Sample Developable Design calculations

In 2008 I wrote a short paper on a mathematical method of developable hull design (which is included in this blog, see entry of Dec. 2008).  I tried to keep it short and discuss only general principles, but some subjects are difficult to understand without providing examples.  Without any examples, I don't think it was very meaningful for most readers.  Recently the subject came up again, concerning the difficulty of developable design and the paucity of explanatory material.  Last night, I wrote a short paper concerning a stepwise approach and actual sample calculations involved in creating a hull shape.  The numbers used are from the runabout building in my shop currently.  The paper is not a comprehensive "how to" but hopefully will make the subject more easily understandable.

Sample computations for Developable Design

To illustrate the generation of many exact points along the chine curve, the following is a list of some coordinates of the chine at 7” intervals for the anterior chine of the hull I am now building:  (0, 0, 24.48), (7, 3.2, 22.56), (14, 6.3, 20.7), (21 ,9.2, 18.96), (28, 11.9, 17.34), (35, 14.4, 15.84), (42, 16.7, 14.46) ,.…. until (119, 28.8, 7.2).  This series is generated from the parabolic curve Y = 28.8- (119- X)squared / 490 and also Z = 0.6(28.8-Y) + 7.2 and is valid for values of X between 7 and 119. 

Where did these equations come from?  I want a boat about 6’ by 18’, so I picked a length of about half that which I could evenly divide into 16 segments [more segments = more accuracy]; 16x7 or 112” will suffice, next add a short straight 7” segment onto that.  Have you ever noticed that when you bend a batten, the curve does not extend all the way to the end of the board?  At the batten end there is no fulcrum to apply torque.  Thus, last few inches have no bend unless confined along its entire length.  So I always add a straight segment at the end of every curve.  A 6’ beam gives a 36” half beam; minus some width for flare of the topsides (4”) and an allowance for a chine flat (3.2” at maximum point) and you end up with 28.8” chine beam which is 25.6” of camber in the 112” length and 3.2” of offset along the 7” straight end segment.

To create the forward keel projection, I most frequently use a parallel projection.  A conic projection, with the apex of the cone forward, will create sharp curvatures in that area.  It may be what you want, but it will also be hard to plank.  A conic apex amidships will give a mild curve at the keel, if that is what is desired.  The parallel projection has a controllable curvature and is easy to calculate.  We already have defined a 28.8” chine beam, and picked a 7.2” deadrise to go with it [based on estimated displacement].  So deciding on the slope for this parallel projection from chine to keel only involves selecting an X intercept from the maximum chine point (119, 28.8, 7.2).  The further forward we select this point, the more pronounced will be the forefoot of the keel with sharper curvature.  I chose a point 52.5” forward of maximum chine beam.  52.5/7.5 = 7; 28.8/7.5 = 3.84; 7.2/7.5 = 0.96; thus we have our slope, X: Y: Z=7: 3.84: 0.96 which coincides with the 7” segment intervals.

Now all anterior keel intercepts can be calculated to define the keel.  At the keel, Y=0, and we solve for X and Z.  X@keel= X@chine- (7/3.84)Y@chine and Z@keel= Z@chine– (0.96/3.84)Y@chine. Aft of the point X=119”, the keel is straight and Y=0 & Z=0 for the keel.  After finding the shape of the keel, we next move on to the shape of the transverse frames below the chine.  To calculate transverse frame shapes below the chine, the same slope or ratio is used.  We set X = 14, 28, 42, etc., or whatever other frame locations desired, and solve for Y and Z.  For every 7” forward we project a chine coordinate, the Y dimension will decrease by 3.84” and the Z dimension will decrease by 0.96”.  Simple math creates the entire shape of each frame.  Just connect the dots.

Creating topsides with some flare to the bow, transitioning into a tumblehome stern, seems to be best accomplished with a conic projection forward, linked to another conic apex further aft to create the transition, then a parallel projection extending aft to finish the tumblehome contour.  Conic projections involve finding a third point on a line give two defined points.  Rather than discuss an entire design, I will list selected apices and show sample calculations.

The apex of the first cone was selected at X, Y, Z = (56, 39, 70.2).  This will give a bow angle which matches the forefoot of the keel and provide moderate flare to the topsides not to exceed a 6’ total beam.  A sample calculation would be to calculate the Y and Z intercepts at the frame location X = 98 for a line between the apex and the chine coordinates (119, 31.9, 7.2).  The calculation is Y = 39– (98- 56) (39– 31.9)/ (119– 56) = 34.27 and Z = 70.2– (98– 56) (70.2– 7.2)/ (119- 56)= 28.2.  The relation is that the change in any one coordinate of a point on a line is proportional to the change in any other coordinate.  Since we choose our X intervals, we can then find Y and Z.

