Mathematical boat design method

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A technique to create perfect curves is simply to decide how much total offset you desire, say B, and over what length, say L, you want that amount of offset distributed. After subdividing the overall length into some convenient number of fractions, say N, create the curve by offsetting (1/N) squared times B at the end of the first fraction; (2/N) squared times B at the second fraction; and so on until at the end of the curve the offset is (N/N) squared times B. This will give you a parabolic, or trajectory, curve. I have used it for creating arches above our garage doors, archways between rooms, arched cabinet doors, and, of course, as a building block for the design of a number of boats.

As a simple example, consider an 8-foot wide garage door with 8 inches of total desired offset for an arch. Thus, I want a 48 inch long curve (half the total width) with 8 inches of total offset. Subdividing into 6 inch segments, the sequential offsets will be 1/8", 1/2", 1 1/8", 2", 3 1/8", 4 1/2", 6 1/8", and 8".

For boat design purposes, the desired curve can be divided into smaller fractions (instead of 1/8s, I usually use 1/100s) and can be defined in three dimensions. The slope at any point can be easily calculated as 2B/L (where B and L are the offset and length at that point), and the length of the curve to that point is available by putting appropriate values into an integrated equation. You don't need much of a batten to smoothly connect points that are only 3-4 inches apart (if needed) along a ten-foot length. Mathematical conic projection is greatly facilitated when a large number of exactly defined points can each be projected to define the new curve at an intersection of planes. Multiple projections can be easily combined using defined ruling lines.