2b:null 2c:["$","title",null,{"children":"Extrapolating the spectral locus (for Sprague interpolation) | Eric Portis | Observable"}] 31:T74cd,// CIE 1931 2 Degree Standard Observer // http://www-cvrl.ucsd.edu/cmfs.htm // I got it from https://github.com/facelessuser/coloraide/blob/bcfcdb5a6f75964f2af3cde66605c6dbec3cc636/coloraide/cmfs.py#L38 CIE_1931_2DEG = ({ 360: [0.000129900000, 0.000003917000, 0.000606100000], 361: [0.000145847000, 0.000004393581, 0.000680879200], 362: [0.000163802100, 0.000004929604, 0.000765145600], 363: [0.000184003700, 0.000005532136, 0.000860012400], 364: [0.000206690200, 0.000006208245, 0.000966592800], 365: [0.000232100000, 0.000006965000, 0.001086000000], 366: [0.000260728000, 0.000007813219, 0.001220586000], 367: [0.000293075000, 0.000008767336, 0.001372729000], 368: [0.000329388000, 0.000009839844, 0.001543579000], 369: [0.000369914000, 0.000011043230, 0.001734286000], 370: [0.000414900000, 0.000012390000, 0.001946000000], 371: [0.000464158700, 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0.005723000000, 0.000000000000], 696: [0.014790640000, 0.005343059000, 0.000000000000], 697: [0.013831320000, 0.004995796000, 0.000000000000], 698: [0.012948680000, 0.004676404000, 0.000000000000], 699: [0.012129200000, 0.004380075000, 0.000000000000], 700: [0.011359160000, 0.004102000000, 0.000000000000], 701: [0.010629350000, 0.003838453000, 0.000000000000], 702: [0.009938846000, 0.003589099000, 0.000000000000], 703: [0.009288422000, 0.003354219000, 0.000000000000], 704: [0.008678854000, 0.003134093000, 0.000000000000], 705: [0.008110916000, 0.002929000000, 0.000000000000], 706: [0.007582388000, 0.002738139000, 0.000000000000], 707: [0.007088746000, 0.002559876000, 0.000000000000], 708: [0.006627313000, 0.002393244000, 0.000000000000], 709: [0.006195408000, 0.002237275000, 0.000000000000], 710: [0.005790346000, 0.002091000000, 0.000000000000], 711: [0.005409826000, 0.001953587000, 0.000000000000], 712: [0.005052583000, 0.001824580000, 0.000000000000], 713: [0.004717512000, 0.001703580000, 0.000000000000], 714: [0.004403507000, 0.001590187000, 0.000000000000], 715: [0.004109457000, 0.001484000000, 0.000000000000], 716: [0.003833913000, 0.001384496000, 0.000000000000], 717: [0.003575748000, 0.001291268000, 0.000000000000], 718: [0.003334342000, 0.001204092000, 0.000000000000], 719: [0.003109075000, 0.001122744000, 0.000000000000], 720: [0.002899327000, 0.001047000000, 0.000000000000], 721: [0.002704348000, 0.000976589600, 0.000000000000], 722: [0.002523020000, 0.000911108800, 0.000000000000], 723: [0.002354168000, 0.000850133200, 0.000000000000], 724: [0.002196616000, 0.000793238400, 0.000000000000], 725: [0.002049190000, 0.000740000000, 0.000000000000], 726: [0.001910960000, 0.000690082700, 0.000000000000], 727: [0.001781438000, 0.000643310000, 0.000000000000], 728: [0.001660110000, 0.000599496000, 0.000000000000], 729: [0.001546459000, 0.000558454700, 0.000000000000], 730: [0.001439971000, 0.000520000000, 0.000000000000], 731: [0.001340042000, 0.000483913600, 0.000000000000], 732: [0.001246275000, 0.000450052800, 0.000000000000], 733: [0.001158471000, 0.000418345200, 0.000000000000], 734: [0.001076430000, 0.000388718400, 0.000000000000], 735: [0.000999949300, 0.000361100000, 0.000000000000], 736: [0.000928735800, 0.000335383500, 0.000000000000], 737: [0.000862433200, 0.000311440400, 0.000000000000], 738: [0.000800750300, 0.000289165600, 0.000000000000], 739: [0.000743396000, 0.000268453900, 0.000000000000], 740: [0.000690078600, 0.000249200000, 0.000000000000], 741: [0.000640515600, 0.000231301900, 0.000000000000], 742: [0.000594502100, 0.000214685600, 0.000000000000], 743: [0.000551864600, 0.000199288400, 0.000000000000], 744: [0.000512429000, 0.000185047500, 0.000000000000], 745: [0.000476021300, 0.000171900000, 0.000000000000], 746: [0.000442453600, 0.000159778100, 0.000000000000], 747: [0.000411511700, 0.000148604400, 0.000000000000], 748: [0.000382981400, 0.000138301600, 0.000000000000], 749: [0.000356649100, 0.000128792500, 0.000000000000], 750: [0.000332301100, 0.000120000000, 0.000000000000], 751: [0.000309758600, 0.000111859500, 0.000000000000], 752: [0.000288887100, 0.000104322400, 0.000000000000], 753: [0.000269539400, 0.000097335600, 0.000000000000], 754: [0.000251568200, 0.000090845870, 0.000000000000], 755: [0.000234826100, 0.000084800000, 0.000000000000], 756: [0.000219171000, 0.000079146670, 0.000000000000], 757: [0.000204525800, 0.000073858000, 0.000000000000], 758: [0.000190840500, 0.000068916000, 0.000000000000], 759: [0.000178065400, 0.000064302670, 0.000000000000], 760: [0.000166150500, 0.000060000000, 0.000000000000], 761: [0.000155023600, 0.000055981870, 0.000000000000], 762: [0.000144621900, 0.000052225600, 0.000000000000], 763: [0.000134909800, 0.000048718400, 0.000000000000], 764: [0.000125852000, 0.000045447470, 0.000000000000], 765: [0.000117413000, 0.000042400000, 0.000000000000], 766: [0.000109551500, 0.000039561040, 0.000000000000], 767: [0.000102224500, 0.000036915120, 0.000000000000], 768: [0.000095394450, 0.000034448680, 0.000000000000], 769: [0.000089023900, 0.000032148160, 0.000000000000], 770: [0.000083075270, 0.000030000000, 0.000000000000], 771: [0.000077512690, 0.000027991250, 0.000000000000], 772: [0.000072313040, 0.000026113560, 0.000000000000], 773: [0.000067457780, 0.000024360240, 0.000000000000], 774: [0.000062928440, 0.000022724610, 0.000000000000], 775: [0.000058706520, 0.000021200000, 0.000000000000], 776: [0.000054770280, 0.000019778550, 0.000000000000], 777: [0.000051099180, 0.000018452850, 0.000000000000], 778: [0.000047676540, 0.000017216870, 0.000000000000], 779: [0.000044485670, 0.000016064590, 0.000000000000], 780: [0.000041509940, 0.000014990000, 0.000000000000], 781: [0.000038733240, 0.000013987280, 0.000000000000], 782: [0.000036142030, 0.000013051550, 0.000000000000], 783: [0.000033723520, 0.000012178180, 0.000000000000], 784: [0.000031464870, 0.000011362540, 0.000000000000], 785: [0.000029353260, 0.000010600000, 0.000000000000], 786: [0.000027375730, 0.000009885877, 0.000000000000], 787: [0.000025524330, 0.000009217304, 0.000000000000], 788: [0.000023793760, 0.000008592362, 0.000000000000], 789: [0.000022178700, 0.000008009133, 0.000000000000], 790: [0.000020673830, 0.000007465700, 0.000000000000], 791: [0.000019272260, 0.000006959567, 0.000000000000], 792: [0.000017966400, 0.000006487995, 0.000000000000], 793: [0.000016749910, 0.000006048699, 0.000000000000], 794: [0.000015616480, 0.000005639396, 0.000000000000], 795: [0.000014559770, 0.000005257800, 0.000000000000], 796: [0.000013573870, 0.000004901771, 0.000000000000], 797: [0.000012654360, 0.000004569720, 0.000000000000], 798: [0.000011797230, 0.000004260194, 0.000000000000], 799: [0.000010998440, 0.000003971739, 0.000000000000], 800: [0.000010253980, 0.000003702900, 0.000000000000], 801: [0.000009559646, 0.000003452163, 0.000000000000], 802: [0.000008912044, 0.000003218302, 0.000000000000], 803: [0.000008308358, 0.000003000300, 0.000000000000], 804: [0.000007745769, 0.000002797139, 0.000000000000], 805: [0.000007221456, 0.000002607800, 0.000000000000], 806: [0.000006732475, 0.000002431220, 0.000000000000], 807: [0.000006276423, 0.000002266531, 0.000000000000], 808: [0.000005851304, 0.000002113013, 0.000000000000], 809: [0.000005455118, 0.000001969943, 0.000000000000], 810: [0.000005085868, 0.000001836600, 0.000000000000], 811: [0.000004741466, 0.000001712230, 0.000000000000], 812: [0.000004420236, 0.000001596228, 0.000000000000], 813: [0.000004120783, 0.000001488090, 0.000000000000], 814: [0.000003841716, 0.000001387314, 0.000000000000], 815: [0.000003581652, 0.000001293400, 0.000000000000], 816: [0.000003339127, 0.000001205820, 0.000000000000], 817: [0.000003112949, 0.000001124143, 0.000000000000], 818: [0.000002902121, 0.000001048009, 0.000000000000], 819: [0.000002705645, 0.000000977058, 0.000000000000], 820: [0.000002522525, 0.000000910930, 0.000000000000], 821: [0.000002351726, 0.000000849251, 0.000000000000], 822: [0.000002192415, 0.000000791721, 0.000000000000], 823: [0.000002043902, 0.000000738090, 0.000000000000], 824: [0.000001905497, 0.000000688110, 0.000000000000], 825: [0.000001776509, 0.000000641530, 0.000000000000], 826: [0.000001656215, 0.000000598090, 0.000000000000], 827: [0.000001544022, 0.000000557575, 0.000000000000], 828: [0.000001439440, 0.000000519808, 0.000000000000], 829: [0.000001341977, 0.000000484612, 0.000000000000], 830: [0.000001251141, 0.000000451810, 0.000000000000] })32:T67b,To get a fake point, we multiply a matrix of the six closest "real" points by a matrix of each fake point's specified coefficients (here labeled