When calculations are complete for X between 0 and 126, we select a second apex (91, 35.5, 38.7) which lies on the ruling line, halfway between the first apex and the point of maximum chine beam (to the outside of the chine flat) which is (126, 32, 7.2).  This new apex is then used to calculate points aft to the chine location (189, 32, 7.2).  From there a parallel projection is used with the slope 7: 0.25: 2.25 and defined points every 7” along an extended chine equation to X= 294.  Although the actual chine ends at X= 213.5, the extended portion will determine the shape of the tumblehome at the stern when projected forward.  As more curvature is included in this extended curve, the tumblehome will also increase.

The entire shape is not yet designed, but sequence and type of calculations needed should be understandable.  The results, when finished, are full-size measurements in three dimensions with fair curves and excellent accuracy using simple tools.  Enough offsets are generated that all you have to do is connect the conveniently-spaced dots.

To the deck and onward

With the interior almost complete, the deck has been my focus recently.  First came an initial sheathing with 6mm okoume ply.  Then a second layer of okoume ply was shaped to the deck edge (about 9" wide) and bonded in place.  This was then stained with a deep brown water-based stain, called "Expresso".  Then, major parts of the fore and aft decks were covered with 6mm thick African mahogany planking.  Next a 1/2' by 1/2' channel was refined with a router at the deck-topside junction (sheer) and an appropriate piece of African mahogany was bonded into this sheer joint and sanded to provide a smooth rounded lip to the deck edge.

At this point the stained section of the deck needs another coat of Expresso stain and then it will be ready to be sealed with fiberglass and epoxy.  The recessed portion of the stern, incl. splash well and engine mount, will also need to be glassed.  But we can now also start thinking about the windshield design.  Today I cobbled together a quick mock-up, using cheap ply and duct tape, of a proposed windshield design.  Looks okay for a first attempt, but undoubtedly will be modified in the final version.  We are thinking of having the windshield frame fabricated in metal.  That would require finding a metal fabrication shop willing, equipped, and reasonably priced to do the job.  That may be a task in itself.

My biggest problem holding things up is that I need to sell my current boat to make room in the garage.  The new boat is bigger, heavier, and would benefit from a more powerful engine.  The existing engine, controls, hydraulic steering, and trailer could be transferred to the new boat, but would make the existing boat even less marketable.  When I built sailboats, they were so much cheaper to equip that I could sell them much more cheaply.  I never count my labor as a cost; it is just something I enjoy doing.  It replaces golf and a gym membership.            

Slow but Steady Progress

 I have been working on the boat steadily, but it is a project I do in my spare time, so progress has been slow.  My job, snow skiing, a couple trips elsewhere, honey-do tasks, and such come first.  Work has been focused on the boat interior.  I bonded another layer of wood to the inside bottom of the hull.  Where it would show, mahogany planks were used; where it wouldn't show, 6mm. okoume plywood was used; and in the bow where to curvature is very significant, a layer of 8 oz. fiberglass was bonded into place.  I built in five shelves in the cockpit under the side decks. The instrument panel and the throttle mounting plate were designed and built. 

The seats were designed and built; they are sectional for ease of removal and will be upholstered when we get closer to completion.  I ordered the fuel tank (18 gallon with gauge sending unit) which will be mounted under the front seats and built a fuel tank support platform.  Removable floor panels were built as well as a shelf to go under the foredeck and a battery support for under the aft deck.   Then the inside of the hull was cleaned up by removing resin blebs and sanding.

Most recently the entire interior has been stained.  Next will be the application of three coats of varnish to the interior.   For staining and varnish, it is very preferred to keep the shop windows open for ventilation.  We have had snow as recent as ten days ago, so the weather hasn't been warm enough for open windows.  To stall for time, the plywood panels which will be the initial layer of decking have been already cut to exact size and are ready for installation as soon as the interior is done.  Also need to cut the flotation foam which will be under the aft deck around the OB engine.

When you design your own boat, each step in the building process requires finalizing the design as well as the actual building.  You learn from your previous efforts and hope to improve.  Functional parts I design myself, but for esthetics I get my wife's input.  She is not a boat person, so her viewpoint is unbiased by traditional nautical convention.  With a convenient shop space and no deadline for completion of the boat, the building process is very enjoyable, a combination of engineering, artwork, and exercise.  I already am already working on a design for a "next" boat.

 

   

Finally the hull is turned over!

