I recently read an article concerning conversion of developable surfaces into flat sheet patterns. In previous posts, I really havent discussed how to do that using a mathematical approach. It can be done; I used this approach when I built a 20 1/2 foot sharpie design many years ago. The only reason I havent expanded on the subject is that, once I have a hull framework constructed, it is easier to simply make a stiff rosin paper pattern by laying a roll of paper over the framework and tracing the edges. But, for those who do not want to construct a full hull framework, being able to go directly to laying out patterns for sheathing may be worthwhile.
The method utilizes what may be best called external triangulation. It is external because we are not calculating distances just between points within the design surface but instead are using our conic apex, located at some distance from the surface borders, as a reference point for many measurements. Using the coordinates of the apex in conic development and the coordinates of consecutive points along the chine of the hull, we can calculate the length of radiating (ruling) lines emanating from that apex. The chine of the hull is created using a mathematical equation; thus, by integrating this equation (the power of calculus) we can find an equation [it is a long but standard equation] to calculate the distance along the curve between any two designated points on the chine. Those points along the chine provide the third distance in creating a series of triangles which, when plotted in sequence, will give us the shape of the developed surface. However, part of the chine curvature may be contained within the outline of the surface itself; thus the sequential points of the chine are best first plotted along a fair batten which is then laid in place and connected to the ruling lines from the apex. For parallel projections, instead of an apex, we would be plotting parallel ruling lines. If you calculate the distance between two sets of coordinates in a developed surface NOT connected by a ruling (straight) line by using simply the formula square root of (x squared plus y squared plus z squared), the distance will be distorted by lack of consideration for the curvature present.
In conclusion, yes, I can create flat surface patterns for sheathing. But is it really worth the effort when a roll of rosin paper will give me the same thing?
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