An understanding of mathematics seems necessary to scientific research or engineering, but not to the cinematographer as artist. Perhaps this is why math is such a topic of incredible loathing in cinematography classes. I sympathize, after all, because at the end of high school I suffered through Calculus with a kind and brilliant but completely inept teacher. My final school experience of math is inextricably linked to two years of bumbling explanations and my frustrations to understand them. With many years of emotional distance I see that my dislike of the Calculus had nothing to do with the actual subject, but my poor experience. As a result I’ve decided to re-kindle my understanding of mathematics because it has become increasingly necessary in my research.
One semester I had a student who struggled with any homework that involved math. This is not new or unusual which is why I try to spell out the steps clearly in examples attached to the homework. Despite my efforts the student still approached me for help with the flustered response “I never took Calculus!” Each time we encountered math he would echo this statement in knee-jerk response. I always wanted to reply in exasperation “This is trigonometry!” or more often “This is arithmetic!” His personal block toward applying his mind to the math was so great I felt that if I jumped out of a dark alley and shouted “What’s 2+2?!?” he would scream “I never took Calculus!” Many semesters after he graduated I was pleased to hear from another student the following story as they collaborated on a project. Turns out they needed to solve a particularly difficult color temperature problem involving the choice of color correction gel. The student who never learned Calculus remembered that he had learned the concept of MIREDs and together the two of them worked through the problem. The student who relayed the story to me said that his friend’s fear of math had been replaced by exhilaration in recalling that he knew how to solve the problem, he just had to locate the formula and put in the numbers. Apparently, the gel with the MIRED they calculated worked perfectly in their film.
I attribute this change in attitude to math on the student’s part as not due to any brilliant teaching insight on my part. I simply helped him when asked and neither displayed any emotion toward his lack of understanding or fear of the subject, nor any emotion toward the subject itself. I remember just keeping our interactions simple and calm. I found nearly the exact opposite recently while reading David Stump’s Digital Cinematography where he trips over himself to warn the reader whenever there is math.
From pages 15 and 40 of Digital Cinematography by David Stump, ASC
This attitude does no one any favors since treating anything with numbers or formulas as subjects of terror only reinforces this attitude to students and colleagues. A heightened language will invoke a heightened response. I personally encounter this when realizing I can solve a decision on set by an equation or some mathematics. Perhaps I come across as too overjoyed because the response of other crew is sometimes a dismissive "It's ultimately art so why bother with the numbers?" or more aggravating a condescending statement like "Oh look, the nerds are at work!"
Despite the fact I'm not terribly good at math and am less certain what a nerd is and why I am one there must be a benefit to understanding and applying math at work? To better explore this problem I believe two central questions must be considered:
1 – Where is the math in cinematography?
2 – What benefits would be conferred upon a cinematographer who understands the math in his work?
In an effort to keep this post short I feel I should address the first question with a few examples which I have literally sketched out. I will inevitably expand each example out in a future post in which the answer to the second question will emerge.
Exposure
The relationship between light intensity, exposure time, sensitivity of the sensor, and the iris of the lens is held in tight reciprocal bonds. (Except for reciprocity failure, but in this case I’m speaking about the most generally common shooting conditions.) All of these coordinate scales rise and fall by a factor of two, which is very simple mathematic relationship. There is some complexity in the fact that each of these scales are in different units thus requiring a student to be conversant with the meaning of the numbers. Nonetheless, the decisions made to achieve correct exposure are entirely mathematically base.
The graphic method of a cinematographer practicing his scales.
There has been a consistent observation by other cinematography professors and myself that the understanding and application of these scales purely mentally is more commonly seen in those with an analog background. This is most likely due to the fact that analog photography requires attention and memory or else you are punished when the print is projected. Now that exposure can be achieved visually on a calibrated monitor many younger students have trouble keeping track of their technical settings and fluidly relating them in their mind.
Optics
Geometry and trigonometry are the prevalent mathematics in the science of optics. I have used trigonometry a number of times in work to calculating angle of view to focal length. One class a group of students approached with a video that they found online that showed footage of a rock climber scaling a particularly vertical cliff in the desert. The technical information claimed that the footage was shot one mile away and the focal length listed sounded too small to frame the climber so close from such a long distance. Unfortunately, I never kept track of the video but we can analyze a similar situation with this beautiful video of Dean Potter tightrope walking against the Moon that was photographed by Mikey Schaefer. What focal length lens would be required to film this event from one mile away? My work is as follows:
This frame could be accomplished with a piece of optics from the astronomy industry or with the use of a 600mm or 800mm prime lens with 2x or 1.4x lens extenders. Regardless, the use of trigonometry goes beyond just being able to prove to my students the accuracy of a claim found online, but gives one the tools to make the correct technical decision in order to successfully achieve an artistic vision.
Another, perhaps more common use is the geometry needed to calculate angular field of view because of its relation to format size and therefore the choice of focal length of lens. Currently there is constant shifting of format size unlike the analog era where the choice of format was largely confined to 16mm or 35mm. Now digital cameras have APS sized sensors, S-35mm sized sensors, full frame 35mm sensors, and camera manufacturers such as Red are constantly increasing the area of the format as they increase resolution. Moving between these cameras means becoming conversant with problem of picking focal lengths to achieve an equivalent field of view, especially when different cameras are working on the same set. Surprisingly this must still be explained to not only students, but often professionals working in the field.