I didn't realize it has been so long since I updated my current boat building.  When I went to my local lumber store(last summer) to buy planking stock, I noticed that they had some beautiful African mahogany for not-much-more than the cost of cypress.  It also came in sizes more amenable to re-sawing with less potential waste, so I went for it.  My wife has pretty much blessed the project, saying, "Don't go cheap; get whatever is best."  She liked my last boat and the compliments it gets (here in Colorado boat-building is quite rare, so almost anything impresses the onlookers), so she fully supports my current project.

At home I used a bandsaw to slice and re-slice the mahogany into strips of about 5.5 mm. thick by 3 inches wide with assorted lengths so that I could stagger the plank junctions.  A bandsaw leaves some roughness and thickness variation in the planks which I corrected with a planer.  Planks were laid out on the rosin paper patterns, which I had used to outline-shape the plywood sheathing and numbered in sequence for placement on the hull.  Then the task was to tack the planks in place on the hull with temporary staples after coating the hull and plank underside with a thin layer of slightly thickened epoxy.  Mahogany is heavier, denser, stiffer than cypress.  I discovered that the staples I had used for the cypress on my previous boat were not quite strong enough to penetrate and hold the mahogany in place on the ply sheathing, but I switched to using brads (airgun, pad under the head for easy removal later) where necessary and got the job done.  I was glad that I hadn't made the planks any thicker.

 Next is (of course) a preliminary sanding to get rid of any resin blebs and planking variations.  I used an orbital sander with a canister to collect the fine mahogany dust, as well as a long board.  Then came a layer of fiberglass cloth (8 oz. on the bottom, 6 oz. on topsides) and five coats of epoxy resin to completely bury the cloth.  Did I mention that sanding epoxy is my least favorite task?  The bottom wasn't too difficult because the surface is almost horizontal, and I was able to use filler to achieve the correct non-slumping mix of resin.  So, the resin coat came out fairly even.  But the plan was to finish the topsides clear to show the underlying mahogany grain.  This meant that I was working on a somewhat vertical surface and could use no fillers.  Even when the epoxy coats were quite thin, runs could happen easily and show up after you thought you were finished and had walked away.  Those tiny little brad holes that I thought would cover easily didn't always fill like I had hoped.  Thus, I sanded and re-coated several more times.  I used the fine mahogany sanding dust (from the sander canister) as a filler to a limited extent.  The color match was perfect, but the particles weren't uniform enough to create a true smooth layer.  I should have used a flour sifter.  Lessons:  1.  Avoid clear finish on the vertical hull surfaces.  2.  Fill every imperfection, no matter how tiny, before applying epoxy.  3.  When creating your own wood flour, use a fine sifter to get uniform particle size.

With the winter holidays coming up, it was time to paint the bottom and let it thoroughly dry for several weeks before turning the hull over.  I washed the epoxy to get rid of any wax, lightly sanded, then cleaned the bottom again and let it dry.  The paint went down beautifully in a nice even coat.  Then, I waited for it to dry.  We went away for Thanksgiving, and when we came back the bottom was still tacky.  That paint never did dry on the hull; it dried on the mixing paddle and in the application pan, but not on the hull.  I scraped and sanded to take all the non-completely hardened paint off.  I had heard that with epoxy, you didn't need a primer.  Well, now I put down a primer coat, then paint from a fresh can.  And it dried nicely.  I use a laser to mark my intended waterline.  I designed the hull; I know its displacement; only a few minutes for me to mark three points along the side of the hull and then connect them with a laser and tape it with painters' tape.  I am sure I will come back to this task for a touch-up and plan to add a boot stripe.

Last night, with Christmas and New Years past and the decorations put away, my wife and I turned the hull over.  First, I constructed a sling; when the hull was supported from overhead, I released it from the building strongback.  Then the hull was rotated in its sling and lowered down unto a cradle I had constructed.  Now comes more enjoyable and creative steps in building.  I have learned to start from the bottom.  Do those things which are most difficult to access before covering over with decking, seating, and other surface coverings.  Next week is the Denver boat show.  It is not too early to start looking at outboard motors, debating between 60 and 75 hp. 

       

Hull sheathing

The new hull has now been sheathed in 6mm marine plywood.  This entailed making eight scarf joints.  A 9/1 scarf slope was used with all panels being beveled using a hand plane.  For me at least, the hardest part of beveling the scarfs seems to be to hold the plywood absolutely flat.  Thin plywood has a tendency to bow slightly either up or down, so I built an 8-foot jig and used multiple clamps to hold the plywood straight and level.  Four of the scarf joints (the short pieces) were bonded on the jig and four were bonded on the hull itself with equally successful results.  I did not want to try to position, epoxy bond, and clamp 19-foot panels by myself on the hull frame.

