What a great question. I wish my Geometry teacher had detached himself from his protective podium long enough to ask us to open our eyes and have a look around instead of pouring sand from the desert called proofs down our throats everyday.
Geometry is everywhere in surfboard construction. It starts perhaps when we sight down the stringer of our fresh blank and check to see that stringer is suggests a straight line. If not, we make some marks on the two ends and draw one ourselves to serve as a reference line. From here, depending on whether we have a template or not, we draw a series of reference lines perpendicular to the stringer or our reference line we drew because the stringer was cattywhompus. We mark our nose, widepoint, and tail widths. Then, we place our template on one side of the stringer and trace out half an outline before reflecting in our mirror line stringer, flipping the template, and tracing the other half of our outline. From here we cut out our template. We then use a t-square and sanding block, planer perhaps, to square up or true our outline.
Perhaps the board we are building was inspired by a picture we saw in a magazine or on the internet. If so, we used our knowledge of geometric proportions to enlarge the picture into a functional outline. Perhaps we are building a front foot driving hull and notice how, on some of the most fine tuned examples, all the curves, outline, rocker, and hull, appear to coincide around something called The Golden Section, approximately 2/3 up from the tail. Same for many of the boards that are meant to be driven with the back foot, except for 2/3 back from the nose.
Once we have our outline cut out we might stand it up and check the outline. Is is too parallel or not parallel enough? We keep in mind for whom the board is being built and maintain a certain volume in the blank. We start shaping the rocker and notice the entry rocker on many boards accelerating 1/3 back from the nose through the tip. We’ll adjust the crown of the rocker to coincide with the surfer’s preference for front foot or back foot surfing. Depending on what type of surf the board is being desinged for, we’ll measure 1/4 up or 1/3 up from the tail for our tail rocker break. The curves then blend, we don’t necessarily want 3 different planes in our bottom curve.
We might think about how to increase the surface area of the bottom of our surfboard without making it “too” wide, and reveal channels, or concaves, or hull. If we’re shaping a hull we might sight down the straight line stringer and measure out to The Golden Section, make a mark and have our hull accelerate from there out to the rail. Same thing for the hump in the s-deck or for marking the start of a step-deck.
Once the board is shaped it’s then up to the glasser to place the fin box perpendicular to the stringer or perpendicular to the plane that is the bottom of the surfboard. Perhaps the board is a tri-fin, suggestive of a triangle, and the glasser must place the center fin perpendicular to the stringer while the side fins are placed at two different angles, toe-in and cant.
Now, if we’re big time, once the board is finished, we might depend in part on advertising to market and sell our surfboards. We notice in the best ads, the ads that seem to whisper in our ears and encourage us to reach for and extract money from our wallets, that our boards are placed in areas on the page that correspond in one way or another to The Golden Section.
This barely scratches the surface. There are others here with much more knowledge who might add a thought or two.
Good luck with your project.