2315:
56:
1724:
2428:
1713:
1020:
20:
1240:
447:
2412:
1708:{\displaystyle {\begin{aligned}y&=y_{0}\left(1-{\frac {x-x_{0}}{x_{1}-x_{0}}}\right)+y_{1}\left(1-{\frac {x_{1}-x}{x_{1}-x_{0}}}\right)\\&=y_{0}\left(1-{\frac {x-x_{0}}{x_{1}-x_{0}}}\right)+y_{1}\left({\frac {x-x_{0}}{x_{1}-x_{0}}}\right)\\&=y_{0}\left({\frac {x_{1}-x}{x_{1}-x_{0}}}\right)+y_{1}\left({\frac {x-x_{0}}{x_{1}-x_{0}}}\right)\end{aligned}}}
1015:{\displaystyle {\begin{aligned}y&=y_{0}+(x-x_{0}){\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}\\&={\frac {y_{0}(x_{1}-x_{0})}{x_{1}-x_{0}}}+{\frac {y_{1}(x-x_{0})-y_{0}(x-x_{0})}{x_{1}-x_{0}}}\\&={\frac {y_{1}x-y_{1}x_{0}-y_{0}x+y_{0}x_{0}+y_{0}x_{1}-y_{0}x_{0}}{x_{1}-x_{0}}}\\&={\frac {y_{0}(x_{1}-x)+y_{1}(x-x_{0})}{x_{1}-x_{0}}},\end{aligned}}}
2231:
Linear interpolation has been used since antiquity for filling the gaps in tables. Suppose that one has a table listing the population of some country in 1970, 1980, 1990 and 2000, and that one wanted to estimate the population in 1994. Linear interpolation is an easy way to do this. It is believed
59:
In this geometric visualisation, the value at the green circle multiplied by the horizontal distance between the red and blue circles is equal to the sum of the value at the red circle multiplied by the horizontal distance between the green and blue circles, and the value at the blue circle
2222:
That is, the approximation between two points on a given function gets worse with the second derivative of the function that is approximated. This is intuitively correct as well: the "curvier" the function is, the worse the approximations made with simple linear interpolation become.
2218:
1078:
This formula can also be understood as a weighted average. The weights are inversely related to the distance from the end points to the unknown point; the closer point has more influence than the farther point. Thus, the weights are
2057:
367:
1245:
452:
2072:
1235:
1156:
1898:
1067:
228:
155:
108:
2294:
Lerp operations are built into the hardware of all modern computer graphics processors. They are often used as building blocks for more complex operations: for example, a
1911:
1818:
255:
398:
2380:
Linear interpolation as described here is for data points in one spatial dimension. For two spatial dimensions, the extension of linear interpolation is called
442:
422:
250:
181:
370:
2246:
2737:
2392:
of the spatial coordinates, rather products of linear functions; this is illustrated by the clearly non-linear example of
1843:
2358:
function is insufficient, for example if the process that has produced the data points is known to be smoother than
2648:" is the "alpha value"), and the formula may be extended to blend multiple components of a vector (such as spatial
2476:// This method is monotonic. This form may be used when the hardware has a native fused multiply-add instruction.
2298:
can be accomplished in three lerps. Because this operation is cheap, it's also a good way to implement accurate
1237:, which are normalized distances between the unknown point and each of the end points. Because these sum to 1,
2818:
2828:
2810:
2460:
in the closed unit interval . Signatures between lerp functions are variously implemented in both the forms
2431:
A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom)
2213:{\displaystyle |R_{T}|\leq {\frac {(x_{1}-x_{0})^{2}}{8}}\max _{x_{0}\leq x\leq x_{1}}\left|f''(x)\right|.}
1161:
1082:
2800:
2823:
2805:
2696:
2551:// Precise method, which guarantees v = v1 when t = 1. This method is monotonic only when v0 * v1 < 0.
2473:// Imprecise method, which does not guarantee v = v1 when t = 1, due to floating-point arithmetic error.
2794:
1727:
Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines).
2729:
Science and
Civilisation in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth
2785:
2405:
23:
Given the two red points, the blue line is the linear interpolant between the points, and the value
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2369:
2288:
1026:
187:
114:
67:
2385:
2351:
1790:
2850:
2681:
2423:
values 0, 1, 1, and 0.5 as indicated. Interpolated values in between are represented by colour.