The hull was designed with flat cross-sectional panel sections located eight feet from the stern with backing plates (3/4" lumber) for the future joining of fore and aft panels.  That way I was able to split the sheathing task into forward 11-foot panels and aft 8-foot panels.   Each piece needs to be positioned precisely after being pre-coated with fresh epoxy; it needs to be done somewhat quickly so that the epoxy is still in an easily flowable and penetrating state, and then cleanup to remove squeeze-out and drips needs to be completed.  Especially at the hull forefoot and topside stern tumblehome, the panels need to easily bend to significant curvatures.

Thus, these panels were soaked in a shallow basin of water for two hours and then temporarily clamped in place until they dried out.  In this way they adopted a pre-warp to more easily conform to the curvature and required much less clamping force when permanently bonded.  The plywood sheathing will provide the foundation for the next layer of hull coating, 5mm thick planking.  This will be followed by fiberglass cloth and more epoxy resin.

The origin of the "Boat Bowl"

A few months ago, my wife showed me a picture of a wooden bowl from a catalog and asked me if I could create something similar. I had a concept in my mind that I thought I would try "someday", so this was the opportunity to use it. I quickly drew and fabricated a bowl to prove the technique. It came out okay, so I made some bowls slightly bigger, realizing that bigger was actually easier when laminating and sanding the inside. Then she asked for a longer, more flared design, and I made several of those, including for a friend who liked the design. Well, as one who enjoys designing and building boats, you may guess what happened next. I took the design for the next boat I plan to build and simplified and scaled it to create a "boat bowl".

Another year, another boat

A new boating season is beginning, and it is time to recharge the battery, refill the gas tanks and get out on the water.  However, after last season I thought of all the ways I could learn and improve from my last boatbuilding experience and create an even better design.  I wanted something of the same size but more closely resembling a classic runabout and with a better integrated sun shade.  After calculating the offsets and sketching out the results, I started drawing out full-size patterns.

At about Christmas, I started putting together wooden frames.  By themselves, the frames look like just a pile of wood scraps, but when you mount them on a strongback, align them using a laser, and start adding the keel and other longitudinal members, it begins to look like a boat.  With the experience of building something similar, I have been able to move forward more confidently this time.  Instead of searching for answers, I know how to handle the design and construction challenges at each step.  Having purchased a new table saw has been a great help also.

I am now ready to make patterns for the exterior sheathing.  The 6mm marine plywood will arrive soon.  I hope to finish the sheathing (plywood, cypress planking, glass and epoxy) and sanding during the summer months when I can leave the windows open.  Next fall will be time for an upholstered interior and other finishing details.  Changes from the previous design include a constant deadrise (12 degree) hull bottom, chine flats, a straight sheer profile with the foredeck raised 5 inches, and reduced deck camber.  This hull is meant for slightly more power / higher speed.


 

First season report

Boating season is about over for this year but it has been very enjoyable. The only changes I have made so far to the boat are to move the throttle control over about 1 1/2 inches away from the steering wheel, to allow more clearance, and to replace the plexiglass windshield with auto safety glass for a clearer view. After trying three propellors, we have boat speed up to 33 mph (at a 7500' altitude). The engine trim control allows us to bring the bow down in waves and trim it up in smooth water for more speed. My wife enjoys the boat which is important. She spent more time driving it than I did on this last trip. That is great; my great joy is to start with a vision, create a plan and then make it a reality. This winter I will start on a new boat.

A Trial Run

We took the boat out this weekend and had a great time. We were part of an Antique & Classic Boat Society club get-together. We were surrounded by beautiful traditional mahogany runabouts. I had changed the engine propeller to one with a more aggressive pitch and the result was more speed- 28 mph. The engine was still achieving 5950 rpm, at the top end of its recommended rpm range, and I am wondering if an even higher pitch will bring the rpms down a little and improve speed further. You can see in the attached image that the boat's bow is out of the water at speed. This is due to the slight rocker in the run. By calculation, the bow should rise about ten inches to achieve its planing attitude, and I don't think that is far off. From a visibility standpoint, the bow rise is not an impediment. At low speed the boat rests at its marked waterline. The ETEC engine is so smooth and quiet, and we are still using our first six-gallon tank of gas. My wife is not much of a boater but seems to really like this one. With the intense sun in Colorado, our bimini top is an essential.

Finally in the water!

Springtime in Colorado. After ten days of cold and light snow, the temperatures finally got up in the 60's, and we were able to launch the new boat this week. Everything went perfectly. I had re-enforced the deck at the bimini support attachment points, put flotation foam in the stern, installed a new battery, and readied my list of supplies. Amazingly nothing was forgotten; everything performed as designed; and the setup-launch-retrieval was easier than expected.

