## Friday, May 01, 2020

### Creating a Developable Surface

This has nothing to do with any computer program and its underlying algorithms; instead, let's consider basic geometry, some algebra, a touch of trigonometry, and perhaps some optional calculus.  Developable surfaces cannot include curvature in multiple directions at a single point.  A straight line must pass through any point on such a surface; these are called "ruling lines".  A unique line can be defined by either two distinct points in space, or one point and a defined slope.  Surfaces can be created by projection from points: parallel in two dimensions (flat), parallel in three dimensions (think of a cylinder), or conic- radiating from a single point which I will call the focus or focal point (FP).  Multiple projections can be connected to create a fair surface as long as those projections are linked by common ruling lines.

In this sketch of the topsides of a previous boat, I used four projections. I used a parallel entry for only a short distance.  Then I switched to a conic projection (using a common ruling line) to create a flaring bow.  Next, I moved to a different conic projection for a transition to stern tumblehome.  Finally, a parallel projection was used to finish the tumblehome.  You can see how projections are combined, using common ruling lines, to create varied surfaces and the desired shape.  All projections were done from defined coordinates along the chine or extensions of the chine.

First, let's concentrate on ruling lines.  On a segment of any line between two points in an X-Y-Z-defined space, the change in X is proportional to the change in Y and the change in Z according to the slope of the line.  The slope can be calculated as (X1-X2)/(Y1-Y2)/(Z1-Z2) if we know the coordinates of two points on the line.  We can calculate the coordinates of another point (or multiple points) on that line (X3,Y3,Z3) using (X1,Y1,Z1), the calculated slope, and any one of the X,Y,or Z coordinates for point 3.  Let's assume we know X3, then the formula for Y3 would be the following:

Y3= Y1-(X1-X3)(Y1-Y2)/X1-X2)
Z3= Z1+ (X1-X3)(Z1-Z3)/(X1-X3)

In words, the change in Y is proportional to the change in X, and that change is then subtracted (or added, depending on direction) from the initial Y value.

Here comes the big concept which makes this whole system work:  In order to make this work, we need some defined points in space to start describing our desired surface.  We can't just draw a line on paper and say that this defines one edge of our surface.  We need numerically exact discrete points.  A mathematically-defined curve provides the solution.  I generally use a form of trajectory curve, also called a parabolic curve, for this purpose.  An example below:

Y=14.4-(72-X3)squared (14.4) /5184          (5184=72 squared)

Z= 4.8+(72-X3)squared (5.76)/5184

The X-Y slope at any point =2(14.4-Y3)/(72-X3) where X is measured from the bow and Y from the midline

If you integrate the curve (which I will not do here), you can calculate the length of the curve or a section of it.  I actually did the integration calculations for the hull I am building.  It eliminated the need for a strongback frame as the parts were self-aligning.

Using these particular curves, I insert values of X3=66,60,54,48,.... etc. and solve for Y and Z values at each of these X intercepts.  This provides a table of exact coordinates, spaced every six inches along a chine curve.  I could calculate values every three inches if needed, but the extra accuracy is unneeded.

Let's say I am designing a guide boat with a frame every 12 inches (which I did).  We have the chine; what should the bottom look like?  Displacement, stability, and construction considerations help decide this.  I decided on a midships deadrise slope of  1:3 and a plank keel 7.2 inches wide (can use standard 1x8 lumber).  Now for the projection.....

By setting Y=0  (width=0), we can solve for X and Z (length and height) and plot the bow profile  Note that as the focal point (FP) is located nearer to the surface being described, the resulting projected curvature increases.  This is generally true.  Note also that the ratio of Y to Z at the FP is the same as the midships deadrise, z/y=1:3.  This has to be true to create a common ruling line.  Shifting the X coordinate fore or aft also affects the slope.  If the X coordinate matches the stem half angle, it will create a plumb bow.

By setting Z=1.2, we can solve for X and Y (length and width) and create the plank keel profile with a width of 7.2 inches.  Using a conic projection results in a longer keel with slightly more abrupt entry profile.

We have a choice of projections for creating the frames at X=,12,24,36, etc.  This drawing was created using a projected parallel profile, X:Y:Z= 6:2.4:0.8.

These frame profiles were created using a conic projection.  the FP is located at (-20,-9,-3)  Recent experience suggests that the conic projection may be superior in this application; it depends on what characteristics you value most.  A conic projection appears to shift some of the surface curvature near the stem from the Y-Z plane to the X-Y plane.  At least that is my current best guess.  The parallel projection creates a more aesthetic bow profile curvature, but a shorter waterline length.

Demonstrating the alignment of frames using the plank keel, and the calculated length chine and sheer as references.  These pieces will only fit together in one relationship when aligned.  No strongback needed.

Because this hull is designed as a developable surface, compound curvatures are avoided when planking.

At 8:17 PM, Unknown said...

I'm having trouble understanding how you drew the bow profile. For instance where you have FP=(14,6,2). What equations are you using to come up with the curve? Thank you for sharing your work on developable surfaces.

At 5:58 PM, Wayne said...

I appreciate your interest. First, I'll check to see if I have made any errors. Then, I'll post an answer to your question. Give me an hour or so to create a clear explanation.

At 5:30 PM, Unknown said...

Very much appreciated! It's been forty+ years since I had this in school, and this math page helped:
https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfLines.aspx
I may have a few more questions later, after I review some more!

At 4:15 PM, Unknown said...

Hi, same person as in above comments, I was able to draw the bottom for a boat I had been playing with, by solving for Y=0, as you have shown in your example, worked out great, thank you.
In drawing the sides, when using a conic projection, like you have shown, is there any simple way to solve the intersection with the sheer? I wasn't sure how to do it, so I solved for a fixed Z that was above the sheer, and then I can do it graphically. Making sure I'm not missing anything, thanks again!

At 6:21 PM, Wayne said...

To Unknown
Glad I could help. In my example, Y decreases and Z increases toward the ends of the hull during calculations and that is what I used to establish my sheer. However, in runabout hulls I have built, when I wanted a different sheer & deck curvature, I would create a mathematical sheer description (fore & aft plus from centerline to outward to full beam) and then graphically find the intersection, as you did. Because I only had to plot those few points near the intersection point, I would graph at two or three times the actual size which results in great accuracy in fixing the intersection.