# Example 889 — The hues were averaged across the wrap `the_hues_were_averaged_across_the_wrap.eml` - A palette tool picks an image's representative colour by averaging the hue of every pixel, and the average is exact over every pixel. What the mean of two hues near the seam of the colour wheel is, is computed below. ## EML ```eml # Self-authored for the EML case corpus (no external origin). A palette tool # picks an image's representative colour by averaging the hue of every pixel, # and the average is exact over every pixel. What the mean of two hues near the # seam of the colour wheel is, is computed below. # # The pick is careful. It reads the real hue of every pixel, not a sample; it # uses the engine's own arithmetic mean; it weights every pixel equally; and the # intent is exactly 'the hue that represents the image'. # # Hue is an angle on a circle that wraps at 360, and two reds - 350 and 10 - # average arithmetically to 180, which is cyan, the one hue farthest from both. 350 => hue_a_degrees 10 => hue_b_degrees 0 => circular_mean_degrees hue_a_degrees + hue_b_degrees => hue_sum int(hue_sum / 2) => arithmetic_mean_degrees arithmetic_mean_degrees - circular_mean_degrees => degrees_from_the_true_mean int(degrees_from_the_true_mean * 10000 / 180) => share_of_the_farthest_possible_error_per_myriad "hue A : " + str(hue_a_degrees) + " degrees, a red" ^0 "hue B : " + str(hue_b_degrees) + " degrees, a red" ^0 "arithmetic mean : " + str(arithmetic_mean_degrees) + " degrees, cyan" ^0 "circular mean : " + str(circular_mean_degrees) + " degrees, red" ^0 "error of the arithmetic mean : " + str(degrees_from_the_true_mean) + " degrees" ^0 "of the farthest possible error : " + str(share_of_the_farthest_possible_error_per_myriad) + " per ten thousand" ^0 "" ^0 # ---- what the pick verified ---- "the representative-hue pick" ^0 " reads : the real hue of every pixel" ^0 " mean : the engine's own arithmetic mean" ^0 " weighting : every pixel equal" ^0 " intent : the hue that represents the image" ^0 " pixels omitted : 0" ^0 " verdict : MEAN HUE IS 180" ^0 "" ^0 " taking the exact arithmetic mean of every pixel's real hue" ^0 " is the part done right here, and it is why 180 is" ^0 " precisely the mean of the numbers" ^0 "" ^0 # ---- what hue is ---- "hue as a number" ^0 " what hue is : an angle around a wheel, 0 and 360 the same" ^0 " colour" ^0 " where 350 and 10 sit : 20 degrees apart, across the seam" ^0 " what the arithmetic mean does : treats 350 and 10 as 340" ^0 " apart and lands halfway, at 180" ^0 " what 180 is : the hue opposite both inputs" ^0 " so the representative hue of a red image : cyan" ^0 "" ^0 # ---- what the caller got ---- "the palette" ^0 " image : reds, 350 and 10" ^0 " representative hue chosen : " + str(arithmetic_mean_degrees) + ", cyan" ^0 " is the mean miscomputed : no; it is exact" ^0 " is a mean of angles the same as a mean of numbers : no;" ^0 " across the seam it is the opposite" ^0 "" ^0 # ---- null control ---- # The same hues averaged as directions (mean of the unit vectors, then the angle # of the result), which has no seam. 180 => nc_mean_as_numbers 0 => nc_mean_as_directions 180 => nc_degrees_the_circular_mean_corrects "null control - average the directions, not the numbers" ^0 " mean as numbers : " + str(nc_mean_as_numbers) + " degrees" ^0 " mean as directions : " + str(nc_mean_as_directions) + " degrees" ^0 " degrees the circular mean corrects : " + str(nc_degrees_the_circular_mean_corrects) ^0 " no pixel and no hue changed; the mean stopped crossing" ^0 " the seam" ^0 "" ^0 # ---- the rule ---- "what an arithmetic mean of hues guarantees" ^0 " the result is the mean of the hue numbers : exactly, every" ^0 " pixel, the engine's own mean" ^0 " the result is a hue between the inputs : not addressed;" ^0 " hue wraps at 360, and two reds at " + str(hue_a_degrees) + " and " + str(hue_b_degrees) + " average to " + str(arithmetic_mean_degrees) + "," ^0 " the hue opposite both - " + str(share_of_the_farthest_possible_error_per_myriad) + " per ten thousand of the worst" ^0 " possible error" ^0 "" ^0 "a circle has no smallest number, and a mean that assumes one puts the middle" ^0 "of two neighbours on the far side of the wheel; the same seam that broke" ^0 "longitude at 180 breaks hue at 360, for the same reason" ^0 "" ^0 "It takes the exact arithmetic mean of every pixel's real hue - 180 is the mean" ^0 "of the numbers. But hue is an angle, and " + str(hue_a_degrees) + " and " + str(hue_b_degrees) + " are two reds 20 degrees" ^0 "apart across the seam; their arithmetic mean is " + str(arithmetic_mean_degrees) + ", cyan, " + str(degrees_from_the_true_mean) + " degrees off, until" ^0 "the hues are averaged as directions." ^0 ``` ## Python (deterministic transpilation) ```python hue_a_degrees = 350 hue_b_degrees = 10 circular_mean_degrees = 0 hue_sum = hue_a_degrees + hue_b_degrees arithmetic_mean_degrees = int(hue_sum / 2) degrees_from_the_true_mean = arithmetic_mean_degrees - circular_mean_degrees share_of_the_farthest_possible_error_per_myriad = int(degrees_from_the_true_mean * 10000 / 180) print("hue A : " + str(hue_a_degrees) + " degrees, a red") print("hue B : " + str(hue_b_degrees) + " degrees, a red") print("arithmetic mean : " + str(arithmetic_mean_degrees) + " degrees, cyan") print("circular mean : " + str(circular_mean_degrees) + " degrees, red") print("error of the arithmetic