Out of the Shop

I had planned on using beer and thirsty friends to get the boat out of the shop and onto a trailer, but then a I reconsidered the logistics of getting six friends here at the same time with late fall weather cooperating and devised a new plan on the first warm day.   As when I previously turned the hull over, I suspended ratcheting tie-down straps from the ceiling beams and lifted the boat up off its building cradle. Then I moved a small cart under the center of the hull and lowered the boat onto the padded cart. I backed the 8' wide trailer up to the 6' wide shop door; pushed the hull out the door, rolling on its cart, until about 8' of hull length was out the door and on the trailer; then simply finished winching the hull onto the trailer.

Crowley Marine was ready to install the engine, steering gear, engine controls, and wiring, and they recommended a canvas shop for the travel cover and bimini. So, in a short period of time, the boat is now basically ready for the water. Of course, there are always more finishing projects. I need to (1) reinforce the deck underside where the bimini fittings are fastened; (2) place flotation foam in the bow and stern; (3) touch up the cetol finish in spots along the sheer where it was disturbed when the sheer strip was placed. After pricing simple custom upholstery for the plywood-framed cockpit seats, I found that I could purchase quality standard bucket seats for less than half the price. I should get the seats in the next two weeks.

But with winter's snow in our yard and the ski slopes beckoning, that will all wait until warmer weather.

Ready to leave the shop

Painting is not my favorite thing, but I have been doing quite a bit of it to complete this build. I still need to do some touch up but I have been trying to get this boat out of the shop and onto a trailer so I can get the engine and controls installed. Of course, the boat will still need cushions, bimini top, and a travel cover. The weather forecast mentions possible snow next week, so I am just a little late to launch this season.

Finishing the Stern, Onto the Windshield

Finally was able to get the transom and splash pan covered in glass and resin. I wasn't looking forward to this step. It was like developing the pattern for a garment; I had to make patterns for the complex adjoining surfaces using six different sections. Had to decide which edges to put tabs on so that I had overlapped joints, and had to decide the exact order of placement to minimize interference. All concave junctions for adjoining surfaces were first filled with a fresh fillet of resin putty to prevent dry spots where the cloth couldn't adapt to the abrupt bend. Most of the surfaces were vertical, so the cloth needed to be taped in place while I spread resin and tried to avoid drips and runs. Because the glass cloth distorts easily, I had to note reference points for careful placement. This covering helps not only to protect the wood, but also to add considerable strength to this area which will receive maximum stress from the engine. To my relief, this session went well. Next up will be covering the entire deck with glass and resin.

Thinking ahead, I started to finalize the windshield design. No experience, so I started looking at pictures of old launches to see how it was done traditionally and what might work on the narrow, cambered foredeck. I sat in the cockpit and tried to imagine how a bimini top will join to the top of the windshield. Because the entire boat surface is developable and mathematically created, I was able to select x,y,z coordinates for all points defining the windshield and then calculate lengths and angles for creation of a cheap plywood mock-up of the windshield. Looks pretty good, so now to select real wood and put together the actual windshield. This will be a nice side project while waiting for resin to dry on the deck.

Wood Decking

The plywood under-decking was bonded in place with temporary staples to hold it until the epoxy hardened. Then the plywood was trimmed around the sheer and cockpit. The process was repeated with cypress planks, about 3/16" thick. Now the planks have been sanded. Finally starting to look like more than just the hope of a boat. I need to glass the stern, recessed transom, and splash pan before glassing the deck itself.

Interior Details

Sometimes you can be working diligently on the boat, and progress seems slow. This is one of those times. I want to put the deck on (a big visual change), but first I needed to consider everything under the deck while access was good.  Thus, I have been working on the instrument panel, floorboards, seats, and interior finish. Also placed reinforcements in the frame where deck cleats, chocks, and bimini supports will be positioned later. One photo shows the deck plywood panels resting in place. Those will be bonded and stapled (temporary), then a layer of cypress will be added, then 6 oz. fiberglass. And then it will get painted. I know that the cypress wood grain is beautiful, but this is Colorado, and the sun's intensity would vaporize that varnish. I think I have selected paint colors. I have made an engine selection; expect to use hydraulic steering because of its compact installation; and need to select gauges for the instrument panel. I am debating a couple alternatives for the windshield design; I'll be making some mockups to help visualize the effect.

