} ,\;f(x)=mx+c}
6890:
6647:
6473:
6337:
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999),
6183:
5083:
2364:
The advantage of using homogeneous coordinates is that one can
966:, this function is one-to-one, and so, has an inverse function
195:
of that projective space that leave the hyperplane at infinity
6508:
6910:
6596:
6132:
1610:{\displaystyle {\mathbf {w}}={\overrightarrow {c\sigma (c)}}}
1321:
and formally needs to be distinguished from the affine space
83:
6382:
Elementary
Mathematics from an Advanced Standpoint: Geometry
2291:
under the operation of composition of functions, called the
180:, and compositions of them in any combination and sequence.
6634:
5518:
6144:
6120:
6108:
6096:
6084:
486:
here, as usual, the subtraction of two points denotes the
97:(Euclidean spaces are specific affine spaces), that is, a
3391:
is positive. In the last case this is in 3D the group of
4276:{\displaystyle f\colon {\mathcal {A}}\to {\mathcal {B}}}
3976:{\displaystyle g\colon {\mathcal {A}}\to {\mathcal {B}}}
3444:{\displaystyle f\colon {\mathcal {A}}\to {\mathcal {B}}}
2132:{\displaystyle \mathbf {y} =A\mathbf {x} +\mathbf {b} .}
152:
Examples of affine transformations include translation,
5437:
Affine transformations in two real dimensions include:
3371:
affine transformations: those where the determinant of
1050:{\displaystyle m_{c}^{-1}({\textbf {v}})={\vec {v}}(c)}
6173:
6171:
5970:
5941:
5905:
5876:
5525:. Drawing out a whole grid of parallelograms based on
5486:
To visualise the general affine transformation of the
5272:
5100:
4977:
4877:
4754:
4657:
2617:
2546:
2495:
2461:
2056:
1975:
105:
an affine space onto itself while preserving both the
5870:
5838:
5784:
5762:
5742:
5722:
5655:
5266:
5094:
4971:
4871:
4748:
4651:
4559:
4441:
4385:
4358:
4293:
4249:
4225:
4201:
4170:
4115:
4023:
3989:
3949:
3914:
3827:
3795:
3756:
3736:
3706:
3634:
3548:
3509:
3488:
3466:
3417:
3377:
3327:
3299:
3266:
3246:
3129:
3098:
3012:
2948:
2902:
2856:
2829:
2802:
2775:
2748:
2534:
2455:
2305:
2255:
2228:
2154:
2101:
1969:
1946:
1854:
1829:
1809:
1787:
1767:
1649:
1572:
1374:
1204:
1076:
991:
874:
830:
778:
548:
500:
424:
3063:{\displaystyle {\overrightarrow {f(p_{3})f(p_{4})}}}
2999:{\displaystyle {\overrightarrow {f(p_{1})f(p_{2})}}}
907:
are two interchangeable notations for an element of
6168:
6156:
6072:
6060:
199:, restricted to the complement of that hyperplane.
6505:, R. Fisher, S. Perkins, A. Walker and E. Wolfart.
6390:
6359:
5999:
5853:
5804:, are precisely the affine transformations of the
5796:
5770:
5748:
5728:
5708:
5358:
5211:
5063:
4940:
4840:
4717:
4565:
4541:
4420:
4368:
4344:
4275:
4235:
4211:
4176:
4154:{\displaystyle {\vec {b}}={\overrightarrow {O'B}}}
4153:
4097:
4006:
3975:
3935:
3896:
3810:
3781:
3742:
3722:
3685:
3614:
3531:
3494:
3472:
3443:
3383:
3345:. Such transformations form a subgroup called the
3333:
3305:
3278:
3252:
3221:
3104:
3062:
2998:
2934:
2888:
2842:
2815:
2788:
2761:
2696:
2520:
2341:
2279:
2241:
2191:
2142:The above-mentioned augmented matrix is called an
2131:
2081:
1952:
1901:
1837:
1815:
1795:
1773:
1719:
1609:
1501:
1304:
1188:
1049:
899:
860:
798:
621:
534:
478:
6336:
6189:
6025:– artistic applications of affine transformations
4345:{\displaystyle \{(a_{i},\lambda _{i})\}_{i\in I}}
1734:is the composition of a linear transformation of
6984:
5820:A simple affine transformation on the real plane
1781:and the translation as the addition of a vector
1738:(viewed as a vector space) and a translation of
6450:Geometry of Classical Groups over Finite Fields
6243:Schneider, Philip K.; Eberly, David H. (2003).
5521:, there is a unique such affine transformation
5476:combined with a homothety and a translation, or
4597:attributes the term "affine transformation" to
1730:That is, an arbitrary affine transformation of
1061:into a vector space (with respect to the point
183:Viewing an affine space as the complement of a
89:More generally, an affine transformation is an
5581:respects scalar multiples of vectors based at
2935:{\displaystyle {\overrightarrow {p_{3}p_{4}}}}
2889:{\displaystyle {\overrightarrow {p_{1}p_{2}}}}
1632:, there exists a unique linear transformation
6560:
6429:
6150:
6138:
6126:
6114:
6102:
6090:
490:from the second point to the first one, and "
6366:(New ed.), Cambridge University Press,
6354:
6201:
5482:combined with a homothety and a translation.
5456:, and combined with translation it includes
4421:{\displaystyle \sum _{i\in I}\lambda _{i}=1}
4327:
4294:
4184:consists of a translation and a linear map.
6430:Snapper, Ernst; Troyer, Robert J. (1989) ,
2383:
6567:
6553:
5678:
5841:
5764:
5671:
5663:
4187:
3341:is 1 or −1 then the transformation is an
6287:
5823:
5815:
5644:
5428:
4109:followed by the translation by a vector
2449:that achieves the affine transformation
2280:{\displaystyle \operatorname {GL} (n,K)}
1916:
1535:fixed. It is a linear transformation of
900:{\displaystyle {\vec {v}}={\textbf {v}}}
29:
5632:areas (as defined, for example, by the
4608:
3686:{\displaystyle f(Q)-f(P)=\varphi (Q-P)}
3360:Each of these groups has a subgroup of
2711:
1539:, viewed as a vector space with origin
14:
6985:
6503:Geometric Operations: Affine Transform
6410:
6388:
6315:
6204:"Affine transformations and convexity"
6078:
6066:
3789:, then this means that for any vector
3782:{\displaystyle f(O)\in {\mathcal {B}}}
1057:. These functions can be used to turn
764:By the definition of an affine space,
622:{\displaystyle f(y)-f(x)=f(y')-f(x').}
6574:
6548:
6509:
6379:
6269:
6263:
6246:Geometric Tools for Computer Graphics
6177:
4164:The conclusion is that, intuitively,
3532:{\displaystyle P,Q\in {\mathcal {A}}}
191:, the affine transformations are the
6972:List of computer graphics algorithms
5811:
3936:{\displaystyle O'\in {\mathcal {B}}}
2716:An affine transformation preserves:
2357:, the higher dimensional space is a
6633:
6447:
6273:Introductio in analysin infinitorum
6195:
6162:
5452:) or negative; the latter includes
4591:Introductio in analysin infinitorum
3723:{\displaystyle O\in {\mathcal {A}}}
3481:determines a linear transformation
3113:
2203:(as it can also be used to perform
1912:
1015:
892:
868:. Here we use the convention that
756:, that is, it maps lines to lines.
384:be its associated vector space. An
295:is an affine map if there exists a
24:
4361:
4268:
4258:
4243:, over the same field, a function
4228:
4204:
3968:
3958:
3928:
3774:
3715:
3524:
3503:such that, for any pair of points
3436:
3426:
3076:of weighted collections of points.
206:of an affine transformation is an
25:
7009:
6540:Affine Transformation with MATLAB
6466:
5389:
3455:is a map on the points that acts
2342:{\displaystyle (0,0,\dotsc ,0,1)}
2210:This representation exhibits the
1748:
707:are parallel affine subspaces of
479:{\displaystyle g(y-x)=f(y)-f(x);}
6472:
5854:{\displaystyle \mathbb {R} ^{2}}
5407:
5400:
5368:
5221:
5073:
4950:
4850:
4727:
2641:
2622:
2570:
2551:
2499:
2465:
2201:projective transformation matrix
2122:
2114:
2103:
2060:
2017:
1979:
1892:
1884:
1870:
1856:
1831:
1789:
1671:
1575:
1562:and consider the translation of
1550:be any affine transformation of
824:. We can denote this action by
799:{\displaystyle (x,\mathbf {v} )}
789:
692:-dimensional affine subspace of
418:is well defined by the equation
6290:'Digital Image Processing, 3rd'
6281:
6249:. Morgan Kaufmann. p. 98.
