The Easy Way to Create Fair Curves

Picturing a hypothetical hull type in my mind, the first portion I start to formulate is the major chine.  I have described this process in previous entries, but recently, looking back on those entries, it didn't seem as clear and simple as when first described.  Over many years, I have mentally simplified the process, by skipping steps and not completely constructing a mathematical formula.  Let me illustrate this easy method.

Recently, I saw a photo of a kayak hull with very long slender ends (which appeared to be straight lines) and decided to try to mathematically design a similar shape.  "X" is always the dimension I use for length, starting at the bow.  "Y" is the width offset measured from the midline.  For a kayak or canoe-type hull, I like to create offsets spaced about six inches apart, to be adjusted later.  A hull of this type will have a fine entry, slope x/y about 3/1 to 4/1 ratio.  The overall intended width would be between 26 and 30 inches; at the chine it should be several inches narrower.

With this information, I can start playing with numbers.  I say 'playing' because it usually takes several iterations to find harmony between width, length, bow half angle, and uncomplicated curve coefficients.  I will only discuss the 'Y" width dimension here, but similar considerations occur in creating the "Z" dimension, the upward sweep at the hull entry.

The curve we are trying to develop is controlled by the slope coefficient, curve length, and the number of segments dividing that curve.  For hulls of this type, I often choose a slope coefficient of 0.1 in calculating Y.  For "Z", I often use a smaller coefficient (i.e. 0.0625 or 1/16).  A blunter hull entry will require a larger slope coefficient.  Rather than taking you through the initial iterations I tried, let me describe the winning solution.  

A curve 48 inches long divided into eight (8) segments each six (6) inches long.  The total delta Y (change in Y) will be 6.4 inches.  The end slope of this curve =2(6.4)/48 which condenses to x/y=1.6/6.  Each X segment is 6 inches long.  To the end of this curve, I added three more (constant slope) segments, 18 inches X length and 4.8 inches Y width. 

The total chine (curve plus constant slope) curve is 66 inches long with a total offset (width) of 11.2 inches.  That curve is slightly finer than desired; thus, I shortened the segments to 5.5 inches long- total length 60.5 inches; then I added a straight parallel segment at the midsection of the hull to arrive at the final hull length.

Hopefully, the included diagram below will help clarify the procedure. 

As you can see, our major chine curve consists of two segments with matching slopes at their point of junction.  Here are the X and Y offsets for each segment. (0, 0), (18, 4.8), (24, 6.3), (30, 7.6), (36, 8.7), (42, 9.6), (48, 10.3), (54, 10.8), (60, 11.1), (66, 11.2)

"B" squared (1, 4, 9, 16, 25, 36, 49, 64) is multiplied by 0.1 and subtracted from the total offset of 11.2 inches.  That is a very easy computation to create a fair curve.