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of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an
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1651:{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.}
2101:{\displaystyle S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}},}
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In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an
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316:. For example, doubling distances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scale factor for volume of one half. The basic equation for it is image over preimage.
753:{\displaystyle S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.}
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In the field of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio of any two corresponding lengths in two similar geometric figures is also called a scale.
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In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (
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will retain their direction. A vector that is the sum of two or more non-zero vectors belonging to different eigenspaces will be tilted towards the eigenspace with largest eigenvalue.
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Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor
1801:{\displaystyle S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.}
1333:{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.}
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945:. As a special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a
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along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis
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are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers
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486:{\displaystyle S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.}
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shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a
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264:(scaling about a point). In most cases, the homothetic transformations are non-linear transformations.
206:) is obtained when at least one of the scaling factors is different from the others; a special case is
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When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called
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234:. When the scale factor is a positive number smaller than 1, scaling is sometimes also called
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176:(in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that
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by a factor between the smallest and the largest product of two scale factors, and the
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is usually a decimal which scales, or multiplies, some quantity. In the equation
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contains triangles related to the next iteration by a scale factor of 1/2
2197:{\displaystyle {\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.}
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1665:(uniform scaling) can be accomplished by using this scaling matrix:
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As shown below, the multiplication will give the expected result:
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As shown below, the multiplication will give the expected result:
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Non-uniform scaling is accomplished by multiplication with any
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that enlarges (increases) or shrinks (diminishes) objects by a
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of the matrix are the scale factors, and the corresponding
816:, scaling increases the area of any surface by a factor of
925:, that is, multiplying each coordinate of each point by
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of an object by a factor between the scale factors, the
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2363:{\displaystyle {\begin{cases}x'=mx\\y'=ny\end{cases}}}
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with a separate scale factor for each axis direction.
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214:(in one direction). Non-uniform scaling changes the
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62:. Unsourced material may be challenged and removed.
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2496:{\displaystyle y=nf\left({\frac {x}{m}}\right).}
949:whose entries on the diagonal are all equal to
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328:A scaling can be represented by a scaling
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122:Learn how and when to remove this message
1056:{\displaystyle v_{1},v_{2},\ldots v_{n}}
2879:. Massachusetts Institute of Technology
2864:
2618:, the transformation is vertical; when
2404:{\displaystyle m,n\in \mathbb {R} ^{+}}
1160:), each homogeneous coordinate vector
7:
3353:List of computer graphics algorithms
172:). The result of uniform scaling is
168:that is the same in all directions (
60:adding citations to reliable sources
188:of a building, car, airplane, etc.
2910:The Wolfram Demonstrations Project
25:
2838:Scaling in statistical estimation
2208:Function dilation and contraction
2755:
874:{\displaystyle \mathbb {R} ^{n}}
36:
2823:Scale factor (computer science)
1134:, points are represented using
828:Scaling in arbitrary dimensions
782:the scaling factors are equal (
47:needs additional citations for
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881:, uniform scaling by a factor
1:
3310:3D computer graphics software
2111:which would be equivalent to
1122:Using homogeneous coordinates
775:by the product of all three.
3125:Hidden-surface determination
2771:2D_computer_graphics#Scaling
2415:Therefore, given a function
763:Such a scaling changes the
332:. To scale an object by a
306:constant of proportionality
3390:
2741:, the transformation is a
1138:. To scale an object by a
3374:Transformation (function)
2791:Homothetic transformation
2563:, it is a dilation, when
2290:{\displaystyle P'(x',y')}
1181:projective transformation
262:homothetic transformation
2906:Understanding 3D Scaling
2902:Understanding 2D Scaling
2828:Scale factor (cosmology)
304:, and may be called the
288:is the scale factor for
260:, and a special case of
27:Geometric transformation
3337:Vector graphics editors
3332:Raster graphics editors
2916:Scale Factor Calculator
2781:Dilation (metric space)
2670:, it is a contraction.
2644:it is a dilation, when
2589:, it is a contraction.