6236:
6220:
6190:Brannan, Esplen & Gray 1999
5424:
3085:As an affine transformation is
2092:is equivalent to the following
1527:is an affine transformation of
861:{\displaystyle {\vec {v}}(x)=y}
6534:Wolfram Demonstrations Project
6341:, Cambridge University Press,
6278:Book II, sect. XVIII, art. 442
5897:
5688:
5682:
5667:
5176:
5170:
5159:
5153:
5135:
5129:
5115:
5109:
4643:(transform to original image)
4536:
4523:
4369:{\displaystyle {\mathcal {A}}}
4323:
4297:
4263:
4236:{\displaystyle {\mathcal {B}}}
4212:{\displaystyle {\mathcal {A}}}
4122:
4092:
4089:
4083:
4074:
4057:
4054:
4051:
4045:
4030:
3993:
3963:
3891:
3888:
3882:
3873:
3861:
3858:
3855:
3849:
3834:
3802:
3766:
3760:
3680:
3668:
3659:
3653:
3644:
3638:
3609:
3591:
3576:
3570:
3561:
3555:
3431:
3406:
3177:
3051:
3038:
3032:
3019:
2987:
2974:
2968:
2955:
2336:
2306:
2274:
2262:
2218:affine transformations as the
1874:
1866:
1757:to represent linear maps, and
1706:
1700:
1697:
1685:
1659:
1653:
1598:
1592:
1493:
1487:
1473:
1453:
1450:
1444:
1431:
1399:
1393:
1390:
1378:
1256:
1250:
1150:
1144:
1128:
1122:
1044:
1038:
1032:
1020:
1010:
881:
849:
843:
837:
793:
779:
613:
602:
593:
582:
573:
567:
558:
552:
470:
464:
455:
449:
440:
428:
13:
1:
6929:3D computer graphics software
6309:
5589:transforms the grid based on
5541:is determined by noting that
2706:
2416:that define a non-degenerate
1343:, we can define the function
1317:This vector space has origin
364:
6744:Hidden-surface determination
6362:Affine Differential Geometry
5771:{\displaystyle \mathbb {R} }
5569:applied to the line segment
5557:applied to the line segment
5367:
5260:
5220:
5088:
4965:
4949:
4865:
4742:
4726:
4645:
2145:affine transformation matrix
1838:{\displaystyle \mathbf {x} }
1796:{\displaystyle \mathbf {b} }
816:there is associated a point
759:
7:
6491:Encyclopedia of Mathematics
6016:
5639:
4849:
4007:{\displaystyle O\mapsto O'}
2353:. If the original space is
10:
7014:
5624:(respect orientation), or
4580:
3811:{\displaystyle {\vec {x}}}
3291:similarity transformations
2205:projective transformations
919:one can define a function
772:, so that, for every pair
373:be an affine space over a
273:the respective associated
257:be two affine spaces with
193:projective transformations
129:can be represented as the
6998:Transformation (function)
6964:
6919:
6786:
6675:
6605:
6582:
6288:Gonzalez, Rafael (2008).
6276:(in Latin). Vol. II.
6151:Snapper & Troyer 1989
6139:Snapper & Troyer 1989
6127:Snapper & Troyer 1989
6115:Snapper & Troyer 1989
6103:Snapper & Troyer 1989
6091:Snapper & Troyer 1989
4613:In their applications to
3080:
637:semiaffine transformation
535:{\displaystyle y-x=y'-x'}
66:, "connected with") is a
6532:by Bernard Vuilleumier,
6270:Euler, Leonhard (1748).
6054:
4615:digital image processing
3495:{\displaystyle \varphi }
3399:and pure translations).
3353:of the plane taken with
3293:form the subgroup where
2384:Example augmented matrix
741:preserves parallelism".
654:onto itself satisfying:
68:geometric transformation
34:An image of a fern-like
6956:Vector graphics editors
6951:Raster graphics editors
6514:"Affine Transformation"
6486:"Affine transformation"
6389:Samuel, Pierre (1988),
5797:{\displaystyle m\neq 0}
2942:is the same as that of
2351:homogeneous coordinates
2199:, the matrix becomes a
1531:which leaves the point
399:onto itself that is an
42:) that exhibits affine
6839:Checkerboard rendering
6448:Wan, Zhe-xian (1993),
6432:Metric Affine Geometry
6415:. New York: Springer.
6411:Sharpe, R. W. (1997).
6380:Klein, Felix (1948) ,
6233:, volume 2, pp. 105–7.
6001:
5855:
5829:
5821:
5798:
5772:
5750:
5730:
5710:
5434:
5360:
5213:
5065:
4942:
4842:
4719:
4567:
4543:
4422:
4370:
4352:of weighted points in
4346:
4277:
4237:
4213:
4188:Alternative definition
4178:
4155:
4099:
4008:
3977:
3937:
3898:
3812:
3783:
3744:
3724:
3687:
3616:
3533:
3496:
3474:
3445:
3385:
3335:
3307:
3280:
3254:
3223:
3222:{\displaystyle \left.}
3106:
3064:
3000:
2936:
2890:
2844:
2817:
2790:
2763:
2698:
2522:
2343:
2281:
2243:
2193:
2133:
2083:
1954:
1929:
1903:
1845:can be represented as
1839:
1817:
1797:
1775:
1721:
1611:
1503:
1306:
1190:
1051:
901:
862:
800:
623:
536:
480:
185:hyperplane at infinity
78:, but not necessarily
47:
6794:Affine transformation
6773:Surface triangulation
6717:Anisotropic filtering
6479:Affine transformation
6358:; Sasaki, S. (1994),
6002:
5856:
5827:
5819:
5799:
5773:
5751:
5731:
5711:
5645:Over the real numbers
5432:
5361:
5214:
5066:
4943:
4843:
4720:
4568:
4544:
4423:
4371:
4347:
4278:
4238:
4214:
4179:
4156:
4100:
4009:
3978:
3938:
3899:
3813:
3784:
3745:
3725:
3688:
3617:
3534:
3497:
3475:
3446:
3393:rigid transformations
3386:
3336:
3313:is a scalar times an
3308:
3281:
3255:
3224:
3114:matrix representation
3107:
3065:
3001:
2937:
2891:
2845:
2843:{\displaystyle p_{4}}
2818:
2816:{\displaystyle p_{3}}
2791:
2789:{\displaystyle p_{2}}
2764:
2762:{\displaystyle p_{1}}
2699:
2523:
2359:real projective space
2344:
2282:
2244:
2242:{\displaystyle K^{n}}
2194:
2192:{\displaystyle \left}
2134:
2084:
1955:
1938:matrix multiplication
1927:
1904:
1840:
1818:
1798:
1776:
1755:matrix multiplication
1722:
1612:
1504:
1334:linear transformation
1307:
1191:
1052:
911:. By fixing a point
902:
863:
801:
624:
537:
481:
403:; this means that a
386:affine transformation
135:linear transformation
56:affine transformation
33:
6481:at Wikimedia Commons
6323:, Berlin: Springer,
5868:
5836:
5782:
5760:
5740:
5720:
5653:
5505:of the plane taking
5264:
5092:
4969:
4869:
4746:
4649:
4627:Transformation name
4609:Image transformation
4557:
4439:
4383:
4356:
4291:
4247:
4223:
4199:
4168:
4113:
4021:
3987:
3947:
3912:
3825:
3793:
3754:
3734:
3704:
3632:
3546:
3507:
3486:
3464:
3415:
3375:
3325:
3297:
3264:
3244:
3238:general linear group
3127:
3096:
3010:
2946:
2900:
2854:
2827:
2800:
2773:
2746:
2712:Properties preserved
2532:
2453:
2303:
2253:
2226:
2152:
2099:
1967:
1944:
1852:
1827:
1807:
1785:
1765:
1647:
1570:
1372:
1202:
1074:
989:
872:
828:
776:
635:is at least two, a
631:If the dimension of
546:
498:
422:
6809:Collision detection
6737:Global illumination
6452:, Chartwell-Bratt,
6397:, Springer-Verlag,
6393:Projective Geometry
6231:Projective Geometry
3279:{\displaystyle n+1}
1823:acting on a vector
1422:
1269: for all
1231:
1163: for all
1106:
1009:
265:the point sets and
212:affine homomorphism
174:hyperbolic rotation
80:Euclidean distances
6859:Scanline rendering
6653:Parallax scrolling
6643:Isometric graphics
6511:Weisstein, Eric W.
6202:Reinhard Schultz.