Finally the Hull is Upright

Progress has been slow due to the intervention of ski season and work, but today finally I was able to turn over the hull so that work on the deck and interior can begin. In the past few months I have been sanding, putting down epoxy and 6-ounce fiberglass cloth, and then painting. All that just takes time. On the vertical surfaces each coat must be thin to avoid runs but the epoxy must be built up enough to fully fill the weave and provide a sand-able surface coat. Because the cloth and resin are clear, it is difficult to spot small surface irregularities. I chose Pettit Easypoxy for the bottom of this to-be-trailered hull. It went down fairly well with a roll-and-tip technique; the critical element seemed to be getting each coat just the right thickness. I used a laser to strike the waterline, connecting three points on each side of the hull which I estimate to be about where the hull should float. I may later have to experiment in placement of the battery and fuel tank to achieve proper flotation. On the topsides I elected to use Sikkens Cetol. It seemed easy to apply. The directions say to apply it "liberally" for three coats. I got a bit sloppy (lighting wasn't good enough) on the second coat and experienced some sags, but it should be easy to fix.

The hull was easy to turn over. I had anticipated using beer and multiple neighbors to get the job done. Instead, I used C-clamps to suspend ratcheting tie-downs from the overhead. Tightening the tie-downs lifted the boat up in a suspended sling. Rotating the hull in the sling is fairly simple. I removed the strongback and substituted a simple plywood support cradle which I then lowered the hull onto. Although I didn't weigh it, I noted that the hull is light enough for two people to lift at this point: very good for an 18.5-foot hull.

I have been looking at outboard engines. All the dealers are accustomed to fiberglass hulls which weight much more than the wood-epoxy. Thus, they have been recommending at least 60 hp, but I think that 40 hp will probably reach my powering goals.

Planking the hull

Planking the hull has been fairly straightforward. First, I had to assemble my new band saw and learn to properly adjust and check it. That proved a bit more difficult than expected since it was shipped to me minus a screw (which took a while to discover) needed for proper tensioning. I also needed to choose which wood to use. Surprisingly, I found a local source in Colorado for cypress at moderate cost. I resawed the boards into a 3" by 3/16" cross-section. Most boards have some kind of curve to them; so, I had to be careful to mark and maintain a straight line when laying them out on the curved hull surface. Then it was simply a case of coating each board and the underlying hull plywood with epoxy and temporarily stapling them in place with minimum mess. I used a bit of excess to make sure that they were thoroughly wetted and bonded which then required that I sand the entire hull to remove blebs and discontinuities.

initial hull sheathing

Once all the frames and the anterior keel are mounted and aligned on the strongback, the entire framework was then faired so that panels would lay flat instead of merely touching one edge of each frame. This provides more panel support and more bonding/nailing surface. Once faired, rosin paper patterns were made for each proposed panel of the hull sheathing. I was planning to make thin ply patterns but found that stiff paper was accurate enough in this case.

I had to decide the order of panel placement and the position of junctions between panels. I wanted to place the panels which required the most bending first, and I wanted to join separate panels in places where hull curvature was at a minimum. On my previous boat I scarfed all the panels together before placement, but found that handling an 18+' long, narrow panel wet with epoxy resin, precisely placing it, and getting it securely fastened in place quickly was a difficult task. Fortunately, this hull has convenient panel juncture points where either a butt plate could be placed or a nearby frame could back an in-place scarf joint.

Those plywood panels which required significant bending, the bottom forefoot and stern side tumblehome, were submerged in a shallow basin of water for several hours and then quickly clamped on all edges into place of the frame and left to dry for a few days. The panels were then removed, trimmed for a more exact fit, and bonded into place. As more panels were placed with adjoining edges, clamps could no longer be used on all edges. For these locations I used a nail gun with a 3/4" or 1" 18-gauge nail and scrap 1/4" ply placed under the head to facilitate later removal.

At this point, the entire hull is sheathed, and we can begin to see the full shape of the hull. I now want to cover the entire hull with a second layer; 6 mm. just isn't enough thickness for me. Sure, I could use multiple layers of resin and cloth, but that stuff is heavy and expensive. The strength-to-weight ratio for wood is excellent, so I will add about 4 mm. of thin wood planks; sand it as needed, and finish with a layer of resin and cloth. I've had to research band saws and blades to get the right setup for "resawing", cutting a normal plank into 3-5 mm. thicknesses. The band saw I bought was missing an important set screw, and I was unable to properly adjust it until I figured out that the mechanism was not functioning properly. Looks like it is now properly set, and producing thin planks is my next task. I have picked cypress wood for the planks- available, relatively cheap, fairly light, and rot resistant.

The Design Becomes a Reality

I have already posted on the design of this boat, so I will skip that subject other than to mention that my initial "plan' consists of tables of computed offsets in an X,Y,Z three-dimensional framework. I am having computer issues with importing photos, so they will get posted a little later.

1. First job was to convert those dimensions into full-size patterns for all the frames and the curved keel forefoot. I used stiff rosin paper for the patterns, the largest pattern is about 36" by 68". The offsets only define the outline of each part; thus, each frame was then drawn as a series of boards with joining angles and gussets. A horizontal reference line was drawn at the same level for each frame; this was used to align them all vertically when mounted on the strongback. The centerline can be used to align them horizontally.