mean : " + str(degrees_from_the_true_mean) + " degrees") print("of the farthest possible error : " + str(share_of_the_farthest_possible_error_per_myriad) + " per ten thousand") print("") print("the representative-hue pick") print(" reads : the real hue of every pixel") print(" mean : the engine's own arithmetic mean") print(" weighting : every pixel equal") print(" intent : the hue that represents the image") print(" pixels omitted : 0") print(" verdict : MEAN HUE IS 180") print("") print(" taking the exact arithmetic mean of every pixel's real hue") print(" is the part done right here, and it is why 180 is") print(" precisely the mean of the numbers") print("") print("hue as a number") print(" what hue is : an angle around a wheel, 0 and 360 the same") print(" colour") print(" where 350 and 10 sit : 20 degrees apart, across the seam") print(" what the arithmetic mean does : treats 350 and 10 as 340") print(" apart and lands halfway, at 180") print(" what 180 is : the hue opposite both inputs") print(" so the representative hue of a red image : cyan") print("") print("the palette") print(" image : reds, 350 and 10") print(" representative hue chosen : " + str(arithmetic_mean_degrees) + ", cyan") print(" is the mean miscomputed : no; it is exact") print(" is a mean of angles the same as a mean of numbers : no;") print(" across the seam it is the opposite") print("") nc_mean_as_numbers = 180 nc_mean_as_directions = 0 nc_degrees_the_circular_mean_corrects = 180 print("null control - average the directions, not the numbers") print(" mean as numbers : " + str(nc_mean_as_numbers) + " degrees") print(" mean as directions : " + str(nc_mean_as_directions) + " degrees") print(" degrees the circular mean corrects : " + str(nc_degrees_the_circular_mean_corrects)) print(" no pixel and no hue changed; the mean stopped crossing") print(" the seam") print("") print("what an arithmetic mean of hues guarantees") print(" the result is the mean of the hue numbers : exactly, every") print(" pixel, the engine's own mean") print(" the result is a hue between the inputs : not addressed;") print(" hue wraps at 360, and two reds at " + str(hue_a_degrees) + " and " + str(hue_b_degrees) + " average to " + str(arithmetic_mean_degrees) + ",") print(" the hue opposite both - " + str(share_of_the_farthest_possible_error_per_myriad) + " per ten thousand of the worst") print(" possible error") print("") print("a circle has no smallest number, and a mean that assumes one puts the middle") print("of two neighbours on the far side of the wheel; the same seam that broke") print("longitude at 180 breaks hue at 360, for the same reason") print("") print("It takes the exact arithmetic mean of every pixel's real hue - 180 is the mean") print("of the numbers. But hue is an angle, and " + str(hue_a_degrees) + " and " + str(hue_b_degrees) + " are two reds 20 degrees") print("apart across the seam; their arithmetic mean is " + str(arithmetic_mean_degrees) + ", cyan, " + str(degrees_from_the_true_mean) + " degrees off, until") print("the hues are averaged as directions.") ``` ## stdout (executed) ```text hue A : 350 degrees, a red hue B : 10 degrees, a red arithmetic mean : 180 degrees, cyan circular mean : 0 degrees, red error of the arithmetic mean : 180 degrees of the farthest possible error : 10000 per ten thousand the representative-hue pick reads : the real hue of every pixel mean : the engine's own arithmetic mean weighting : every pixel equal intent : the hue that represents the image pixels omitted : 0 verdict : MEAN HUE IS 180 taking the exact arithmetic mean of every pixel's real hue is the part done right here, and it is why 180 is precisely the mean of the numbers hue as a number what hue is : an angle around a wheel, 0 and 360 the same colour where 350 and 10 sit : 20 degrees apart, across the seam what the arithmetic mean does : treats 350 and 10 as 340 apart and lands halfway, at 180 what 180 is : the hue opposite both inputs so the representative hue of a red image : cyan the palette image : reds, 350 and 10 representative hue chosen : 180, cyan is the mean miscomputed : no; it is exact is a mean of angles the same as a mean of numbers : no; across the seam it is the opposite null control - average the directions, not the numbers mean as numbers : 180 degrees mean as directions : 0 degrees degrees the circular mean corrects : 180 no pixel and no hue changed; the mean stopped crossing the seam what an arithmetic mean of hues guarantees the result is the mean of the hue numbers : exactly, every pixel, the engine's own mean the result is a hue between the inputs : not addressed; hue wraps at 360, and two reds at 350 and 10 average to 180, the hue opposite both - 10000 per ten thousand of the worst possible error a circle has no smallest number, and a mean that assumes one puts the middle of two neighbours on the far side of the wheel; the same seam that broke longitude at 180 breaks hue at 360, for the same reason It takes the exact arithmetic mean of every pixel's real hue - 180 is the mean of the numbers. But hue is an angle, and 350 and 10 are two reds 20 degrees apart across the seam; their arithmetic mean is 180, cyan, 180 degrees off, until the hues are averaged as directions. ``` ## Round-trip `ok: true` — round-trip fixpoint reached (python1 == python2) ## Trace event types eml:run:start · eml:assign · eml:output · eml:run:done