5997:
5991:
5956:
5930:
5891:
5851:
5830:
5822:
5794:
5768:
5746:
5726:
5706:
5470:and a translation,
5441:pure translations,
5435:
5419:radial distortions
5380:image registration
5356:
5350:
5209:
5203:
5061:
5055:
4938:
4932:
4838:
4832:
4715:
4709:
4563:
4539:
4509:
4465:
4418:
4401:
4366:
4342:
4273:
4233:
4209:
4174:
4151:
4095:
4004:
3973:
3933:
3894:
3808:
3779:
3750:denotes its image
3740:
3720:
3683:
3612:
3529:
3492:
3470:
3441:
3381:
3355:Euclidean distance
3331:
3303:
3276:
3250:
3219:
3210:
3102:
3060:
2996:
2932:
2886:
2840:
2813:
2786:
2759:
2694:
2676:
2605:
2518:
2512:
2478:
2339:
2277:
2239:
2220:semidirect product
2189:
2183:
2129:
2079:
2073:
2045:
1992:
1950:
1930:
1899:
1835:
1813:
1793:
1771:
1717:
1607:
1499:
1405:
1302:
1214:
1186:
1089:
1047:
992:
897:
858:
796:
619:
532:
476:
52:Euclidean geometry
48:
6980:
6979:
6921:Graphics software
6814:Planar projection
6799:Back-face culling
6671:
6670:
6615:Alpha compositing
6576:Computer graphics
6477:Media related to
6441:978-0-486-66108-7
6373:978-0-521-44177-3
6348:978-0-521-59787-6
6256:978-1-55860-594-7
6165:, pp. 19–20.
5812:In plane geometry
5749:{\displaystyle c}
5729:{\displaystyle m}
5593:to that based in
5585:. Geometrically
5415:
5414:
5376:
5375:
4566:{\displaystyle f}
4494:
4450:
4386:
4287:for every family
4283:is an affine map
4177:{\displaystyle f}
4149:
4125:
4086:
4048:
3885:
3852:
3805:
3743:{\displaystyle B}
3607:
3583:
3566:
3473:{\displaystyle f}
3384:{\displaystyle A}
3347:equi-affine group
3343:equiareal mapping
3334:{\displaystyle A}
3315:orthogonal matrix
3306:{\displaystyle A}
3253:{\displaystyle n}
3185:
3180:
3112:appearing in its
3105:{\displaystyle A}
3058:
2994:
2930:
2884:
2370:computer graphics
1953:{\displaystyle A}
1925:
1816:{\displaystyle f}
1774:{\displaystyle A}
1605:
1490:
1294:
1286:
1278:
1270:
1178:
1164:
1035:
1017:
959:
894:
884:
840:
62:(from the Latin,
16:(Redirected from
7005:
6906:Volume rendering
6778:Wire-frame model
6631:
6630:
6569:
6562:
6555:
6546:
6545:
6529:Affine Transform
6524:
6523:
6499:
6476:
6462:
6444:
6426:
6407:
6396:
6385:
6376:
6365:
6351:
6333:
6304:
6303:
6292:. Pearson Hall.
6285:
6279:
6277:
6267:
6261:
6260:
6240:
6234:
6224:
6218:
6217:
6215:
6213:
6208:
6199:
6193:
6187:
6181:
6175:
6166:
6160:
6154:
6148:
6142:
6141:, p. 76,87.
6136:
6130:
6124:
6118:
6112:
6106:
6100:
6094:
6088:
6082:
6076:
6070:
6064:
6006:
6004:
6003:
5998:
5996:
5995:
5961:
5960:
5935:
5934:
5896:
5895:
5860:
5858:
5857:
5852:
5850:
5849:
5844:
5803:
5801:
5800:
5795:
5777:
5775:
5774:
5769:
5767:
5755:
5753:
5752:
5747:
5735:
5733:
5732:
5727:
5715:
5713:
5712:
5707:
5674:
5666:
5490:, take labelled
5466:combined with a
5458:glide reflection
5411:
5404:
5397:
5396:
5372:
5365:
5363:
5362:
5357:
5355:
5354:
5314:
5313:
5289:
5288:
5249:
5247:
5245:
5244:
5241:
5238:
5225:
5218:
5216:
5215:
5210:
5208:
5207:
5077:
5070:
5068:
5067:
5062:
5060:
5059:
5024:
5023:
4989:
4988:
4954:
4947:
4945:
4944:
4939:
4937:
4936:
4854:
4847:
4845:
4844:
4839:
4837:
4836:
4806:
4805:
4776:
4775:
4731:
4724:
4722:
4721:
4716:
4714:
4713:
4624:
4623:
4572:
4570:
4569:
4564:
4553:In other words,
4548:
4546:
4545:
4540:
4535:
4534:
4519:
4518:
4508:
4490:
4486:
4485:
4484:
4475:
4474:
4464:
4427:
4425:
4424:
4419:
4411:
4410:
4400:
4375:
4373:
4372:
4367:
4365:
4364:
4351:
4349:
4348:
4343:
4341:
4340:
4322:
4321:
4309:
4308:
4282:
4280:
4279:
4274:
4272:
4271:
4262:
4261:
4242:
4240:
4239:
4234:
4232:
4231:
4218:
4216:
4215:
4210:
4208:
4207:
4183:
4181:
4180:
4175:
4160:
4158:
4157:
4152:
4150:
4145:
4141:
4132:
4127:
4126:
4118:
4104:
4102:
4101:
4096:
4088:
4087:
4079:
4067:
4050:
4049:
4041:
4013:
4011:
4010:
4005:
4003:
3982:
3980:
3979:
3974:
3972:
3971:
3962:
3961:
3942:
3940:
3939:
3934:
3932:
3931:
3922:
3903:
3901:
3900:
3895:
3887:
3886:
3878:
3854:
3853:
3845:
3817:
3815:
3814:
3809:
3807:
3806:
3798:
3788:
3786:
3785:
3780:
3778:
3777:
3749:
3747:
3746:
3741:
3729:
3727:
3726:
3721:
3719:
3718:
3692:
3690:
3689:
3684:
3621:
3619:
3618:
3613:
3608:
3603:
3595:
3584:
3579:
3564:
3550:
3538:
3536:
3535:
3530:
3528:
3527:
3501:
3499:
3498:
3493:
3479:
3477:
3476:
3471:
3450:
3448:
3447:
3442:
3440:
3439:
3430:
3429:
3397:proper rotations
3390:
3388:
3387:
3382:
3340:
3338:
3337:
3332:
3312:
3310:
3309:
3304:
3285:
3283:
3282:
3277:
3259:
3257:
3256:
3251:
3236:, which has the
3228:
3226:
3225:
3220:
3215:
3211:
3183:
3182:
3181:
3173:
3170:
3169:
3153:
3151:
3150:
3137:
3111:
3109:
3108:
3103:
3069:
3067:
3066:
3061:
3059:
3054:
3050:
3049:
3031:
3030:
3014:
3005:
3003:
3002:
2997:
2995:
2990:
2986:
2985:
2967:
2966:
2950:
2941:
2939:
2938:
2933:
2931:
2926:
2925:
2924:
2915:
2914:
2904:
2895:
2893:
2892:
2887:
2885:
2880:
2879:
2878:
2869:
2868:
2858:
2849:
2847:
2846:
2841:
2839:
2838:
2822:
2820:
2819:
2814:
2812:
2811:
2795:
2793:
2792:
2787:
2785:
2784:
2768:
2766:
2765:
2760:
2758:
2757:
2703:
2701:
2700:
2695:
2690:
2689:
2681:
2680:
2656:
2655:
2644:
2631:
2630:
2625:
2610:
2609:
2585:
2584:
2573:
2560:
2559:
2554:
2527:
2525:
2524:
2519:
2517:
2516:
2502:
2483:
2482:
2468:
2448:
2444:
2432:
2423:
2415:
2403:
2394:
2348:
2346:
2345:
2340:
2286:
2284:
2283:
2278:
2248:
2246:
2245:
2240:
2238:
2237:
2198:
2196:
2195:
2190:
2188:
2184:
2138:
2136:
2135:
2130:
2125:
2117:
2106:
2088:
2086:
2085:
2080:
2078:
2077:
2063:
2050:
2046:
2020:
2014:
2008:
1997:
1996:
1982:
1959:
1957:
1956:
1951:
1934:augmented matrix
1926:
1913:Augmented matrix
1908:
1906:
1905:
1900:
1895:
1887:
1873:
1859:
1844:
1842:
1841:
1836:
1834:
1822:
1820:
1819:
1814:
1803:, an affine map
1802:
1800:
1799:
1794:
1792:
1780:
1778:
1777:
1772:
1741:
1737:
1733:
1726:
1724:
1723:
1718:
1713:
1709:
1676:
1675:
1674:
1639:
1635:
1631:
1627:
1616:
1614:
1613:
1608:
1606:
1601:
1584:
1579:
1578:
1565:
1561:
1557:
1553:
1549:
1542:
1538:
1534:
1530:
1526:
1508:
1506:
1505:
1500:
1492:
1491:
1486:
1478:
1460:
1456:
1443:
1442:
1421:
1413:
1364:
1342:
1338:
1324:
1320:
1311:
1309:
1308:
1303:
1295:
1292:
1287:
1284:
1279:
1276:
1271:
1268:
1263:
1259:
1249:
1248:
1230:
1222:
1195:
1193:
1192:
1187:
1179:
1176:
1165:
1162:
1157:
1153:
1143:
1142:
1121:
1120:
1105:
1097:
1064:
1060:
1056:
1054:
1053:
1048:
1037:
1036:
1028:
1019:
1018:
1008:
1000:
984:
965:
961:
960:
957:
937:
918:
914:
910:
906:
904:
903:
898:
896:
895:
886:
885:
877:
867:
865:
864:
859:
842:
841:
833:
823:
819:
815:
805:
803:
802:
797:
792:
771:
767:
751:
747:
740:
732:
721:
710:
706:
702:
695:
691:
685:
674:
670:
663:
653:
645:
641:
634:
628:
626:
625:
620:
612:
592:
541:
539:
538:
533:
531:
520:
485:
483:
482:
477:
417:
413:
409:
398:
394:
383:
379:
372:
360:
356:
352:
316:
294:
280:
272:
268:
264:
260:
256:
240:
224:
189:projective space
148:
140:
128:
124:
111:affine subspaces
21:
7013:
7012:
7008:
7007:
7006:
7004:
7003:
7002:
6993:Affine geometry
6983:
6982:
6981:
6976:
6960:
6915:
6782:
6727:Fluid animation
6667:
6629:
6601:
6592:Diffusion curve
6584:Vector graphics
6578:
6573:
6484:
6469:
6460:
6442:
6423:
6405:
6374:
6356:Nomizu, Katsumi
6349:
6331:
6312:
6307:
6300:
6286:
6282:
6268:
6264:
6257:
6241:
6237:
6225:
6221:
6211:
6209:
6206:
6200:
6196:
6188:
6184:
6176:
6169:
6161:
6157:
6149:
6145:
6137:
6133:
6125:
6121:
6113:
6109:
6101:
6097:
6089:
6085:
6077:
6073:
6065:
6061:
6057:
6044:Flat (geometry)
6029:Affine geometry
6019:
5990:
5989:
5980:
5979:
5966:
5965:
5955:
5954:
5948:
5947:
5937:
5936:
5929:
5928:
5923:
5917:
5916:
5911:
5901:
5900:
5890:
5889:
5883:
5882:
5872:
5871:
5869:
5866:
5865:
5845:
5840:
5839:
5837:
5834:
5833:
5814:
5783:
5780:
5779:
5763:
5761:
5758:
5757:
5741:
5738:
5737:
5721:
5718:
5717:
5670:
5662:
5654:
5651:
5650:
5647:
5642:
5537:) of any point
5488:Euclidean plane
5480:squeeze mapping
5427:
5392:
5349:
5348:
5343:
5338:
5332:
5331:
5326:
5321:
5309:
5305:
5302:
5301:
5296:
5284:
5280:
5278:
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5261:
5242:
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5228:
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5196:
5191:
5185:
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5162:
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5118:
5096:
5095:
5093:
5090:
5089:
5054:
5053:
5048:
5043:
5037:
5036:
5031:
5019:
5015:
5013:
5007:
5006:
5001:
4996:
4984:
4980:
4973:
4972:
4970:
4967:
4966:
4931:
4930:
4925:
4920:
4914:
4913:
4908:
4903:
4897:
4896:
4891:
4886:
4873:
4872:
4870:
4867:
4866:
4831:
4830:
4825:
4820:
4814:
4813:
4801:
4797:
4795:
4790:
4784:
4783:
4771:
4767:
4765:
4760:
4750:
4749:
4747:
4744:
4743:
4708:
4707:
4702:
4697:
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4685:
4680:
4674:
4673:
4668:
4663:
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4650:
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4611:
4583:
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4476:
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4445:
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4390:
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4380:
4360:
4359:
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4353:
4330:
4326:
4317:
4313:
4304:
4300:
4292:
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4267:
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4224:
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4220:
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4190:
4169:
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4165:
4134:
4133:
4131:
4117:
4116:
4114:
4111:
4110:
4078:
4077:
4060:
4040:
4039:
4022:
4019:
4018:
3996:
3988:
3985:
3984:
3967:
3966:
3957:
3956:
3948:
3945:
3944:
3927:
3926:
3915:
3913:
3910:
3909:
3877:
3876:
3844:
3843:
3826:
3823:
3822:
3797:
3796:
3794:
3791:
3790:
3773:
3772:
3755:
3752:
3751:
3735:
3732:
3731:
3730:is chosen, and
3714:
3713:
3705:
3702:
3701:
3633:
3630:
3629:
3596:
3594:
3551:
3549:
3547:
3544:
3543:
3523:
3522:
3508:
3505:
3504:
3487:
3484:
3483:
3465:
3462:
3461:
3435:
3434:
3425:
3424:
3416:
3413:
3412:
3409:
3376:
3373:
3372:
3326:
3323:
3322:
3298:
3295:
3294:
3265:
3262:
3261:
3245:
3242:
3241:
3209:
3208:
3203:
3198:
3193:
3187:
3186:
3172:
3171:
3162:
3158:
3152:
3143:
3139:
3134:
3130:
3128:
3125:
3124:
3097:
3094:
3093:
3083:
3045:
3041:
3026:
3022:
3015:
3013:
3011:
3008:
3007:
2981:
2977:
2962:
2958:
2951:
2949:
2947:
2944:
2943:
2920:
2916:
2910:
2906:
2905:
2903:
2901:
2898:
2897:
2874:
2870:
2864:
2860:
2859:
2857:
2855:
2852:
2851:
2850:, the ratio of
2834:
2830:
2828:
2825:
2824:
2807:
2803:
2801:
2798:
2797:
2780:
2776:
2774:
2771:
2770:
2753:
2749:
2747:
2744:
2743:
2714:
2709:
2682:
2675:
2674:
2669:
2664:
2658:
2657:
2645:
2640:
2639:
2637:
2632:
2626:
2621:
2620:
2613:
2612:
2611:
2604:
2603:
2598:
2593:
2587:
2586:
2574:
2569:
2568:
2566:
2561:
2555:
2550:
2549:
2542:
2541:
2533:
2530:
2529:
2511:
2510:
2504:
2503:
2498:
2491:
2490:
2477:
2476:
2470:
2469:
2464:
2457:
2456:
2454:
2451:
2450:
2446:
2443:
2434:
2431:
2425:
2421:
2414:
2405:
2402:
2396:
2389:
2386:
2374:computer vision
2304:
2301:
2300:
2254:
2251:
2250:
2233:
2229:
2227:
2224:
2223:
2182:
2181:
2176:
2171:
2166:
2159:
2155:
2153:
2150:
2149:
2121:
2113:
2102:
2100:
2097:
2096:
2072:
2071:
2065:
2064:
2059:
2052:
2051:
2044:
2043:
2038:
2033:
2028:
2022:
2021:
2016:
2013:
2005:
2001:
1991:
1990:
1984:
1983:
1978:
1971:
1970:
1968:
1965:
1964:
1945:
1942:
1941:
1917:
1915:
1891:
1883:
1869:
1855:
1853:
1850:
1849:
1830:
1828:
1825:
1824:
1808:
1805:
1804:
1788:
1786:
1783:
1782:
1766:
1763:
1762:
1759:vector addition
1751:
1739:
1735:
1731:
1681:
1677:
1670:
1669:
1665:
1648:
1645:
1644:
1637:
1633:
1629:
1626:
1618:
1585:
1583:
1574:
1573:
1571:
1568:
1567:
1563:
1559:
1555:
1554:. Pick a point
1551:
1547:
1540:
1536:
1532:
1528:
1513:
1479:
1477:
1476:
1438:
1434:
1427:
1423:
1414:
1409:
1373:
1370:
1369:
1344:
1340:
1336:
1322:
1318:
1291:
1285: and
1283:
1275:
1267:
1244:
1240:
1236:
1232:
1223:
1218:
1203:
1200:
1199:
1175:
1161:
1138:
1134:
1116:
1112:
1111:
1107:
1098:
1093:
1075:
1072:
1071:
1065:) by defining:
1062:
1058:
1027:
1026:
1014:
1013:
1001:
996:
990:
987:
986:
975:
967:
963:
956:
947:
939:
928:
920:
916:
912:
908:
891:
890:
876:
875:
873:
870:
869:
832:
831:
829:
826:
825:
821:
817:
807:
788:
777:
774:
773:
769:
765:
762:
749:
745:
738:
723:
712:
708:
704:
700:
693:
687:
676:
672:
668:
666:affine subspace
659:
651:
643:
639:
632:
605:
585:
547:
544:
543:
524:
513:
499:
496:
495:
423:
420:
419:
415:
411:
407:
396:
392:
381:
377:
370:
367:
358:
354:
326:
318:
307:
299:
282:
278:
277:over the field
270:
266:
262:
258:
242:
226:
222:
146:
138:
126:
122:
70:that preserves
44:self-similarity
40:Barnsley's fern
28:
23:
22:
18:Affine function
15:
12:
11:
5:
7011:
7001:
7000:
6995:
6978:
6977:
6975:
6974:
6968:
6966:
6962:
6961:
6959:
6958:
6953:
6948:
6947:
6946:
6941:
6936:
6925:
6923:
6917:
6916:
6914:
6913:
6908:
6903:
6898:
6893:
6888:
6883:
6878:
6876:Shadow mapping
6873:
6868:
6863:
6862:
6861:
6856:
6851:
6846:
6841:
6836:
6831:
6821:
6816:
6811:
6806:
6801:
6796:
6790:
6788:
6784:
6783:
6781:
6780:
6775:
6770:
6769:
6768:
6758:
6751:
6746:
6741:
6740:
6739:
6729:
6724:
6719:
6714:
6709:
6703:
6698:
6692:
6687:
6681:
6679:
6673:
6672:
6669:
6668:
6666:
6665:
6660:
6655:
6650:
6645:
6639:
6637:
6628:
6627:
6622:
6617:
6611:
6609:
6603:
6602:
6600:
6599:
6594:
6588:
6586:
6580:
6579:
6572:
6571:
6564:
6557:
6549:
6543:
6542:
6537:
6525:
6506:
6500:
6482:
6468:
6467:External links
6465:
6464:
6463:
6458:
6445:
6440:
6427:
6421:
6408:
6403:
6386:
6377:
6372:
6352:
6347:
6334:
6329:
6317:Berger, Marcel
6311:
6308:
6306:
6305:
6298:
6280:
6262:
6255:
6235:
6219:
6194:
6182:
6167:
6155:
6143:
6131:
6119:
6107:
6095:
6083:
6071:
6058:
6056:
6053:
6052:
6051:
6046:
6041:
6036:
6031:
6026:
6018:
6015:
6008:
6007:
5994:
5988:
5985:
5982:
5981:
5978:
5975:
5972:
5971:
5969:
5964:
5959:
5953:
5950:
5949:
5946:
5943:
5942:
5940:
5933:
5927:
5924:
5922:
5919:
5918:
5915:
5912:
5910:
5907:
5906:
5904:
5899:
5894:
5888:
5885:
5884:
5881:
5878:
5877:
5875:
5848:
5843:
5813:
5810:
5793:
5790:
5787:
5766:
5745:
5725:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5681:
5677:
5673:
5669:
5665:
5661:
5658:
5649:The functions
5646:
5643:
5641:
5638:
5620:may either be
5614:
5613:
5492:parallelograms
5484:
5483:
5477:
5471:
5461:
5442:
5426:
5423:
5413:
5412:
5405:
5391:
5390:Affine warping
5388:
5374:
5373:
5366:
5353:
5347:
5344:
5342:
5339:
5337:
5334:
5333:
5330:
5327:
5325:
5322:
5320:
5317:
5312:
5308:
5304:
5303:
5300:
5297:
5295:
5292:
5287:
5283:
5279:
5277:
5274:
5273:
5271:
5259:
5251:
5250:
5219:
5206:
5200:
5197:
5195:
5192:
5190:
5187:
5186:
5183:
5180:
5178:
5175:
5172:
5169:
5166:
5163:
5161:
5158:
5155:
5152:
5149:
5146:
5145:
5142:
5139:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5117:
5114:
5111:
5108:
5105:
5102:
5101:
5099:
5087:
5079:
5078:
5071:
5058:
5052:
5049:
5047:
5044:
5042:
5039:
5038:
5035:
5032:
5030:
5027:
5022:
5018:
5014:
5012:
5009:
5008:
5005:
5002:
5000:
4997:
4995:
4992:
4987:
4983:
4979:
4978:
4976:
4964:
4956:
4955:
4948:
4935:
4929:
4926:
4924:
4921:
4919:
4916:
4915:
4912:
4909:
4907:
4904:
4902:
4899:
4898:
4895:
4892:
4890:
4887:
4885:
4882:
4879:
4878:
4876:
4864:
4856:
4855:
4848:
4835:
4829:
4826:
4824:
4821:
4819:
4816:
4815:
4812:
4809:
4804:
4800:
4796:
4794:
4791:
4789:
4786:
4785:
4782:
4779:
4774:
4770:
4766:
4764:
4761:
4759:
4756:
4755:
4753:
4741:
4733:
4732:
4725:
4712:
4706:
4703:
4701:
4698:
4696:
4693:
4692:
4689:
4686:
4684:
4681:
4679:
4676:
4675:
4672:
4669:
4667:
4664:
4662:
4659:
4658:
4656:
4644:
4635:
4634:
4631:
4630:Affine matrix
4628:
4610:
4607:
4582:
4579:
4562:
4551:
4550:
4538:
4533:
4529:
4525:
4522:
4517:
4513:
4507:
4504:
4501:
4497:
4493:
4489:
4483:
4479:
4473:
4469:
4463:
4460:
4457:
4453:
4448:
4444:
4430:
4429:
4417:
4414:
4409:
4405:
4399:
4396:
4393:
4389:
4363:
4339:
4336:
4333:
4329:
4325:
4320:
4316:
4312:
4307:
4303:
4299:
4296:
4285:if and only if
4270:
4265:
4260:
4255:
4252:
4230:
4206:
4189:
4186:
4173:
4148:
4144:
4140:
4137:
4130:
4124:
4121:
4107:
4106:
4094:
4091:
4085:
4082:
4076:
4073:
4070:
4066:
4063:
4059:
4056:
4053:
4047:
4044:
4038:
4035:
4032:
4029:
4026:
4002:
3999:
3995:
3992:
3970:
3965:
3960:
3955:
3952:
3930:
3925:
3921:
3918:
3906:
3905:
3893:
3890:
3884:
3881:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3851:
3848:
3842:
3839:
3836:
3833:
3830:
3804:
3801:
3776:
3771:
3768:
3765:
3762:
3759:
3739:
3717:
3712:
3709:
3695:
3694:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3623:
3622:
3611:
3606:
3602:
3599:
3593:
3590:
3587:
3582:
3578:
3575:
3572:
3569:
3563:
3560:
3557:
3554:
3526:
3521:
3518:
3515:
3512:
3491:
3469:
3438:
3433:
3428:
3423:
3420:
3411:An affine map
3408:
3405:
3380:
3330:
3302:
3275:
3272:
3269:
3249:
3230:
3229:
3218:
3214:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3188:
3179:
3176:
3168:
3165:
3161:
3157:
3154:
3149:
3146:
3142:
3138:
3136:
3133:
3101:
3082:
3079:
3078:
3077:
3071:
3057:
3053:
3048:
3044:
3040:
3037:
3034:
3029:
3025:
3021:
3018:
2993:
2989:
2984:
2980:
2976:
2973:
2970:
2965:
2961:
2957:
2954:
2929:
2923:
2919:
2913:
2909:
2883:
2877:
2873:
2867:
2863:
2837:
2833:
2810:
2806:
2783:
2779:
2756:
2752:
2740:
2737:extreme points
2730:
2724:
2713:
2710:
2708:
2705:
2693:
2688:
2685:
2679:
2673:
2670:
2668:
2665:
2663:
2660:
2659:
2654:
2651:
2648:
2643:
2638:
2636:
2633:
2629:
2624:
2619:
2618:
2616:
2608:
2602:
2599:
2597:
2594:
2592:
2589:
2588:
2583:
2580:
2577:
2572:
2567:
2565:
2562:
2558:
2553:
2548:
2547:
2545:
2540:
2537:
2515:
2509:
2506:
2505:
2501:
2497:
2496:
2494:
2489:
2486:
2481:
2475:
2472:
2471:
2467:
2463:
2462:
2460:
2438:
2429:
2409:
2400:
2385:
2382:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2236:
2232:
2187:
2180:
2177:
2175:
2172:
2170:
2167:
2165:
2162:
2161:
2158:
2140:
2139:
2128:
2124:
2120:
2116:
2112:
2109:
2105:
2090:
2089:
2076:
2070:
2067:
2066:
2062:
2058:
2057:
2055:
2049:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2023:
2019:
2015:
2012:
2009:
2007:
2004:
2000:
1995:
1989:
1986:
1985:
1981:
1977:
1976:
1974:
1949:
1914:
1911:
1910:
1909:
1898:
1894:
1890:
1886:
1882:
1879:
1876:
1872:
1868:
1865:
1862:
1858:
1833:
1812:
1791:
1770:
1750:
1749:Representation
1747:
1728:
1727:
1716:
1712:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1680:
1673:
1668:
1664:
1661:
1658:
1655:
1652:
1622:
1604:
1600:
1597:
1594:
1591:
1588:
1582:
1577:
1566:by the vector
1510:
1509:
1498:
1495:
1489:
1485:
1482:
1475:
1472:
1469:
1466:
1463:
1459:
1455:
1452:
1449:
1446:
1441:
1437:
1433:
1430:
1426:
1420:
1417:
1412:
1408:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1315:
1314:
1313:
1312:
1301:
1298:
1293: in
1290:
1282:
1277: in
1274:
1266:
1262:
1258:
1255:
1252:
1247:
1243:
1239:
1235:
1229:
1226:
1221:
1217:
1213:
1210:
1207:
1197:
1185:
1182:
1177: in
1174:
1171:
1168:
1160:
1156:
1152:
1149:
1146:
1141:
1137:
1133:
1130:
1127:
1124:
1119:
1115:
1110:
1104:
1101:
1096:
1092:
1088:
1085:
1082:
1079:
1046:
1043:
1040:
1034:
1031:
1025:
1022:
1012:
1007:
1004:
999:
995:
971:
943:
924:
889:
883:
880:
857:
854:
851:
848:
845:
839:
836:
795:
791:
787:
784:
781:
761:
758:
735:
734:
697:
618:
615:
611:
608:
604:
601:
598:
595:
591:
588:
584:
581:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
530:
527:
523:
519:
516:
512:
509:
506:
503:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
366:
363:
322:
303:
216:affine mapping
204:generalization
26:
9:
6:
4:
3:
2:
7010:
6999:
6996:
6994:
6991:
6990:
6988:
6973:
6970:
6969:
6967:
6963:
6957:
6954:
6952:
6949:
6945:
6942:
6940:
6937:
6935:
6932:
6931:
6930:
6927:
6926:
6924:
6922:
6918:
6912:
6909:
6907:
6904:
6902:
6899:
6897:
6894:
6892:
6889:
6887:
6884:
6882:
6881:Shadow volume
6879:
6877:
6874:
6872:
6869:
6867:
6864:
6860:
6857:
6855:
6852:
6850:
6847:
6845:
6842:
6840:
6837:
6835:
6832:
6830:
6827:
6826:
6825:
6822:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6800:
6797:
6795:
6792:
6791:
6789:
6785:
6779:
6776:
6774:
6771:
6767:
6764:
6763:
6762:
6759:
6756:
6755:Triangle mesh
6752:
6750:
6747:
6745:
6742:
6738:
6735:
6734:
6733:
6730:
6728:
6725:
6723:
6720:
6718:
6715:
6713:
6710:
6707:
6704:
6702:
6699:
6697:
6693:
6691:
6688:
6686:
6685:3D projection
6683:
6682:
6680:
6678:
6674:
6664:
6661:
6659:
6656:
6654:
6651:
6649:
6646:
6644:
6641:
6640:
6638:
6636:
6632:
6626:
6625:Text-to-image
6623:
6621:
6618:
6616:
6613:
6612:
6610:
6608:
6604:
6598:
6595:
6593:
6590:
6589:
6587:
6585:
6581:
6577:
6570:
6565:
6563:
6558:
6556:
6551:
6550:
6547:
6541:
6538:
6535:
6531:
6530:
6526:
6521:
6520:
6515:
6512:
6507:
6504:
6501:
6497:
6493:
6492:
6487:
6483:
6480:
6475:
6471:
6470:
6461:
6459:0-86238-326-9
6455:
6451:
6446:
6443:
6437:
6433:
6428:
6424:
6422:0-387-94732-9
6418:
6414:
6409:
6406:
6404:0-387-96752-4
6400:
6395:
6394:
6387:
6383:
6378:
6375:
6369:
6364:
6363:
6357:
6353:
6350:
6344:
6340:
6335:
6332:
6330:3-540-11658-3
6326:
6322:
6318:
6314:
6313:
6301:
6299:9780131687288
6295:
6291:
6284:
6275:
6274:
6266:
6258:
6252:
6248:
6247:
6239:
6232:
6228:
6227:Oswald Veblen
6223:
6205:
6198:
6192:, p. 53.
6191:
6186:
6180:, p. 70.
6179:
6174:
6172:
6164:
6159:
6153:, p. 86.
6152:
6147:
6140:
6135:
6129:, p. 59.
6128:
6123:
6117:, p. 69.
6116:
6111:
6105:, p. 66.
6104:
6099:
6093:, p. 65.
6092:
6087:
6081:, p. 11.
6080:
6075:
6069:, p. 38.
6068:
6063:
6059:
6050:
6049:Bent function
6047:
6045:
6042:
6040:
6037:
6035:
6034:3D projection
6032:
6030:
6027:
6024:
6021:
6020:
6014:
6011:
5992:
5986:
5983:
5976:
5973:
5967:
5962:
5957:
5951:
5944:
5938:
5931:
5925:
5920:
5913:
5908:
5902:
5892:
5886:
5879:
5873:
5864:
5863:
5862:
5846:
5826:
5818:
5809:
5807:
5791:
5788:
5785:
5743:
5723:
5703:
5700:
5697:
5694:
5691:
5685:
5679:
5675:
5659:
5656:
5637:
5636:of vectors).
5635:
5634:cross product
5631:
5627:
5623:
5619:
5611:
5607:
5603:
5602:
5601:
5598:
5596:
5592:
5588:
5584:
5580:
5576:
5572:
5568:
5564:
5560:
5556:
5552:
5548:
5544:
5540:
5536:
5532:
5528:
5524:
5520:
5516:
5512:
5508:
5504:
5500:
5496:
5493:
5489:
5481:
5478:
5475:
5474:shear mapping
5472:
5469:
5465:
5462:
5459:
5455:
5451:
5446:
5443:
5440:
5439:
5438:
5431:
5422:
5420:
5410:
5406:
5403:
5399:
5398:
5395:
5387:
5385:
5381:
5371:
5351:
5345:
5340:
5335:
5328:
5323:
5318:
5315:
5310:
5306:
5298:
5293:
5290:
5285:
5281:
5275:
5269:
5258:
5257:
5253:
5252:
5231:
5224:
5204:
5198:
5193:
5188:
5181:
5173:
5167:
5164:
5156:
5150:
5147:
5140:
5132:
5126:
5123:
5120:
5112:
5106:
5103:
5097:
5086:
5085:
5081:
5080:
5076:
5072:
5056:
5050:
5045:
5040:
5033:
5028:
5025:
5020:
5016:
5010:
5003:
4998:
4993:
4990:
4985:
4981:
4974:
4963:
4962:
4958:
4957:
4953:
4933:
4927:
4922:
4917:
4910:
4905:
4900:
4893:
4888:
4883:
4880:
4874:
4863:
4862:
4858:
4857:
4853:
4833:
4827:
4822:
4817:
4810:
4807:
4802:
4798:
4792:
4787:
4780:
4777:
4772:
4768:
4762:
4757:
4751:
4740:
4739:
4735:
4734:
4730:
4710:
4704:
4699:
4694:
4687:
4682:
4677:
4670:
4665:
4660:
4654:
4642:
4641:
4637:
4636:
4632:
4629:
4626:
4625:
4622:
4620:
4619:interpolation
4616:
4606:
4604:
4600:
4596:
4592:
4588:
4578:
4576:
4560:
4531:
4527:
4520:
4515:
4511:
4505:
4502:
4499:
4495:
4491:
4487:
4481:
4477:
4471:
4467:
4461:
4458:
4455:
4451:
4446:
4442:
4435:
4434:
4433:
4415:
4412:
4407:
4403:
4397:
4394:
4391:
4387:
4379:
4378:
4377:
4337:
4334:
4331:
4318:
4314:
4310:
4305:
4301:
4286:
4253:
4250:
4195:
4194:affine spaces
4185:
4171:
4162:
4146:
4142:
4138:
4135:
4128:
4119:
4080:
4071:
4068:
4064:
4061:
4042:
4036:
4033:
4027:
4024:
4017:
4016:
4015:
4000:
3997:
3990:
3953:
3950:
3923:
3919:
3916:
3908:If an origin
3879:
3870:
3867:
3864:
3846:
3840:
3837:
3831:
3828:
3821:
3820:
3819:
3799:
3769:
3763:
3757:
3737:
3710:
3707:
3700:If an origin
3698:
3677:
3674:
3671:
3665:
3662:
3656:
3650:
3647:
3641:
3635:
3628:
3627:
3626:
3604:
3600:
3597:
3588:
3585:
3580:
3573:
3567:
3558:
3552:
3542:
3541:
3540:
3519:
3516:
3513:
3510:
3502:
3489:
3480:
3467:
3458:
3454:
3453:affine spaces
3421:
3418:
3404:
3400:
3398:
3394:
3378:
3370:
3366:
3364:
3358:
3356:
3352:
3348:
3344:
3328:
3320:
3316:
3300:
3292:
3287:
3273:
3270:
3267:
3247:
3239:
3235:
3216:
3212:
3205:
3200:
3195:
3190:
3174:
3166:
3163:
3159:
3155:
3147:
3144:
3140:
3131:
3123:
3122:
3121:
3119:
3115:
3099:
3092:
3091:square matrix
3088:
3075:
3072:
3055:
3046:
3042:
3035:
3027:
3023:
3016:
2991:
2982:
2978:
2971:
2963:
2959:
2952:
2927:
2921:
2917:
2911:
2907:
2881:
2875:
2871:
2865:
2861:
2835:
2831:
2808:
2804:
2781:
2777:
2754:
2750:
2741:
2738:
2734:
2731:
2728:
2725:
2722:
2719:
2718:
2717:
2704:
2691:
2686:
2683:
2677:
2671:
2666:
2661:
2652:
2649:
2646:
2634:
2627:
2614:
2606:
2600:
2595:
2590:
2581:
2578:
2575:
2563:
2556:
2543:
2538:
2535:
2513:
2507:
2492:
2487:
2484:
2479:
2473:
2458:
2441:
2437:
2428:
2419:
2412:
2408:
2399:
2392:
2381:
2379:
2375:
2371:
2367:
2362:
2360:
2356:
2352:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2296:
2294:
2290:
2287:. This is a
2271:
2268:
2265:
2259:
2256:
2234:
2230:
2221:
2217:
2213:
2208:
2206:
2202:
2185:
2178:
2173:
2168:
2163:
2156:
2147:
2146:
2126:
2118:
2110:
2107:
2095:
2094:
2093:
2074:
2068:
2053:
2047:
2040:
2035:
2030:
2025:
2010:
2002:
1998:
1993:
1987:
1972:
1963:
1962:
1961:
1960:is a matrix,
1947:
1939:
1935:
1896:
1888:
1880:
1877:
1863:
1860:
1848:
1847:
1846:
1810:
1768:
1760:
1756:
1746:
1743:
1714:
1710:
1703:
1694:
1691:
1688:
1682:
1678:
1666:
1662:
1656:
1650:
1643:
1642:
1641:
1625:
1621:
1617:, denoted by
1602:
1595:
1589:
1586:
1580:
1544:
1524:
1520:
1516:
1496:
1483:
1480:
1470:
1467:
1464:
1461:
1457:
1447:
1439:
1435:
1428:
1424:
1418:
1415:
1410:
1406:
1402:
1396:
1387:
1384:
1381:
1375:
1368:
1367:
1366:
1363:
1359:
1355:
1351:
1347:
1335:
1330:
1328:
1299:
1296:
1288:
1280:
1272:
1264:
1260:
1253:
1245:
1241:
1237:
1233:
1227:
1224:
1219:
1215:
1211:
1208:
1205:
1198:
1183:
1180:
1172:
1169:
1166:
1158:
1154:
1147:
1139:
1135:
1131:
1125:
1117:
1113:
1108:
1102:
1099:
1094:
1090:
1086:
1083:
1080:
1077:
1070:
1069:
1068:
1067:
1066:
1041:
1029:
1023:
1005:
1002:
997:
993:
983:
979:
974:
970:
955:
951:
946:
942:
936:
932:
927:
923:
887:
878:
855:
852:
846:
834:
814:
810:
785:
782:
757:
755:
742:
733:are parallel.