2416:
2396:
in the figure below. Other extensions of linear interpolation can be applied to other kinds of
2393:
2381:
2295:
1830:
1778:
2727:
2755:"A chronology of interpolation: from ancient astronomy to modern signal and image processing"
2686:
2365:
1796:
1072:
2318:
Comparison of linear and bilinear interpolation some 1- and 2-dimensional interpolations.
16:
Method of curve fitting to construct new data points within the range of known data points
8:
2241:
1786:
377:
369:
which can be derived geometrically from the figure on the right. It is a special case of
2314:
2240:(second century BC). A description of linear interpolation can be found in the ancient
2062:
427:
407:
235:
166:
47:
to construct new data points within the range of a discrete set of known data points.
2733:
2264:
55:
44:
2401:
2766:
2701:
2441:
2389:
2339:
dots correspond to the interpolated point and neighbouring samples, respectively.
2263:
The basic operation of linear interpolation between two values is commonly used in
2303:
2233:
2706:
2052:{\displaystyle p(x)=f(x_{0})+{\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{0}).}
2641:
1782:
1723:
2835:
2427:
362:{\displaystyle {\frac {y-y_{0}}{x-x_{0}}}={\frac {y_{1}-y_{0}}{x_{1}-x_{0}}},}
2844:
40:
2789:
2397:
2299:
2754:
2236:(last three centuries BC) and by the Greek astronomer and mathematician
60:
multiplied by the horizontal distance between the green and red circles.
2237:
1905:
19:
2770:
2252:
1829:
Linear interpolation is often used to approximate a value of some
2257:
2069:
has a continuous second derivative, then the error is bounded by
1785:
interpolants between each pair of data points. This results in a
50:
2554:// Lerping between same values might not produce the same value
2411:
1022:
which is the formula for linear interpolation in the interval
252:
along the straight line is given from the equation of slopes
1836:
using two known values of that function at other points. The
2836:
Lerp smoothing is broken - a journey of decay and delta time
2291:
lerps incrementally between the two endpoints of the line."
2445:
2284:
2280:
1824:
1715:
yielding the formula for linear interpolation given above.
2341:
Their heights above the ground correspond to their values.
2388:. Notice, though, that these interpolants are no longer
1789:, with a discontinuous derivative (in general), thus of
163:
is the straight line between these points. For a value
1164:
1085:
2400:
such as triangular and tetrahedral meshes, including
2075:
1914:
1846:
1799:
1243:
1029:
450:
430:
410:
380:
258:
238:
190:
169:
117:
70:
64:
If the two known points are given by the coordinates
2404:. These may be defined as indeed higher-dimensional
2364:, it is common to replace linear interpolation with
2795:
Well-behaved interpolation for numbers and pointers
1071:Outside this interval, the formula is identical to
2267:. In that field's jargon it is sometimes called a
2212:
2051:
1892:
1812:
1707:
1229:
1150:
1061:
1014:
436:
416:
392:
361:
244:
222:
175:
149:
102:
2452:), returning an interpolation between two inputs
2842:
2145:
2435:
2725:
1718:
2732:. Cambridge University Press. pp. 147–.
1731:Linear interpolation on a set of data points
51:Linear interpolation between two known points
2250:(äąťç« ç®—čˇ“), dated from 200 BC to AD 100 and the
2226:
2719:
2752:
2247:The Nine Chapters on the Mathematical Art
2640:This lerp function is commonly used for
2426:
2410:
2313:
1825:Linear interpolation as an approximation
1722:
54:
18:
2306:without having too many table entries.
2843:
1781:, resulting from the concatenation of
1230:{\textstyle 1-(x_{1}-x)/(x_{1}-x_{0})}
1151:{\textstyle 1-(x-x_{0})/(x_{1}-x_{0})}
31:may be found by linear interpolation.
2279:olation). The term can be used as a
1840:of this approximation is defined as
13:
2444:have a "lerp" helper-function (in
14:
2862:
2779:
2726:Joseph Needham (1 January 1959).
1904:denotes the linear interpolation
2672:colour components) in parallel.