A quick sketch. The angles are not exactly accurate, but close enough to prove the point.
The illustration above relates field of view to focal length for three different film formats through simple geometry. I particularly like how the location of the lens to the image plane according to its focal length clearly demonstrates role of this number in its effect on field of view. The diagram also reveals the simple factor of two relationship between the three variables. Unfortunately, there is nothing quite so simple with the non-standard choice of sensors camera manufacturers currently are making. Nonetheless, I think an angular field of view calculator using these illustrations as a basis would be far simpler to understand and interpret rather than just a fixed diagram with numbers popping up to give an answer.
Photometry
The measurement of intensity of light and its relation to exposure is governed by the math of photometry. I regret not being taught this subject in school because its power of prediction on set is critical. Young filmmakers are routinely guessing what wattage of unit could produce a given f/stop and much of this is erased with knowing even some simple mathematical relationships using the 100:100:2.8 rule of thumb.
To share a very extreme example here is a sketch of a proposed lighting unit given in my Science of Cinematography Final that is wrapped around a column as an architectural feature on set and contains both blacklight and fluorescents in one unit. The questions surrounding this sketch involve picking gels for the regular fluorescents so they appear to match the blacklight bulbs. Finally, they must calculate the exposure given the photometrics and area of unit (total and apparent). The fact one can not only design complex rigs using lights from other industries but actually predict their behavior on set is incredibly liberating. Who would want to show up to set only to discover that their prized construction fails to illuminate the lead actor?
Depth-of-Field
The math behind calculating Depth of Field is perhaps best left to apps or an old mechanical calculator. I teach the most commonly used DoF equation in class but I discuss its use with great caution because I feel DoF and Hyperfocal distance equations have failed me in my work as a focus puller. This is easy to understand because researching the topic I found that behind the formulas are a set of human assumptions. First, the conventional DoF math requires the cinematographer to pick the Circle of Least Confusion which is a human decision to determine the tolerances and best practices for the calculation. The formula is dumb and needs a human in order to assign it an intelligence in order to provide a correct answer.
From Photographic Optics by Arthur Cox; Expanded Edition, 1971. These equations await your intelligence.
Beyond the fact that one cannot blindly follow this formula many people I talk to are surprised that there exist three different ways to calculate Depth of Field. Above is the conventional formulas used for determining the near and far limits of what is "acceptably" in focus. These you will commonly find in textbooks, but few discuss that the formula for Hyperfocal Distance which must be entered into these focus limit formula can be calculated in two different ways. One method calculates the Hyperfocal Distance considering the image enlargement and viewing angle to the subject as a print or projection, whereas the other only considers the size of the Circle of Confusion on the sensor. I wish you luck finding a chart or finder that reveals which method of determining the Hyperfocal Distance was used!
From The Ins and Outs of Focus by Harold Merklinger
To make matters more interesting the engineer Harold Merklinger published a largely unknown book in 1990 called The Ins and Outs of Focus in which he calculated a third and even more rigorous method than the traditional methods. In his case the calculations are run based on what object in front of the lens a photographer desires to be resolved. The approach is so novel as to turn everything one conventionally learns on its head. The math is more simple, but the precision incredibly ruthless. Reading Merklinger's work is a revelation in how he elucidates the problems inherent in the other formulas as well as novel rules of thumb that work much better than Hyperfocal Distance or the 1/3 Rule. So even research into a topic that seems as concrete as DoF equations can reveal that a dogmatic adherence to a formula one does not truly understand can lead to a less than desirable result.
How Does One Engage the Math?
These are just a few examples to demonstrate that math and geometry are inextricable from the work of cinematography. Some may feel quick to point out that a cinematographer does not need to remember the formulas nor carry a calculator. That may be so, I know very successful cinematographers who create amazing images and could never give you any equations or numbers. However, one cannot deny the need to understand the math because if you are using the tools of the craft you are engaging with the math. Just because a DP decides to expand their Depth of Field by pulling out the ND.6 filter, compensating for the four times greater amount of light by stopping down two stops on the iris thereby narrowing rays of incoming light from objects in front of and behind the object in focus so that these pencil rays of light are now placed within an acceptable tolerance of circle of least confusion without ever calculating the results does not mean he or she is not engaging with the math.
What I have realized in my discussions is that there are as many different relationships between a person and math as there are people. There are those who understand the concepts arising from the formulas in a very practical and often intuitive way. There are those who wish to plunge deep into the details. All of these paths and any in between are legitimate. The only fallacy lies in claiming that knowing the math is unnecessary because then one is denying a level of understanding to their craft. This would be like a woodworker claiming they don't need to become better at hammering a nail, as they hammer in a nail and badly at that. My hope is that those working as cinematographers who dislike math put their prejudice aside and delve into these topics because they can reveal real treasures of understanding. I also hope that the next time you see me on set excitedly solving a problem using my phone as a calculator that you understand I'm just doing my job.