Problem 2: (book Page 457) Consider any two valid colors C1 and C2 with coordinates (x1, y1) and (x2,y2)in the chromaticity diagram. Derive the necessary general expressions for computing the relative percentages of colors C1 and C2 composing a given color that is known to lie on the straight line joining these two colors.
Solution:
Let C(x, y) be any point (color) on the
line joining the two given point (colors) C1(x1, y1) and C2(x2, y2). The distance between the two given colors C1 and C2 is given by the distance formula in calculus:
d (c1, c2) = √(x1 – x2)2 + (y1 – y2)2
Distance between C1(x1, y1) and C(x, y) is d1 (c1, c) = = sqrt{(x1 – x)2 + (y1 – y)2} . And distance between C2(x2, y2) and (x, y) is
d2 (c2, c) = sqrt{(x2 – x)2 + (y2 – y)2}.
Let P1 = percentage of color C1(x1, y1) in C(x, y)
and P2 = percentage of color C2(x2, y2) in C(x, y)
then P1= [{d (c1, c2) - d1 (c1, c)}/ d (c1, c2) ]* 100 % .....(i)
and P2 = {d (c1, c2) - d2 (c2, c) }/ d (c1, c2) = (100 – P1) % ......(ii)
Special Cases:
(1) When C = C1 then
P1 = [{d (c1, c2) - 0} / d (c1, c2) ]* 100 % = 1 * 100 % = 100 % and
P2 = {d (c1, c2) - d2 (c2, c1)}/ d (c1, c2) = 0% = (100 – P1) % = (100 – 100)% = 0%
(2)When C = C2 then
P1=[{d (c1, c2) - d1 (c1, c2)}/ d (c1, c2) ]* 100 % = 0% and
P2 = {d (c1, c2) - d2 (c2, c2)}/ d (c1, c2) ]*100 % = 1*100% = 100%
= (100 – P1) % = (100 – 0) % = 100%
Note:
Percentage of the colors C1(x1, y1) and C2(x2, y2) in any given point (color) between C1 and C2 can be calculated by using the equations (i) and (ii) above.
For Example:
Let us consider 380 nm and 520 nm wavelengths. 380 nm wavelength has x and y coordinates as C1 (0.175, 0.003) and 520nm wavelength has x and y coordinated as C2 (0.055, 0.840). We take any point (color) e.g. C(x, y) with x and y coordinates say C (0.115, 0.4215) on the line joining C1 and C2. We can calculate percentage of C1 and C2 in C as follows:
Using (i) above: Percentage of 380nm wavelength in C(x, y) = P1, so
P1= {d (c1, c2) - d1 (c1, c)}/ d (c1, c2) * 100 %
= {√ (0 .175 – 0.055)2 + (0.003 – 0.840)2 - √ (0.175 - 0.115)2 + (0.003 – 0.4215)2} / {√ (0 .175 – 0.055)2 + (0.003 – 0.840)2} * 100 %
= {√0.0144 + 0.700569 - √0.0036 + 0.17514225} / {√0.0144 + 0.700569} * 100 %
= (0.845558395 – 0.422779197)/ (0.845558395) * 100 %
= (0.422779198) / (0.845558395) * 100 %
= 0.5 * 100 %
= 50%
Therefore, Percentage of 380nm wavelength [C1 (0.175, 0.003] in C(x, y) = C (0.115, 0.4215) is 50 %
Hence, percentage of 520nm wavelength [C2 (0.055, 0.840)] in C(x, y)
P2= (100 – P1) % = (100 - 50)% = 50% -----------------------------------------------------------------------------------------
Problem #3 ( Problem 6.3 p 457)
Consider any three valid colors now c1, c2 and c3 with coordinates (x1,y1), (x2, y2), and
(x3,y3) in the chromacity diagram of Fig 6.5. Derive the necessary general expressions for computing the relative percentages of c1, c2 and c3 composing a given color that is known to lie with in the triangle whose vertices are the coordinates of c1, c2 and c3.
Solution:
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