a

–

f

(\begin{matrix} X_{359} & Y_{359} & Z_{359} \end{matrix}) = (\begin{matrix} a_{359} & b_{359} & \dots & f_{359} \end{matrix}) \cdot (\begin{matrix} X_{360} & Y_{360} & Z_{360} \\ X_{361} & Y_{361} & Z_{361} \\ ⋮ \\ X_{365} & Y_{365} & Z_{365} \end{matrix})

33:T5a1,html`

${Plot.plot({ title: "X", width: width/3, height: width/3, marks: [ // known values Plot.dot( Object.keys(CIE_1931_2DEG).slice(0,6), { x: d => parseInt(d), y: d => CIE_1931_2DEG[d][0], fill: 'black' } ), // extrapolated values Plot.dot( [ [ 358, extrapolated_points[358][0] ], [ 359, extrapolated_points[359][0] ] ], { fill: 'red' } ) ] })} ${ Plot.plot({ title: "Y", width: width/3, height: width/3, marks: [ // known values Plot.dot( Object.keys(CIE_1931_2DEG).slice(0,6), { x: d => parseInt(d), y: d => CIE_1931_2DEG[d][1], fill: 'black' } ), // extrapolated values Plot.dot( [ [ 358, extrapolated_points[358][1] ], [ 359, extrapolated_points[359][1] ] ], { fill: 'red' } ) ] }) } ${ Plot.plot({ title: "Z", width: width/3, height: width/3, marks: [ // known values Plot.dot( Object.keys(CIE_1931_2DEG).slice(0,6), { x: d => parseInt(d), y: d => CIE_1931_2DEG[d][2], fill: 'black' } ), // extrapolated values Plot.dot( [ [ 358, extrapolated_points[358][2] ], [ 359, extrapolated_points[359][2] ] ], { fill: 'red' } ) ] }) }