1136:homogeneous coordinates
778:The scaling is uniform
71:"Scaling" geometry
3220:Checkerboard rendering
2908:by Roger Germundsson,
2833:Scales of scale models
2806:Scale (disambiguation)
2796:Orthogonal coordinates
2735:
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2663:{\displaystyle n<1}
2638:
2637:{\displaystyle n>1}
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2582:{\displaystyle m<1}
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2556:{\displaystyle m>1}
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2443:{\displaystyle y=f(x)}
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2297:through the equations
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2240:{\displaystyle P(x,y)}
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138:Each iteration of the
3175:Affine transformation
3154:Surface triangulation
3098:Anisotropic filtering
2848:Transformation matrix
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2734:{\displaystyle n=1/m}
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2700:{\displaystyle m=1/n}
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1103:{\displaystyle v_{i}}
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903:scalar multiplication
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324:Matrix representation
258:linear transformation
184:, or when creating a
160:linear transformation
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2801:Scalar (mathematics)
2786:Homogeneous function
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56:improve this article
3190:Collision detection
3118:Global illumination
2611:{\displaystyle m=1}
2530:{\displaystyle n=1}
1830:, 1) we would have
1128:projective geometry
901:is accomplished by
852:-dimensional space
208:directional scaling
197:Non-uniform scaling
140:Sierpinski triangle
3240:Scanline rendering
3034:Parallax scrolling
3024:Isometric graphics
2843:Scaling in gravity
2763:Mathematics portal
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985:{\displaystyle vI}
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800:In the case where
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3195:Planar projection
3180:Back-face culling
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2996:Alpha compositing
2957:Computer graphics
2874:"Transformations"
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1132:computer graphics
1076:{\displaystyle i}
962:{\displaystyle v}
938:{\displaystyle v}
918:{\displaystyle v}
894:{\displaystyle v}
845:{\displaystyle n}
156:isotropic scaling
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16:(Redirected from
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3159:Wire-frame model
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2507:Particular cases
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997:symmetric matrix
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191:More general is
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1009:
1008:
971:
970:
951:
950:
947:diagonal matrix
927:
926:
907:
906:
883:
882:
859:
854:
853:
834:
833:
830:
813:
809:
805:
795:
791:
787:
740:
739:
729:
719:
716:
715:
705:
695:
692:
691:
681:
671:
664:
653:
652:
642:
639:
638:
628:
625:
624:
614:
607:
599:
598:
588:
586:
581:
575:
574:
569:
559:
557:
551:
550:
545:
540:
530:
523:
506:
501:
500:
473:
472:
462:
460:
455:
449:
448:
443:
433:
431:
425:
424:
419:
414:
404:
397:
383:
378:
377:
371:
367:
363:
352:
348:
344:
326:
270:
268:Uniform scaling
152:uniform scaling
148:affine geometry
128:
117:
111:
108:
65:
63:
53:
41:
28:
23:
22:
15:
12:
11:
5:
3387:
3385:
3377:
3376:
3366:
3365:
3359:
3358:
3356:
3355:
3349:
3347:
3343:
3342:
3340:
3339:
3334:
3329:
3328:
3327:
3322:
3317:
3306:
3304:
3298:
3297:
3295:
3294:
3289:
3284:
3279:
3274:
3269:
3264:
3259:
3257:Shadow mapping
3254:
3249:
3244:
3243:
3242:
3237:
3232:
3227:
3222:
3217:
3212:
3202:
3197:
3192:
3187:
3182:
3177:
3171:
3169:
3165:
3164:
3162:
3161:
3156:
3151:
3150:
3149:
3139:
3132:
3127:
3122:
3121:
3120:
3110:
3105:
3100:
3095:
3090:
3084:
3079:
3073:
3068:
3062:
3060:
3054:
3053:
3050:
3049:
3047:
3046:
3041:
3036:
3031:
3026:
3020:
3018:
3009:
3008:
3003:
2998:
2992:
2990:
2984:
2983:
2981:
2980:
2975:
2969:
2967:
2961:
2960:
2955:
2953:
2952:
2945:
2938:
2930:
2919:
2918:
2913:
2897:
2896:External links
2894:
2891:
2890:
2863:
2862:
2860:
2857:
2856:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2819:
2818:
2813:
2803:
2798:
2793:
2788:
2783:
2778:
2773:
2767:
2766:
2750:
2747:
2730:
2726:
2722:
2719:
2716:
2696:
2692:
2688:
2685:
2682:
2659:
2656:
2653:
2633:
2630:
2627:
2607:
2604:
2601:
2578:
2575:
2572:
2552:
2549:
2546:
2526:
2523:
2520:
2508:
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2503:
2492:
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2480:
2475:
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2468:
2465:
2462:
2439:
2436:
2433:
2430:
2427:
2424:
2413:
2412:
2398:
2393:
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2275:
2271:
2268:
2264:
2260:
2257:
2236:
2233:
2230:
2227:
2224:
2221:
2212:Given a point
2209:
2206:
2205:
2204:
2193:
2188:
2182:
2179:
2178:
2173:
2169:
2165:
2162:
2161:
2156:
2152:
2148:
2145:
2144:
2139:
2135:
2131:
2128:
2127:
2125:
2109:
2108:
2097:
2092:
2084:
2081:
2076:
2075:
2070:
2066:
2062:
2061:
2056:
2052:
2048:
2047:
2042:
2038:
2034:
2033:
2031:
2026:
2021:
2015:
2012:
2011:
2006:
2002:
1998:
1997:
1992:
1988:
1984:
1983:
1978:
1974:
1970:
1969:
1967:
1960:
1952:
1949:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
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1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1906:
1903:
1900:
1898:
1895:
1893:
1890:
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1884:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1862:
1860:
1855:
1852:
1847:
1843:
1826:
1822:
1818:
1809:
1808:
1797:
1792:
1784:
1781:
1776:
1774:
1771:
1769:
1766:
1764:
1761:
1760:
1757:
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1752:
1749:
1747:
1744:
1742:
1739:
1738:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1716:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
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1694:
1692:
1687:
1682:
1678:
1659:
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1647:
1642:
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1632:
1627:
1623:
1617:
1613:
1609:
1608:
1603:
1599:
1593:
1589:
1585:
1584:
1579:
1575:
1569:
1565:
1561:
1560:
1558:
1553:
1548:
1542:
1539:
1538:
1533:
1529:
1525:
1524:
1519:
1515:
1511:
1510:
1505:
1501:
1497:
1496:
1494:
1487:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1462:
1459:
1456:
1452:
1448:
1444:
1442:
1439:
1437:
1434:
1433:
1430:
1427:
1425:
1422:
1418:
1414:
1410:
1408:
1405:
1404:
1401:
1398:
1396:
1393:
1391:
1388:
1384:
1380:
1376:
1375:
1373:
1368:
1365:
1360:
1356:
1341:
1340:
1329:
1324:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1299:
1296:
1293:
1289:
1285:
1281:
1279:
1276:
1274:
1271:
1270:
1267:
1264:
1262:
1259:
1255:
1251:
1247:
1245:
1242:
1241:
1238:
1235:
1233:
1230:
1228:
1225:
1221:
1217:
1213:
1212:
1210:
1205:
1200:
1196:
1175:
1171:
1167:
1156:
1152:
1148:
1123:
1120:
1097:
1093:
1083:by the factor
1072:
1050:
1046:
1042:
1039:
1034:
1030:
1026:
1021:
1017:
981:
978:
958:
934:
914:
890:
868:
863:
841:
829:
826:
811:
807:
803:
793:
789:
785:
780:if and only if
761:
760:
749:
744:
736:
732:
726:
722:
718:
717:
712:
708:
702:
698:
694:
693:
688:
684:
678:
674:
670:
669:
667:
662:
657:
649:
645:
641:
640:
635:
631:
627:
626:
621:
617:
613:
612:
610:
603:
595:
591:
587:
585:
582:
580:
577:
576:
573:
570:
566:
562:
558:
556:
553:
552:
549:
546:
544:
541:
537:
533:
529:
528:
526:
521:
518:
513:
509:
494:
493:
482:
477:
469:
465:
461:
459:
456:
454:
451:
450:
447:
444:
440:
436:
432:
430:
427:
426:
423:
420:
418:
415:
411:
407:
403:
402:
400:
395:
390:
386:
369:
365:
361:
354:), each point
350:
346:
342:
325:
322:
269:
266:
130:
129:
44:
42:
35:
26:
24:
18:Scaling factor
14:
13:
10:
9:
6:
4:
3:
2:
3386:
3375:
3372:
3371:
3369:
3354:
3351:
3350:
3348:
3344:
3338:
3335:
3333:
3330:
3326:
3323:
3321:
3318:
3316:
3313:
3312:
3311:
3308:
3307:
3305:
3303:
3299:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3262:Shadow volume
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3207:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3172:
3170:
3166:
3160:
3157:
3155:
3152:
3148:
3145:
3144:
3143:
3140:
3137:
3136:Triangle mesh
3133:
3131:
3128:
3126:
3123:
3119:
3116:
3115:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3088:
3085:
3083:
3080:
3078:
3074:
3072:
3069:
3067:
3066:3D projection
3064:
3063:
3061:
3059:
3055:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3021:
3019:
3017:
3013:
3007:
3006:Text-to-image
3004:
3002:
2999:
2997:
2994:
2993:
2991:
2989:
2985:
2979:
2976:
2974:
2971:
2970:
2968:
2966:
2962:
2958:
2951:
2946:
2944:
2939:
2937:
2932:
2931:
2928:
2924:
2921:
2917:
2914:
2911:
2907:
2903:
2900:
2899:
2895:
2875:
2868:
2865:
2858:
2854:
2853:Image scaling
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2817:
2814:
2812:
2811:Scale (ratio)
2809:
2808:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2768:
2764:
2758:
2753:
2748:
2746:
2744:
2728:
2724:
2720:
2717:
2714:
2694:
2690:
2686:
2683:
2680:
2671:
2657:
2654:
2651:
2631:
2628:
2625:
2605:
2602:
2599:
2590:
2576:
2573:
2570:
2550:
2547:
2544:
2524:
2521:
2518:
2506:
2490:
2486:
2481:
2478:
2473:
2469:
2466:
2463:
2460:
2453:
2452:
2451:
2434:
2428:
2425:
2422:
2396:
2386:
2383:
2380:
2377:
2350:
2347:
2344:
2340:
2337:
2329:
2326:
2323:
2319:
2316:
2309:
2300:
2299:
2298:
2280:
2277:
2273:
2269:
2266:
2258:
2255:
2231:
2228:
2225:
2219:
2207:
2191:
2186:
2180:
2171:
2167:
2163:
2154:
2150:
2146:
2137:
2133:
2129:
2123:
2114:
2113:
2112:
2095:
2090:
2082:
2079:
2068:
2064:
2054:
2050:
2040:
2036:
2029:
2024:
2019:
2013:
2004:
2000:
1990:
1986:
1976:
1972:
1965:
1958:
1950:
1947:
1940:
1935:
1930:
1923:
1918:
1913:
1908:
1901:
1896:
1891:
1886:
1879:
1874:
1869:
1864:
1858:
1853:
1850:
1845:
1841:
1833:
1832:
1831:
1829:
1814:
1795:
1790:
1782:
1779:
1772:
1767:
1762:
1755:
1750:
1745:
1740:
1733:
1728:
1723:
1718:
1711:
1706:
1701:
1696:
1690:
1685:
1680:
1676:
1668:
1667:
1666:
1664:
1645:
1640:
1634:
1625:
1621:
1615:
1611:
1601:
1597:
1591:
1587:
1577:
1573:
1567:
1563:
1556:
1551:
1546:
1540:
1531:
1527:
1517:
1513:
1503:
1499:
1492:
1485:
1479:
1474:
1469:
1464:
1457:
1450:
1446:
1440:
1435:
1428:
1423:
1416:
1412:
1406:
1399:
1394:
1389:
1382:
1378:
1371:
1366:
1363:
1358:
1354:
1346:
1345:
1344:
1327:
1322:
1316:
1311:
1306:
1301:
1294:
1287:
1283:
1277:
1272:
1265:
1260:
1253:
1249:
1243:
1236:
1231:
1226:
1219:
1215:
1208:
1203:
1198:
1194:
1186:
1185:
1184:
1182:
1178:
1163:
1159:
1144:
1141:
1137:
1133:
1129:
1121:
1119:
1117:
1111:
1095:
1091:
1070:
1048:
1044:
1040:
1037:
1032:
1028:
1024:
1019:
1015:
1006:
1002:
998:
993:
979:
976:
956:
948:
932:
912:
904:
888:
866:
839:
827:
825:
823:
819:
815:
798:
796:
781:
776:
774:
770:
766:
747:
742:
734:
730:
724:
720:
710:
706:
700:
696:
686:
682:
676:
672:
665:
660:
655:
647:
643:
633:
629:
619:
615:
608:
601:
593:
589:
583:
578:
571:
564:
560:
554:
547:
542:
535:
531:
524:
519:
516:
511:
507:
499:
498:
497:
480:
475:
467:
463:
457:
452:
445:
438:
434:
428:
421:
416:
409:
405:
398:
393:
388:
384:
376:
375:
374:
372:
357:
353:
338:
335:
331:
323:
321:
317:
315:
311:
307:
303:
299:
295:
291:
287:
283:
279:
275:
267:
265:
263:
259:
256:Scaling is a
254:
252:
248:
243:
241:
237:
233:
229:
224:
222:
221:oblique angle
217:
213:
209:
205:
203:
198:
194:
189:
187:
183:
179:
175:
171:
170:isotropically
167:
166:
161:
157:
153:
149:
141:
136:
126:
123:
115:
104:
101:
97:
94:
90:
87:
83:
80:
76:
73: –
72:
68:
67:Find sources:
61:
57:
51:
50:
45:This article
43:
39:
34:
33:
30:
19:
3267:Shear matrix
3251:
3230:Path tracing
3215:Cone tracing
3210:Beam tracing
3130:Polygon mesh
3071:3D rendering
2922:
2920:
2883:12 September
2881:. Retrieved
2877:(PowerPoint)
2867:
2776:Digital zoom
2672:
2591:
2510:
2414:
2211:
2110:
1816:
1812:
1810:
1662:
1660:
1342:
1165:
1161:
1146:
1142:
1125:
1112:
1005:eigenvectors
994:
831:
821:
817:
801:
799:
783:
777:
762:
495:
359:
355:
340:
336:
327:
318:
313:
309:
301:
296:is also the
293:
289:
285:
281:
277:
274:scale factor
273:
271:
255:
244:
239:
235:
231:
227:
225:
211:
207:
200:
196:
192:
190:
165:scale factor
163:
155:
151:
145:
118:
109:
99:
92:
85:
78:
66:
54:Please help
49:verification
46:
29:
3282:Translation
3235:Ray casting
3225:Ray tracing
3103:Cel shading
3077:Image-based
3058:3D graphics
3039:Ray casting
2988:2D graphics
2816:Scale (map)
1001:eigenvalues
298:coefficient
236:contraction
232:enlargement
202:anisotropic
186:scale model
3346:Algorithms
3200:Reflection
1116:eigenspace
251:reflection
247:projection
212:stretching
182:photograph
112:April 2008
82:newspapers
3325:rendering
3315:animation
3205:Rendering
2859:Footnotes
2387:∈
1041:…
969:, namely
240:reduction
178:congruent
3368:Category
3320:modeling
3247:Rotation
3185:Clipping
3168:Concepts
3147:Deferred
3113:Lighting
3093:Aliasing
3087:Unbiased
3082:Spectral
2749:See also
2341:′
2320:′
2281:′
2270:′
2259:′
1183:matrix:
765:diameter
280: =
228:dilation
3252:Scaling
3142:Shading
204:scaling
193:scaling
174:similar
158:) is a
96:scholar
3272:Shader
3044:Skybox
3029:Mode 7
3001:Layers
1140:vector
999:. The
773:volume
334:vector
330:matrix
98:
91:
84:
77:
69:
3292:Voxel
3277:Texel
2978:Pixel
905:with
216:shape
103:JSTOR
89:books
3016:2.5D
2904:and
2885:2008
2655:<
2629:>
2574:<
2548:>
2370:for
769:area
154:(or
75:news
2707:or
2673:If
2592:If
2511:If
1825:, p
1821:, p
1815:= (
1174:, p
1170:, p
1164:= (
1155:, v
1151:, v
1145:= (
1126:In
832:In
814:= k
810:= v
806:= v
792:= v
788:= v
368:, p
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