2. With full-size patterns it is relatively easy to build the actual frames, all 14 of them. An outboard engine is to be contained in a semi-well which intersects with three frames, so those internal details had to be included in the frame designs.

3. A strongback was built with vertical support posts attached at each frame position. The frames were sequentially clamped in place using a laser beam to align them. Then longitudinal components, starting with the keel, were bonded in place. I really was anxious to get the engine well fit and bonded in place. Being integrated with the deck, hull bottom, transom, and three frames and structurally important to transmit the power of the engine, its fit was critical.

Hull design using simple mathematics

The question of how to create more than just simple angular shapes from sheet materials caught my interest quite a few years ago when I built my first plywood-sheathed boat. To visualize the problem I was facing, I first used a flashlight to project the shadow of a wire of various curvatures and orientations onto a flat screen, a crude form of conic projection. Then I realized that if I could model the curved wire mathematically, the same thing could be done in abstract space with highly accurate results. From this, I developed an approach that has worked well for me and, to my knowledge, hasn’t been described elsewhere. I hope to describe it here to learn whether anyone else is using a similar approach or if anyone might benefit from it.  Instead of attempting a comprehensive description (which would put everyone to sleep), I’ll go through a sample design (for the boat I am now using) which illustrates the approach while ignoring details.

 The technique does not require a drafting board or a computer. An inexpensive pocket calculator is the minimum equipment needed. The approach relies on two simple concepts: First, a chine description is created from which you can pick off many closely-spaced exact offsets. I use mathematical relationships to do this (think trajectory or parabolic curve), but any method which results in multiple well-distributed exact offsets is adequate. Second, for any line in a three-dimensional x-y-z coordinate system, as we move along such a line the changes in x, y, and z are proportional to each other according to the slope. 

Thus, if we know the change in one coordinate, we can calculate the change in the other two coordinates. By calculating the new coordinates of many exact points when projected into a different plane, we can derive the offsets for as many transverse sections as desired, create another chine, describe the entire length of the keel, and establish the sheer. A table of accurate offsets is the first result; then the hull sections can be drawn or sketched on graph paper to visually confirm what has been created. I use X to designate length starting at the forward end of the chine; Y designates width from the center line; and Z is the height above base, usually selected as the lowest point on the keel.

Bottom: With some idea of the desired length, beam, displacement and other characteristics of the proposed hull; the forward chine is created. For our example hull {see related drawing}, a few of the offsets starting at the bow are (0, 0, 20.25), (14, 6, 17.25), (28, 11.6, 14.45), (42, 16.4, 12.05), until we reach (119, 28.5, 6) at the point of maximum beam. A parabolic relationship is convenient to use, but other mathematical relations may be preferred depending on the purpose.  Integrate the equation and you can calculate the length of the curve or a section of it; differentiate the equation and you can calculate the slope at any point.

For this example, it was convenient to calculate offsets every 7 inches; that could be doubled to every 3.5” if necessary but commonly provides more data than needed. The point of maximum chine beam when projected to establish the keel coordinates becomes the lowest point on the keel. Several different types of projections could be used. I chose a parallel projection (constant slope) instead of a conic projection because, for this situation, the differences were small. The parallel projection is also easier to work with and minimizes severe localized curvature. I set the lowest point on the keel to be located at (63, 0, 0). The bow half angle, desired deadrise, and type of hull play a part in choosing this point.

Realizing that a unique line can be defined by a point and a slope, each offset on the chine gives us a point, and the slope to be used is the difference between the maximum beam at the chine (119, 28.5, 6) and lowest keel point (63, 0, 0); that difference is 56/28.5/6.0. We know Y=0 at the keel; thus, we can calculate the values of X and Z for each chine offset when projected to the keel or center line. A plot of these coordinates provides the keel shape {see area B1}. Similarly, we can find the offsets for all planned frames by using the established values of X for each frame (i.e. 14, 28, 42, etc.) and solving for Y and Z. Dividing the X:Y:Z slope by 4 reduces it to 14/7.125/1.5 which simplifies computations with changes coinciding with most frame intervals.

Having completed offsets for the forward portion of the bottom, the midsection is defined next {see area B2}. For a variable deadrise hull, a conic projection for the midsection helps reduce deadrise aft. This can be done by creating a short curve with the desired slopes at the keel; then creating a proportionally larger/longer parallel version of the same curve (in this case, twice as big) to extend the chine aft. Finding the apex of this cone is not necessary; it is enough to know that proportional parallel curves are part of a conic development. Continuing aft, the keel Z coordinate will increase more rapidly than for the chine because it achieves its end slope sooner; thus, the deadrise will diminish producing a flatter run. At the aft end of these curves the keel and chine have parallel slopes which are then extended by straight lines to the stern to produce a straight run of constant deadrise {see area B3}. For this hull, the deadrise is about 60 degrees at the bow, 20 degrees six feet aft, and 5.7 degrees at the stern.  For a constant deadrise hull this entire paragraph can be omitted; the keel and chine are simply extended aft as straight lines.