`34:T59e,html`

${Plot.plot({ title: "X", width: width/3, height: width/3, marks: [ // known values Plot.dot( Object.keys(CIE_1931_2DEG).slice(-6), { x: d => parseInt(d), y: d => CIE_1931_2DEG[d][0], fill: 'black' } ), // extrapolated values Plot.dot( [ [ 831, extrapolated_points[831][0] ], [ 832, extrapolated_points[832][0] ] ], { fill: 'red' } ) ] })} ${ Plot.plot({ title: "Y", width: width/3, height: width/3, marks: [ // known values Plot.dot( Object.keys(CIE_1931_2DEG).slice(-6), { x: d => parseInt(d), y: d => CIE_1931_2DEG[d][1], fill: 'black' } ), // extrapolated values Plot.dot( [ [ 831, extrapolated_points[831][1] ], [ 832, extrapolated_points[832][1] ] ], { fill: 'red' } ) ] }) } ${ Plot.plot({ title: "Z", width: width/3, height: width/3, marks: [ // known values Plot.dot( Object.keys(CIE_1931_2DEG).slice(-6), { x: d => parseInt(d), y: d => CIE_1931_2DEG[d][2], fill: 'black' } ), // extrapolated values Plot.dot( [ [ 831, extrapolated_points[831][2] ], [ 832, extrapolated_points[832][2] ] ], { fill: 'red' } ) ] }) }

