Centralized mathematical reference for all formulas in the series.
This appendix collects all mathematical formulas, matrices, and primaries from Modules 1-7 in one place for quick reference.
A.1 Colorimetric Calculations
CIE XYZ Tristimulus Values
From Module 2 - The integration formula payoff
\[\begin{align} X &= k \int_{380}^{780} S(\lambda) \cdot R(\lambda) \cdot \bar{x}(\lambda) \, d\lambda \\ Y &= k \int_{380}^{780} S(\lambda) \cdot R(\lambda) \cdot \bar{y}(\lambda) \, d\lambda \\ Z &= k \int_{380}^{780} S(\lambda) \cdot R(\lambda) \cdot \bar{z}(\lambda) \, d\lambda \end{align}\]Where:
- $S(\lambda)$ = Spectral Power Distribution of illuminant
- $R(\lambda)$ = Reflectance spectrum (0-1)
- $\bar{x}(\lambda), \bar{y}(\lambda), \bar{z}(\lambda)$ = CIE Standard Observer color matching functions
- $k$ = Normalizing constant: $k = 100 / \int S(\lambda) \bar{y}(\lambda) d\lambda$
Chromaticity Coordinates (xyY)
\[x = \frac{X}{X + Y + Z}, \quad y = \frac{Y}{X + Y + Z}\]The luminance $Y$ is kept separately. Together, $(x, y, Y)$ fully describes a color.
CIELAB (Lab*) - Perceptually Uniform Space
\[\begin{align} L^* &= 116 \, f(Y/Y_n) - 16 \\ a^* &= 500 \left[ f(X/X_n) - f(Y/Y_n) \right] \\ b^* &= 200 \left[ f(Y/Y_n) - f(Z/Z_n) \right] \end{align}\]Where the piecewise function $f(t)$ is:
\[f(t) = \begin{cases} t^{1/3} & \text{if } t > \left(\frac{6}{29}\right)^3 \\ \frac{1}{3}\left(\frac{29}{6}\right)^2 t + \frac{4}{29} & \text{otherwise} \end{cases}\]And $(X_n, Y_n, Z_n)$ = XYZ values of reference white (typically D65)
Axes:
- $L^*$: Lightness (0 = black, 100 = white)
- $a^*$: Red-Green axis (negative = green, positive = red)
- $b^*$: Blue-Yellow axis (negative = blue, positive = yellow)
A.2 Transfer Functions
Gamma 2.2 (Simple Power Function)
From Module 3 - Display EOTF
\[V_{\text{display}} = V_{\text{encoded}}^{2.2}\]Encoding (OETF): $V_{\text{encoded}} = V_{\text{linear}}^{1/2.2} \approx V_{\text{linear}}^{0.4545}$
sRGB (Piecewise Transfer Function)
\[V_{\text{srgb}} = \begin{cases} 12.92 \times V_{\text{linear}} & \text{if } V_{\text{linear}} \leq 0.0031308 \\ 1.055 \times V_{\text{linear}}^{1/2.4} - 0.055 & \text{otherwise} \end{cases}\]Effective gamma ~2.2, with linear segment for shadow detail preservation.
