Rules for Creating Developable Hull Surfaces

Assume we are working in a mathematical X,Y,Z coordinate system where X define length, Y defines width, and Z defines height.

1.  A straight line can be defined by two points in space or by one point and a constant slope.

2.  A curved line can be defined by an algebraic equation(s) relating X, Y, and Z to each other.

3.  Projections can be in the form of a plane, a cylinder, or a cone.  The cylinder and cone do not need to be circular in cross-section and do not need to form completely closed surfaces.

4.  A plane can be defined by holding one coordinate constant or by using a constant slope in space.

5.  A cylinder can be defined by constant slope projections from an algebraically defined curve.

6.  A cone is defined by projections from a single focal point to a defined curve or other series of selected points.

7.  Projections can be combined by using common ruling lines (lines which are defined in both adjacent projections).

8.  Surfaces can, thus, be built up by combining multiple projections.

9.  Surfaces can be combined by common predefined border lines (straight or curved) or by intersections, either calculated or by graphic solution.

10.  Locating a conic focal point close to a limiting curve or intersection will accentuate curvature in that localized area.  Be sure that is that what you want.

11a.  Cross section shapes (for frames) can be found by holding X constant and solving for Y and Z.

11b. Waterlines can be calculated by holding Z constant and solving for X and Y.

11c.  Longitudinal sections can be calculated by holding Y constant and solving for X and Z.

12.  Mathematical curves defining the relation between X to Y and X to Z can be separate equations.

13.  Lengthwise, a defined curve can be made up of differing segments as long as they have a common slope at the point of juncture.

14.  The easiest curve I have found to use is a parabolic or trajectory curve with coefficients chosen to create the length and curvature desired.  The slope at any point is easy to determine.  The length to any point along the curve can be calculated using a derived formula.

15.  When creating a hull form, defining the midships cross-section shape and major chine is almost always the favored place to start.

The general form of equation I use is the following: Y=B(1-X^/L^) where "^" indicates the term squared. Y is half width, X is length measured from amidships forward, B is the maximum beam, and L is the overall half length.

The exact equation used for the forward width of my kayak is Y=8.1(1-X^/67.5^) where Y is calculated every 7.5 inches from X=67.5 to X=0 resulting in values of Y=8.1, 7.7, 7.2, 6.5, 5.6.... and so on. Note that the decreases are 0.1, 0.4, 0.9, 1.6, 2.5 and so on until Y=0 at the completion of nine segmental computations (9^=81).  The slope at any point on the curve is (the change in Y multiplied by 2 then divided by the change in X).

Thus, the end slope of this curve is 2x8.1/67.5 or 0.24, this can also be expressed as a change of 0.9 inches in width for every 3.75 inches change in length.  At the end of every attempted curve of a wooden board, there is always a short segment at the end where no fulcrum exists to exert torque to continue the curve to the very end and the end piece is straight (but angled). 

I then add that short segment of straight line (wooden plank) to the end of my curve, resulting in an overall length of 71.25 inches and overall beam of 9.0 inches.  The result is the equation and width offsets for the major chine.

At this point, you have either figured the concept out or are totally confused; I will stop here.