Topsides: Locating an apex for conic development of the hull side laterally in line with the bow half-angle results in a plumb bow profile. Moving the apex further laterally creates more flare to the bow section. By moving the apex moderately laterally the bow profile angle can be matched to the rise of the forefoot of the keel, avoiding a knuckle in the profile {see area S1}. Vertically the apex should be located high enough above the proposed sheer level to avoid any localized severe curvature to the sheer.  Longitudinally, the apex should be approximately halfway between the bow and midpoint, adjusted forward or aft depending on where you want maximum topsides flare. After trying out a few alternatives, I settled on locating the apex at (63, 40.5, 66).  Because any two points also can be used to define a unique line, using this apex and the series of chine offsets, many lines can be projected and their intersections with proposed frame locations in the anterior hull calculated.

Because a tumblehome stern was desired, a new apex for conic development was established closer to the sheer line once development from the previous apex reached the midsection. The new apex was connected by a common ruling line to the previous apex in a segment of slight curvature preventing transitional distortion of the hull surface in this area. Using the new apex, conic development was continued to the aft end of the chine curvature {see area S2}. This apex also provides one point, and the series of chine offsets provide second points so that many unique lines are defined. For each line, its intersections with every applicable frame location can be calculated to any desired exactness.

The initially established chine offsets, used to create the bottom, do not have to be used as the final chine position. In this case, aft chine is tapered through using a new curve to define the topsides. In creating a tumblehome stern, the aft chine, instead of being a straight line, is modified by a calculated curve bending toward the midline {see area S3}. As with the original chine, exact offsets are calculated every 7 inches, but this new curve extends in abstract space about six feet past the proposed stern. For projection from this new curve, the slope established by the last ruling line connecting the amidships apex with the aft end of the chine curve was used. Thus, this is a parallel projection. For the tumblehome stern, we are using one known point and a defined constant slope. Choice of projections depends on the desired result.

Deck: Because the deck curvatures were to be determined independently from the hull, several possible deck configurations could be constructed depending on the desired boat character and planned usage. After considering the alternatives, I elected to use a fairly simple cambered deck. A transverse deck curve of the desired camber and defined offsets (2.5“camber in a 40” long curve) was created. This curve was projected fore and aft mainly by parallel projection except for those areas where a planned open cockpit would allow slight longitudinal curvature to be incorporated without creating compound curvature. Y offsets were held constant while Z offsets changed slightly to reflect the fore and aft curvature.

The final determination of the sheer location is thus defined as the intersection of the deck surface with the hull topsides. Since both are fair surfaces, the sheer will also be fair.  Note that a cambered foredeck, if projected to the bow, will result in a slight "powderhorn" shape at its sheer intersection with the topsides.  If that shape is unacceptable, "powderhorn" curvature can be reduced by reducing foredeck camber or using a chosen Z coordinate profile for the sheer and departing from a true developable shape in the foredeck.  Because foredeck camber is comparatively minor, distortion will be minimal.

With the shape finalized, I made one final modification. I elected to change the X spacing from 7” increments to 6.75” increments. This shortened the overall hull length by 8.125” to about 18 ½ feet overall. No frame offsets were affected by this change; however, the anterior keel will need to be redrawn due to changed X dimensions.

Summary: At this point all that has been created is a boat-like shape. Many steps remain in creating a complete boat design. However, the only goal here was to demonstrate a method of mathematical projection to create desired developable shapes with more complexity than just a few simple curves. Developable hull surface design can be a logical sequential process focusing on one portion of the hull at a time. Using an initial set of exact offsets for a chine and some algebraic relations, a highly accurate set of offsets for an entire hull can be created. Conic, parallel and planar projections can be combined to create multiple contours in a single, fair developable surface. Modifications can be made, or alternative shapes considered, by changing mathematical coefficients and recalculating resulting values only for those parts affected. Accurate drafting skills and/or complex computer software are not essential.

Creating a form in abstract space removes physical restrictions on scale or degree of curvature. Routinely, dimensions are calculated to at least 0.01” accuracy, and many are exact. Curvatures are fair; no lofting to detect inaccuracies is required. Accurate dimensions are used to create drawings rather than the reverse sequence. A plan can be communicated using only sketches and a table of offsets/dimensions.