1893:{\displaystyle R_{T}=f(x)-p(x),}
424:, which is the unknown value at
2375:
2786:Equations of the Straight Line
2199:
2193:
2129:
2102:
2092:
2077:
2043:
2024:
1993:
1980:
1971:
1958:
1946:
1933:
1924:
1918:
1884:
1878:
1869:
1863:
1224:
1198:
1190:
1171:
1145:
1119:
1111:
1092:
1056:
1030:
974:
955:
939:
920:
704:
685:
669:
650:
603:
577:
497:
478:
217:
191:
144:
118:
97:
71:
1:
2712:
2309:
1062:{\displaystyle (x_{0},x_{1})}
223:{\displaystyle (x_{0},x_{1})}
150:{\displaystyle (x_{1},y_{1})}
103:{\displaystyle (x_{0},y_{0})}
2436:Programming language support
2419:on the unit square with the
7:
2824:Encyclopedia of Mathematics
2819:"Finite-increments formula"
2806:Encyclopedia of Mathematics
2675:
2408:(see second figure below).
2384:, and in three dimensions,
2345:
1719:Interpolation of a data set
10:
2867:
2406:piecewise linear functions
404:Solving this equation for
2287:for the operation. e.g. "
2753:Meijering, Erik (2002),
2697:de Casteljau's algorithm
2692:Polynomial interpolation
2470:
2370:polynomial interpolation
2232:that it was used in the
2227:History and applications
371:polynomial interpolation
2759:Proceedings of the IEEE
2386:trilinear interpolation
2061:It can be proven using
1791:differentiability class
2801:"Linear interpolation"
2682:Bilinear interpolation
2432:
2424:
2417:bilinear interpolation
2394:bilinear interpolation
2382:bilinear interpolation
2342:
2302:with quick lookup for
2296:bilinear interpolation
2214:
2053:
1894:
1814:
1728:
1709:
1231:
1152:
1063:
1016:
438:
418:
394:
363:
246:
224:
177:
151:
104:
61:
32:
2430:
2414:
2317:
2289:Bresenham's algorithm
2215:
2054:
1895:
1815:
1813:{\displaystyle C^{0}}
1726:
1710:
1232:
1153:
1064:
1017:
439:
419:
395:
364:
247:
225:
178:
152:
105:
58:
22:
2687:Spline interpolation
2366:spline interpolation
2256:(2nd century AD) by
2242:Chinese mathematical
2073:
1912:
1844:
1797:
1241:
1162:
1083:
1073:linear extrapolation
1027:
448:
428:
408:
378:
256:
236:
188:
167:
115:
68:
37:linear interpolation
2440:Many libraries and
2368:or, in some cases,
393:{\displaystyle n=1}
2433:
2425:
2343:
2210:
2179:
2049:
1890:
1810:
1729:
1705:
1703:
1227:
1148:
1059:
1012:
1010:
434:
414:
390:
359:
242:
220:
173:
161:linear interpolant
147:
100:
62:
45:linear polynomials
33:
2739:978-0-521-05801-8
2448:known instead as
2442:shading languages
2265:computer graphics
2144:
2142:
2022:
1695:
1627:
1552:
1483:
1400:
1324:
1003:
895:
733:
632:
552:
437:{\displaystyle x}
417:{\displaystyle y}
354:
297:
245:{\displaystyle y}
176:{\displaystyle x}
2858:
2832:
2814:
2773:
2771:10.1109/5.993400
2744:
2743:
2723:
2702:First-order hold
2644:(the parameter "
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
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2576:
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2570:
2567:
2564:
2561:
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2555:
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2534:
2531:
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2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2467:
2463:
2459:
2456:for a parameter
2455:
2422:
2390:linear functions
2363:
2356:
2338:
2334:
2330:
2326:
2322:
2304:smooth functions
2219:
2217:
2216:
2211:
2206:
2202:
2192:
2178:
2177:
2176:
2158:
2157:
2143:
2138:
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2136:
2127:
2126:
2114:
2113:
2100:
2095:
2090:
2089:
2080:
2068:
2058:
2056:
2055:
2050:
2042:
2041:
2023:
2021:
2020:
2019:
2007:
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1996:
1992:
1991:
1970:
1969:
1953:
1945:
1944:
1903:
1899:
1897:
1896:
1891:
1856:
1855:
1835:
1821:
1819:
1817:
1816:
1811:
1809:
1808:
1787:continuous curve
1779:piecewise linear
1776:
1714:
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1711:
1706:
1704:
1700:
1696:
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1693:
1692:
1680:
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1651:
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1489:
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1457:
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1267:
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1236:
1234:
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1228:
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1210:
1209:
1197:
1183:
1182:
1157:
1155:
1154:
1149:
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1143:
1131:
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1118:
1110:
1109:
1070:
1068:
1066:
1065:
1060:
1055:
1054:
1042:
1041:
1021:
1019:
1018:
1013:
1011:
1004:
1002:
1001:
1000:
988:
987:
977:
973:
972:
954:
953:
932:
931:
919:
918:
908:
900:
896:
894:
893:
892:
880:
879:
869:
868:
867:
858:
857:
845:
844:
835:
834:
822:
821:
812:
811:
796:
795:
783:
782:
773:
772:
757:
756:
746:
738:
734:
732:
731:
730:
718:
717:
707:
703:
702:
684:
683:
668:
667:
649:
648:
638:
633:
631:
630:
629:
617:
616:
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602:
601:
589:
588:
576:
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557:
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549:
537:
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524:
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474:
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443:
441:
440:
435:
423:
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368:
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278:
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260:
251:
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231:
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183:in the interval
182:
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174:
158:
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153:
148:
143:
142:
130:
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109:
107:
106:
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96:
95:
83:
82:
35:In mathematics,
30:
26:
2866:
2865:
2861:
2860:
2859:
2857:
2856:
2855:
2841:
2840:
2817:
2799:
2782:
2777:
2748:
2747:
2740:
2724:
2720:
2715:
2678:
2647:
2638:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
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2601:
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2595:
2592:
2589:
2586:
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2577:
2574:
2571:
2568:
2565:
2562:
2559:
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2553:
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2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2465:
2461:
2457:
2453:
2438:
2420:
2402:BĂ©zier surfaces
2378:
2359:
2352:
2348:
2340:
2336:
2332:
2328:
2324:
2320:
2319:
2312:
2234:Seleucid Empire
2229:
2185:
2184:
2180:
2172:
2168:
2153:
2149:
2148:
2132:
2128:
2122:
2118:
2109:
2105:
2101:
2099:
2091:
2085:
2081:
2076:
2074:
2071:
2070:
2066:
2063:Rolle's theorem
2037:
2033:
2015:
2011:
2002:
1998:
1997:
1987:
1983:
1965:
1961:
1954:
1952:
1940:
1936:
1913:
1910:
1909:
1908:defined above:
1901:
1851:
1847:
1845:
1842:
1841:
1833:
1827:
1804:
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1798:
1795:
1794:
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1178:
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1101:
1084:
1081:
1080:
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1037:
1033:
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1024:
1023:
1009:
1008:
996:
992:
983:
979:
978:
968:
964:
949:
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927:
923:
914:
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909:
907:
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888:
884:
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863:
859:
853:
849:
840:
836:
830:
826:
817:
813:
807:
803:
791:
787:
778:
774:
768:
764:
752:
748:
747:
745:
736:
735:
726:
722:
713:
709:
708:
698:
694:
679:
675:
663:
659:
644:
640:
639:
637:
625:
621:
612:
608:
607:
597:
593:
584:
580:
571:
567:
566:
564:
555:
554:
545:
541:
532:
528:
527:
520:
516:
507:
503:
502:
500:
491:
487:
469:
465:
458:
451:
449:
446:
445:
429:
426:
425:
409:
406:
405:
379:
376:
375:
374:
347:
343:
334:
330:
329:
322:
318:
309:
305:
304:
302:
290:
286:
279:
272:
268:
261:
259:
257:
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2300:lookup tables
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42:
41:curve fitting
38:
21:
2822:
2804:
2790:cut-the-knot
2762:
2758:
2728:
2721:
2707:BĂ©zier curve
2669:
2665:
2661:
2657:
2653:
2649:
2639:
2449:
2439:
2379:
2376:Multivariate
2360:
2353:
2349:
2293:
2276:
2272:
2268:
2262:
2251:
2245:
2244:text called
2230:
2221:
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1828:
1769:
1762:
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1741:
1734:
1730:
1077:
403:
160:
63:
36:
34:
2466:(t, v0, v1)
2462:(v0, v1, t)
2415:Example of
2713:References
2310:Extensions
2238:Hipparchus
1906:polynomial
232:the value
2829:EMS Press
2811:EMS Press
2275:inear int
2166:≤
2160:≤
2116:−
2097:≤
2031:−
2009:−
1975:−
1873:−
1761:), ..., (
1682:−
1657:−
1614:−
1596:−
1539:−
1514:−
1470:−
1445:−
1436:−
1387:−
1369:−
1353:−
1311:−
1286:−
1277:−
1212:−
1185:−
1169:−
1133:−
1099:−
1090:−
990:−
962:−
934:−
882:−
847:−
785:−
762:−
720:−
692:−
673:−
657:−
619:−
591:−
539:−
514:−
485:−
341:−
316:−
284:−
266:−
2845:Category
2676:See also
2660:axes or
2454:(v0, v1)
2346:Accuracy
2253:Almagest
2190:″
2065:that if
1831:function
444:, gives
2831:, 2001
2813:, 2001
2258:Ptolemy
2736:
2596:return
2518:return
2329:yellow
2271:(from
1900:where
43:using
2584:float
2575:float
2566:float
2557:float
2506:float
2497:float
2488:float
2479:float
2350:If a
2333:green
2321:Black
1838:error
373:with
2734:ISBN
2560:lerp
2482:lerp
2464:and
2446:GLSL
2398:mesh
2337:blue
2323:and
2285:noun
2281:verb
2269:lerp
1747:), (
1158:and
159:the
110:and
2788:at
2767:doi
2450:mix
2325:red
2283:or
2277:erp
2146:max
27:at
2847::
2827:,
2821:,
2809:,
2803:,
2763:90
2761:,
2757:,
2668:,
2664:,
2656:,
2652:,
2629:v1
2617:v0
2578:v1
2569:v0
2545:);
2542:v0
2536:v1
2521:v0
2500:v1
2491:v0
2468:.
2372:.
2260:.
1768:,
1754:,
1740:,
1075:.
2774:.
2769::
2742:.
2670:b
2666:g
2662:r
2658:z
2654:y
2650:x
2646:t
2635:}
2632:;
2626:*
2623:t
2620:+
2614:*
2611:)
2608:t
2605:-
2602:1
2599:(
2593:{
2590:)
2587:t
2581:,
2572:,
2563:(
2548:}
2539:-
2533:(
2530:*
2527:t
2524:+
2515:{
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2509:t
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2494:,
2485:(
2458:t
2421:z
2361:C
2354:C
2335:/
2331:/
2327:/
2273:l
2208:.
2204:|
2200:)
2197:x
2194:(
2187:f
2182:|
2174:1
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2163:x
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2134:2
2130:)
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2111:1
2107:x
2103:(
2093:|
2087:T
2083:R
2078:|
2067:f
2047:.
2044:)
2039:0
2035:x
2028:x
2025:(
2017:0
2013:x
2004:1
2000:x
1994:)
1989:0
1985:x
1981:(
1978:f
1972:)
1967:1
1963:x
1959:(
1956:f
1950:+
1947:)
1942:0
1938:x
1934:(
1931:f
1928:=
1925:)
1922:x
1919:(
1916:p
1902:p
1888:,
1885:)
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1879:(
1876:p
1870:)
1867:x
1864:(
1861:f
1858:=
1853:T
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1820:.
1806:0
1802:C
1775:)
1773:n
1770:y
1766:n
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1698:)
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1654:x
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1580:(
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1566:=
1555:)
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1534:1
1530:x
1522:0
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1429:(
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1419:y
1415:=
1404:)
1395:0
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877:1
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801:+
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728:0
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705:)
700:0
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689:x
686:(
681:0
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670:)
665:0
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562:=
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482:x
479:(
476:+
471:0
467:y
463:=
456:y
432:x
412:y
400:.
388:1
385:=
382:n
357:,
349:0
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336:1
332:x
324:0
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311:1
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300:=
292:0
288:x
281:x
274:0
270:y
263:y
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218:)
213:1
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119:(
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93:0
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85:,
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72:(
29:x
25:y
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