ST.2084 PQ (Perceptual Quantizer for HDR)
From Module 3 - Absolute luminance encoding
\[V = \left( \frac{\max[(L/10000)^{m_1} - c_1, 0]}{c_2 - c_3 (L/10000)^{m_1}} \right)^{m_2}\]Constants:
- $m_1 = 0.1593017578125$ (2610/16384)
- $m_2 = 78.84375$ (2523/32)
- $c_1 = 0.8359375$ (3424/4096)
- $c_2 = 18.8515625$ (2413/128)
- $c_3 = 18.6875$ (2392/128)
$L$ = luminance in nits (cd/m²), range: 0.0001 to 10,000 nits
A.3 ACES Primaries & Transforms
ACES 2065-1 (AP0) Primaries
From Module 4 - The archive space with imaginary primaries
| Primary | x | y |
|---|---|---|
| Red | 0.7347 | 0.2653 |
| Green | 0.0000 | 1.0000 |
| Blue | 0.0001 | -0.0770 |
| White Point | 0.32168 (D60) | 0.33767 |
ACEScg (AP1) Primaries
The working space - all real primaries
| Primary | x | y |
|---|---|---|
| Red | 0.713 | 0.293 |
| Green | 0.165 | 0.830 |
| Blue | 0.128 | 0.044 |
| White Point | 0.32168 (D60) | 0.33767 |
Example ARRI Alexa IDT Matrix (Conceptual)
\[\begin{bmatrix} R_{\text{ACES}} \\ G_{\text{ACES}} \\ B_{\text{ACES}} \end{bmatrix} = \begin{bmatrix} 0.680206 & 0.236137 & 0.083657 \\ 0.035735 & 0.950071 & 0.014194 \\ 0.000000 & 0.002932 & 0.997068 \end{bmatrix} \begin{bmatrix} R_{\text{camera}} \\ G_{\text{camera}} \\ B_{\text{camera}} \end{bmatrix}\]Note: Actual IDT includes LogC decoding + chromatic adaptation D65→D60
A.4 Camera Log Curves
LogC3 (ARRI - Classic Curve)
From Module 6 - For ARRI Wide Gamut 3
\[y = \begin{cases} 5.555556 \times x + 0.052272 & \text{if } x < 0.010591 \\ 0.247190 \times \log_{10}(5.555556 \times x + 0.047996) + 0.385537 & \text{otherwise} \end{cases}\]- $x$ = scene linear exposure (normalized 0-1)
- $y$ = LogC3 code value (0-1)
- Middle Grey (18%) → $y = 0.39$ (39%)
- Dynamic range: ~14 stops
LogC4 (ARRI Alexa 35)
\[y = \begin{cases} (x - 0.0011361) / 0.068512 & \text{if } x < 0.0059569 \\ 0.27861 \times \log_{10}(x \times 10.6723 + 1) + 0.33122 & \text{otherwise} \end{cases}\]- Middle Grey (18%) → $y = 0.32$ (32%)
- Dynamic range: ~17 stops
- Shift from 39% to 32% creates room for 2.5 extra highlight stops
Log3G10 (RED IPP2)
\[y = \begin{cases} 15.1927 \times x & \text{if } x < -0.01 \\ 0.224282 \times \log_{10}(x + 0.01) + 0.444666 & \text{otherwise} \end{cases}\]- $x$ = scene linear exposure
- $y$ = Log3G10 code value (0-1)
- “3G10” = 3 stops below middle grey, 10 above = 13 stops total
- Simpler structure than LogC for cleaner ACES transforms
A.5 Camera & Display Gamut Primaries
Camera Gamuts
| Gamut | Red (x, y) | Green (x, y) | Blue (x, y) | White |
|---|---|---|---|---|
| ARRI Wide Gamut 3 | (0.6840, 0.3130) | (0.2210, 0.8480) | (0.0861, -0.1020) | D65 |
| REDWideGamutRGB | (0.7803, 0.3043) | (0.1216, 1.4940) | (0.0956, -0.0846) | D65 |
| S-Gamut3.Cine (Sony) | (0.766, 0.275) | (0.225, 0.800) | (0.089, -0.087) | D65 |
Display Gamuts
| Gamut | Red (x, y) | Green (x, y) | Blue (x, y) | White |
|---|---|---|---|---|
| Rec.709 / sRGB | (0.640, 0.330) | (0.300, 0.600) | (0.150, 0.060) | D65 |
| DCI-P3 | (0.680, 0.320) | (0.265, 0.690) | (0.150, 0.060) | DCI (0.314, 0.351) |
| Rec.2020 | (0.708, 0.292) | (0.170, 0.797) | (0.131, 0.046) | D65 |
Additional Resources
- CIE (2004). Colorimetry, 3rd Edition. CIE Publication 15:2004.
- Poynton, C. (2012). Digital Video and HDTV: Algorithms and Interfaces, 2nd Ed. Morgan Kaufmann.
- SMPTE Standards Collection: ST.2065-1 (ACES), ST.2084 (PQ), ST.2067 series
- ITU-R Recommendations: BT.709, BT.2020, BT.2100
- ARRI, RED, and Sony technical documentation (referenced in Module 6)