I am a cinematographer experimenting with 3D software (Cinema 4D and Redshift renderer) and am trying to figure out how I can understand light in the digital space the same way I understand it in the real world.
My goal is to have a mathematical formula to give me a consistent luminance value on my color chart card no matter what distance the virtual light is from the virtual color chart.
I created a short video showing what I'm trying to achieve. I'm not fluent in math so I hope I'm clear in the video.
https://app.frame.io/f/9c8dbec9-2ff4-4ef0-82a7-1a836ba5dc6f
In the 3D software, I created a light and reproduced the color chart. I set the light at a distance of 48" from the gray card (you can use real world measurements in the software). And adjusted the intensity of the virtual light until the virtual color chart measured at the same luminance: 53 (this number relates to a real-world number -see the bottom of the post if you're interested). I then moved the virtual light 96" from the virtual gray card and adjusted the light's intensity until the gray card's luminance measured 53.
This gives me the following data:
$l_a = 1.944$
$d_a = 48"$
$l_b = 4.48$
$d_b = 96"$
$l_c = 14.54$
$d_c = 192"$
$l_d = 54.8$
$d_d = 384"$
I tried coming up with something on my own:
$$l_b = {l_a \over d_a^2} * d_b^2$$
but it's not returning the proper $l_b$ from the data above.
I appreciate any help you can offer.
Thank you!
PS (how this relates to real-world cinematography) My first step was to get a value from the camera in the real world that I could use in the digital world. This value is 53.
To get this value, I lit and exposed (with a light meter) a gray card which I filmed with my camera (ARRI Alexa Mini with a Panavision 65mm Primo lens). I imported the footage into my computer and measured the luminance value of the image of the gray card. That number is 53. Different camera systems and post-processing will change the value of this number but this is my baseline.
The actual formula would be $$l_b=\frac{l_{a}f(d_b,s)}{f(d_a,s)}, $$ where $s$ is the side length of the square-shaped illumination source and $$f(d,s)=\int_{-s/2}^{s/2}\int_{-s/2}^{s/2}\frac{d^2}{(d^2+x^2+y^2)^2}\,dx\,dy. $$ Your best bet is to evaluate the integral numerically.