730:
726:
719:
715:
698:
690:
683:
679:
667:
664:-dimensional
662:
657:
656:
655:
649:
638:
629:
616:
609:
606:
599:
596:
589:
586:
579:
576:
570:
564:
561:
555:
549:
542:implies that
528:
525:
521:
517:
514:
510:
507:
504:
501:
494:" means that
493:
489:
473:
467:
461:
458:
452:
446:
443:
437:
434:
431:
425:
406:
402:
391:
387:
376:
362:
350:
346:
342:
338:
334:
330:
325:
321:
315:
311:
306:
302:
298:
293:
289:
285:
276:
275:vector spaces
254:
250:
246:
238:
234:
230:
221:
217:
213:
209:
205:
200:
198:
194:
190:
186:
181:
179:
178:shear mapping
175:
171:
167:
163:
159:
155:
150:
144:
136:
132:
119:
116:
112:
108:
104:
100:
96:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
45:
41:
37:
32:
19:
6886:Shear matrix
6849:Path tracing
6834:Cone tracing
6829:Beam tracing
6793:
6749:Polygon mesh
6690:3D rendering
6527:
6517:
6489:
6449:
6431:
6412:
6392:
6381:
6361:
6338:
6320:
6289:
6283:
6272:
6265:
6245:
6238:
6230:
6222:
6210:. Retrieved
6197:
6185:
6158:
6146:
6134:
6122:
6110:
6098:
6086:
6074:
6062:
6023:Anamorphosis
6012:
6009:
5831:
5648:
5629:
5625:
5621:
5617:
5615:
5609:
5605:
5599:
5594:
5590:
5586:
5582:
5578:
5574:
5570:
5566:
5562:
5558:
5554:
5550:
5546:
5542:
5538:
5534:
5530:
5529:, the image
5526:
5522:
5514:
5510:
5506:
5502:
5498:
5494:
5485:
5436:
5425:In the plane
5416:
5393:
5377:
5254:
5229:
5082:
4959:
4859:
4736:
4638:
4612:
4584:
4552:
4431:
4191:
4163:
4108:
3907:
3699:
3696:
3624:
3482:
3460:
3451:between two
3410:
3401:
3368:
3361:
3359:
3346:
3288:
3234:affine group
3231:
3084:
2721:collinearity
2715:
2439:
2435:
2426:
2410:
2406:
2397:
2390:
2387:
2363:
2297:
2293:affine group
2209:
2200:
2143:
2141:
2091:
1931:
1752:
1744:
1729:
1623:
1619:
1545:
1522:
1518:
1514:
1511:
1361:
1357:
1353:
1349:
1345:
1331:
1326:
1316:
981:
977:
972:
968:
953:
949:
944:
940:
934:
930:
925:
921:
812:
808:
763:
754:collineation
743:
736:
728:
724:
717:
713:
688:
681:
677:
660:
636:
630:
492:well-defined
385:
368:
348:
344:
340:
336:
332:
328:
323:
319:
313:
309:
304:
300:
291:
287:
283:
252:
248:
244:
236:
232:
228:
215:
211:
207:
201:
182:
151:
120:
95:affine space
91:automorphism
88:
63:
59:
55:
49:
6901:Translation
6854:Ray casting
6844:Ray tracing
6722:Cel shading
6696:Image-based
6677:3D graphics
6658:Ray casting
6607:2D graphics
6212:27 February
6079:Samuel 1988
6067:Berger 1987
4738:Translation
4595:Felix Klein
4575:barycenters
4376:such that
3983:that sends
3407:Affine maps
3365:-preserving
3363:orientation
3319:determinant
3074:barycenters
2727:parallelism
1640:such that
488:free vector
143:translation
131:composition
76:parallelism
6987:Categories
6965:Algorithms
6819:Reflection
6321:Geometry I
6310:References
6178:Klein 1948
6039:Homography
5608:/ area of
5454:reflection
5450:projection
5386:together.
4861:Reflection
4573:preserves
4192:Given two
3240:of degree
3118:invertible
3087:invertible
2707:Properties
2216:invertible
962:. For any
686:is also a
658:For every
405:linear map
401:affine map
365:Definition
317:such that
297:linear map
208:affine map
166:reflection
162:similarity
6944:rendering
6934:animation
6824:Rendering
6519:MathWorld
6496:EMS Press
6434:, Dover,
5984:−
5974:−
5898:↦
5806:real line
5789:≠
5668:→
5660::
5517:has zero
5468:homothety
5174:θ
5168:
5157:θ
5151:
5133:θ
5127:
5121:−
5113:θ
5107:
4881:−
4512:λ
4503:∈
4496:∑
4468:λ
4459:∈
4452:∑
4404:λ
4395:∈
4388:∑
4335:∈
4315:λ
4264:→
4254::
4147:→
4123:→
4084:→
4072:φ
4055:↦
4046:→
4028::
4014:, namely
3994:↦
3964:→
3954::
3924:∈
3883:→
3871:φ
3859:↦
3850:→
3832::
3803:→
3770:∈
3711:∈
3675:−
3666:φ
3648:−
3605:→
3589:φ
3581:→
3520:∈
3490:φ
3432:→
3422::
3196:…
3178:→
3164:−
3156:−
3145:−
3056:→
2992:→
2928:→
2882:→
2733:convexity
2684:−
2667:⋯
2635:⋯
2596:⋯
2564:⋯
2355:Euclidean
2322:…
2260:
2169:⋯
2031:⋯
1932:Using an
1695:λ
1651:σ
1603:→
1590:σ
1488:→
1471:λ
1429:λ
1416:−
1388:λ
1356:) :
1225:−
1100:−
1033:→
1003:−
985:given by
882:→
838:→
760:Structure
648:bijection
597:−
562:−
522:−
505:−
459:−
435:−
390:bijection
197:invariant
158:homothety
107:dimension
6939:modeling
6866:Rotation
6804:Clipping
6787:Concepts
6766:Deferred
6732:Lighting
6712:Aliasing
6706:Unbiased
6701:Spectral
6339:Geometry
6319:(1987),
6163:Wan 1993
6017:See also
5640:Examples
5626:indirect
5616:A given
5606:A′B′C′D′
5604:area of
5595:A′B′C′D′
5499:A′B′C′D′
5464:rotation
5384:stitched
4640:Identity
4633:Example
4589:'s 1748
4432:we have
4139:′
4065:′
4001:′
3920:′
3457:linearly
3369:positive
3351:isometry
2378:robotics
1332:For any
976: :
929: :
768:acts on
610:′
590:′
529:′
518:′
353:for all
308: :
281:. A map
170:rotation
115:parallel
99:function
60:affinity
6871:Scaling
6761:Shading
6498:, 2001
6384:, Dover
6229:(1918)
5445:scaling
5246:
5234:
4581:History
2433:, ...,
2418:simplex
2404:, ...,
2395:points
2366:combine
2214:of all
711:, then
675:, then
154:scaling
109:of any
64:affinis
36:fractal
6891:Shader
6663:Skybox
6648:Mode 7
6620:Layers
6456:
6438:
6419:
6401:
6370:
6345:
6327:
6296:
6253:
5630:signed
5622:direct
5577:, and
5227:where
5084:Rotate
4599:Möbius
3565:
3184:
3089:, the
3081:Groups
380:, and
225:. Let
141:and a
101:which
93:of an
84:angles
6911:Voxel
6896:Texel
6597:Pixel
6207:(PDF)
6055:Notes
5716:with
5256:Shear
4961:Scale
4603:Gauss
4587:Euler
2289:group
1512:Then
1327:after
752:is a
722:and
650:from
646:is a
410:from
395:from
388:is a
375:field
220:field
187:of a
133:of a
72:lines
54:, an
6635:2.5D
6454:ISBN
6436:ISBN
6417:ISBN
6399:ISBN
6368:ISBN
6343:ISBN
6325:ISBN
6294:ISBN
6251:ISBN
6214:2017
5778:and
5736:and
5610:ABCD
5591:ABCD
5575:A′C′
5563:A′B′
5549:) =
5527:ABCD
5519:area
5515:ABCD
5497:and
5495:ABCD
5248:=30°
4778:>
4601:and
4219:and
3289:The
3006:and
2896:and
2823:and
2769:and
2376:and
2249:and
1546:Let
952:) =
703:and
369:Let
355:x, y
343:) −
335:) =
269:and
261:and
241:and
210:(or
103:maps
82:and
74:and
5987:100
5977:100
5832:In
5756:in
5573:is
5561:is
5509:to
5294:0.5
5165:cos
5148:sin
5124:sin
5104:cos
3625:or
3367:or
3321:of
3116:is
2528:is
2420:in
2393:+ 1
2222:of
2212:set
2207:).