`35:[] 2f:{"id":"50a928636eef7d24","slug":null,"title":"Extrapolating the spectral locus (for Sprague interpolation)","thumbnail":"b9fd814fcb764a21603f990936ee98af8de2792f75f9318567c4306993baac9e","access_level":"public","create_time":"$D2026-03-06T06:03:57.225Z","update_time":"$D2026-03-06T18:26:28.350Z","view_time":null,"first_public_time":"$D2026-03-06T18:20:48.637Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"3301827868803b5f","login":"eeeps","name":"Eric Portis","avatar_url":"https://avatars-next.observableusercontent.com/d9d2bbfe6f3f798e9ec8f3a91e0c10b7f6890314fd52f40e190022e173d838a9","tier":"starter_2024"},"version":291,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"641c0f2eb3a762f9","login":"eeeps","name":"Eric Portis","avatar_url":"https://avatars-next.observableusercontent.com/d9d2bbfe6f3f798e9ec8f3a91e0c10b7f6890314fd52f40e190022e173d838a9","tier":null}],"members":[{"type":"team","id":"3301827868803b5f","login":"eeeps","name":"Eric Portis","avatar_url":"https://avatars-next.observableusercontent.com/d9d2bbfe6f3f798e9ec8f3a91e0c10b7f6890314fd52f40e190022e173d838a9","tier":"starter_2024","role":"editor"},{"type":"individual","id":"641c0f2eb3a762f9","login":"eeeps","name":"Eric Portis","avatar_url":"https://avatars-next.observableusercontent.com/d9d2bbfe6f3f798e9ec8f3a91e0c10b7f6890314fd52f40e190022e173d838a9","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Extrapolating the spectral locus (for Sprague interpolation)","cells":[{"id":0,"value":"# Extrapolating the spectral locus (for Sprague interpolation)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":102,"value":"Here's the [spectral locus](https://en.wikipedia.org/wiki/Spectral_color#In_color_spaces:~:text=known%20as%20the-,spectral%20locus,-.%20For%20example%2C%20the) in [XYZ](https://en.wikipedia.org/wiki/CIE_1931_color_space#CIE_XYZ_color_space), at 1nm increments.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"To interpolate between the last two points on either end using the CIE-specified method ([Sprague interpolation](https://www.sciencedirect.com/science/article/pii/0771050X75900273)), we need to add four points to the dataset: two to the front and two at the back. Essentially these are \"fake\" data points at 358nm, 359nm, 831nm, and 832nm, added just to guide the interpolated line in the \"right direction\" as it moves through 360-362nm and 858-860nm.\n\nThe [document where these extra points are specified](https://www.cie.co.at/publications/recommended-practice-tabulating-spectral-data-use-colour-computations) is behind a 76€ paywall. But information wants to be free, and [various](https://github.com/colour-science/colour/blob/3d1d587264d958f8d5e5ce1186d3fd93fc416bc3/colour/algebra/interpolation.py#L1129) open source [libraries](https://github.com/facelessuser/coloraide/blob/efd7ac1db65f5e597398c19a8ca5c2b9e21e9605/coloraide/algebra.py#L1067) use these four arrays of six integers to derive the points.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":4,"value":"coefficent_numerators = [\n [884, -1960, 3033, -2648, 1080, -180],\n [508, -540, 488, -367, 144, -24],\n [-24, 144, -367, 488, -540, 508],\n [-180, 1080, -2648, 3033, -1960, 884]\n];","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":118,"value":"As far as I can tell, the actual coefficents are all defined as rationals, and the denominator for all of them is 209. Why? I don't know! How were these derived? I don't know! They're magic numbers to me. But they work!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":74,"value":"coefficents_array = coefficent_numerators.map( row => row.map( n => n / 209 ) )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":140,"value":"What do we actually do with these? Each array of six helps us get one fake data point. I'm going to add the fake wavelengths to each, for clarity.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":146,"value":"coefficents = ({\n 358: coefficents_array[0],\n 359: coefficents_array[1],\n 831: coefficents_array[2],\n 832: coefficents_array[3]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":156,"value":"$32","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"So... let's do it!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":34,"value":"multiplyMatrix = function(a, b) {\n \n const rowsA = a.length;\n const colsA = a[0].length;\n const rowsB = b.length;\n const colsB = b[0].length;\n \n if (colsA !== rowsB) {\n throw \"Matrix dimensions do not match for multiplication\";\n }\n\n const result = Array.from({ length: rowsA }, () => Array(colsB).fill(0));\n\n for (let i = 0; i < rowsA; i++) {\n for (let k = 0; k < colsA; k++) {\n const aik = a[i][k];\n for (let j = 0; j < colsB; j++) {\n result[i][j] += aik * b[k][j];\n }\n }\n }\n\n return result;\n \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":208,"value":"extrapolated_points = {\n const first_six_points = Object.values(CIE_1931_2DEG).slice(0, 6);\n const last_six_points = Object.values(CIE_1931_2DEG).slice(-6);\n return {\n 358: multiplyMatrix( [ coefficents[358] ], first_six_points )[0],\n 359: multiplyMatrix( [ coefficents[359] ], first_six_points )[0],\n 831: multiplyMatrix( [ coefficents[831] ], last_six_points )[0],\n 832: multiplyMatrix( [ coefficents[832] ], last_six_points )[0]\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":217,"value":"Add those back to the \"real\" datapoints and you're ready to Sprague interpolate!\n\nP.S.—\nHere's a shorter, probably more efficient, and harder-for-me-to-read bit of Javascript that does the same thing, ported by Issac Muse from [his Python implementation](https://github.com/facelessuser/coloraide/blob/3c69d9587499893bbd1ad1162f2fa01ad8db16cb/coloraide/algebra.py#L1074), after I pestered him with many questions:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":229,"value":"colorAideImplementation = {\n \n let first = Object.values(CIE_1931_2DEG).slice(0, 6);\n let end = Object.values(CIE_1931_2DEG).slice(-6);\n \n let v1 = [0.0, 0.0, 0.0];\n let v2 = [0.0, 0.0, 0.0];\n let v3 = [0.0, 0.0, 0.0];\n let v4 = [0.0, 0.0, 0.0];\n\n for (let i = 0; i < 3; i++) {\n [v1[i], v2[i], v3[i], v4[i]] = coefficent_numerators.map((row, j) => {\n let r = row.map((col, k) => {\n return col * ((j < 2) ? first[k][i] : end[k][i]);\n });\n return r.reduce((accumulator, currentValue) => accumulator + currentValue, 0) / 209;\n })\n }\n\n return [v1, v2, v3, v4];\n\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":247,"value":"P.P.S.— At this point I should probably just pay the 76€, but I remain curious how these points were derived. They're close to linearly-extrapolated from the preceeding two points... but not quite.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":251,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":272,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 30:"$2f:body"