1636:of
1558:in
1365:by
1339:of
1196:and
938:by
915:in
820:in
806:in
699:If
671:of
642:of
414:to
357:in
214:or
145:of
137:on
121:If
58:or
50:In
6989::
6516:.
6494:,
6488:,
6170:^
5808:.
5597:.
5571:AC
5565:,
5559:AB
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5551:A′
5511:A′
5421:.
5232:=
4605:.
4593:.
4577:.
4161:.
3818::
3539::
3357:.
3286:.
2796:,
2442:+1
2413:+1
2380:.
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2361:.
2295:.
2257:GL
1742:.
1543:.
1521:,
1360:→
1352:,
980:→
954:cx
933:→
811:×
361:.
331:−
312:→
290:→
286::
251:,
247:,
235:,
231:,
202:A
176:,
172:,
168:,
164:,
160:,
156:,
86:.
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6694:(
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6561:t
6554:v
6536:.
6522:.
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6302:.
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6216:.
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5968:[
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5945:x
5939:[
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5880:x
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5689:)
5686:x
5683:(
5680:f
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5587:T
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5579:T
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5555:T
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4991:=
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4975:[
4934:]
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4808:=
4803:y
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4752:[
4711:]
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4413:=
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4332:i
4328:}
4324:)
4319:i
4311:,
4306:i
4302:a
4298:(
4295:{
4269:B
4259:A
4251:f
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4205:A
4172:f
4143:B
4136:O
4129:=
4120:b
4105:,
4093:)
4090:)
4081:x
4075:(
4069:+
4062:O
4058:(
4052:)
4043:x
4037:+
4034:O
4031:(
4025:g
3998:O
3991:O
3969:B
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3904:.
3892:)
3889:)
3880:x
3874:(
3868:+
3865:B
3862:(
3856:)
3847:x
3841:+
3838:O
3835:(
3829:f
3800:x
3775:B
3767:)
3764:O
3761:(
3758:f
3738:B
3716:A
3708:O
3693:.
3681:)
3678:P
3672:Q
3669:(
3663:=
3660:)
3657:P
3654:(
3651:f
3645:)
3642:Q
3639:(
3636:f
3610:)
3601:Q
3598:P
3592:(
3586:=
3577:)
3574:Q
3571:(
3568:f
3562:)
3559:P
3556:(
3553:f
3525:A
3517:Q
3514:,
3511:P
3468:f
3437:B
3427:A
3419:f
3395:(
3379:A
3329:A
3301:A
3274:1
3271:+
3268:n
3248:n
3217:.
3213:]
3206:1
3201:0
3191:0
3175:b
3167:1
3160:A
3148:1
3141:A
3132:[
3100:A
3070:.
3052:)
3047:4
3043:p
3039:(
3036:f
3033:)
3028:3
3024:p
3020:(
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2983:2
2979:p
2975:(
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2912:3
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2832:p
2809:3
2805:p
2782:2
2778:p
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2692:.
2687:1
2678:]
2672:1
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2647:n
2642:x
2628:1
2623:x
2615:[
2607:]
2601:1
2591:1
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2576:n
2571:y
2557:1
2552:y
2544:[
2539:=
2536:M
2514:]
2508:1
2500:x
2493:[
2488:M
2485:=
2480:]
2474:1
2466:y
2459:[
2447:M
2440:n
2436:y
2430:1
2427:y
2422:n
2411:n
2407:x
2401:1
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2334:1
2331:,
2328:0
2325:,
2319:,
2316:0
2313:,
2310:0
2307:(
2275:)
2272:K
2269:,
2266:n
2263:(
2235:n
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2186:]
2179:1
2174:0
2164:0
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2127:.
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2119:+
2115:x
2111:A
2108:=
2104:y
2075:]
2069:1
2061:x
2054:[
2048:]
2041:1
2036:0
2026:0
2018:b
2011:A
2003:[
1999:=
1994:]
1988:1
1980:y
1973:[
1948:A
1897:.
1893:b
1889:+
1885:x
1881:A
1878:=
1875:)
1871:x
1867:(
1864:f
1861:=
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1769:A
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1715:.
1711:)
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1701:(
1698:)
1692:,
1689:c
1686:(
1683:L
1679:(
1672:w
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1660:)
1657:x
1654:(
1638:V
1634:λ
1630:c
1624:w
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1593:(
1587:c
1581:=
1576:w
1564:X
1560:X
1556:c
1552:X
1548:σ
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1533:c
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1525:)
1523:λ
1519:c
1517:(
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1497:.
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1440:c
1436:m
1432:(
1425:(
1419:1
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1403:=
1400:)
1397:x
1394:(
1391:)
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1376:L
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1234:(
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1212:=
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1206:r
1184:,
1181:X
1173:y
1170:,
1167:x
1159:,
1155:)
1151:)
1148:y
1145:(
1140:c
1136:m
1132:+
1129:)
1126:x
1123:(
1118:c
1114:m
1109:(
1103:1
1095:c
1091:m
1087:=
1084:y
1081:+
1078:x
1063:c
1059:X
1045:)
1042:c
1039:(
1030:v
1024:=
1021:)
1016:v
1011:(
1006:1
998:c
994:m
982:X
978:V
973:c
969:m
964:c
958:→
950:x
948:(
945:c
941:m
935:V
931:X
926:c
922:m
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913:c
909:V
893:v
888:=
879:v
856:y
853:=
850:)
847:x
844:(
835:v
822:X
818:y
813:V
809:X
794:)
790:v
786:,
783:x
780:(
770:X
766:V
750:f
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729:T
727:(
725:f
720:)
718:S
716:(
714:f
709:X
705:T
701:S
696:.
694:X
689:d
684:)
682:S
680:(
678:f
673:X
669:S
661:d
652:X
644:X
640:f
633:X
617:.
614:)
607:x
603:(
600:f
594:)
587:y
583:(
580:f
577:=
574:)
571:x
568:(
565:f
559:)
556:y
553:(
550:f
526:x
515:y
511:=
508:x
502:y
474:;
471:)
468:x
465:(
462:f
456:)
453:y
450:(
447:f
444:=
441:)
438:x
432:y
429:(
426:g
416:V
412:V
408:g
397:X
393:f
382:V
378:k
371:X
359:X
351:)
349:y
347:(
345:f
341:x
339:(
337:f
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327:(
324:f
320:m
314:W
310:V
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288:X
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279:k
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263:Z
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255:)
253:k
249:W
245:Z
243:(
239:)
237:k
233:V
229:X
227:(
223:k
147:X
139:X
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38:(
20:)
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