Traditional Model Yacht Design

Preface

This article is excerpted from Thomas Mooreís Build a Winning Model Yacht, published in 1928. Moore was a naval architect and this work captures the state of model yacht design in the 1920ís. The procedures he described can still be used today to produce good-looking designs whose balance enables them to be sailed mostly ìhands offî under R/C. I have edited the work with a light hand, removing mostly references to obsolete class rules. Figure numbers are from the book, and are not necessarily in sequence. In order to provide fast loading for the casual reader, the diagrams are highly compressed. Each diagram is a thumbnail; clicking on the diagram will bring up a much larger version.  You can save that version and view or print it using your favorite picture editing program. If you set the resolution to 600 ppi x 600 ppi you will have an actual size diagram from the book. You can then use your browser's "back" button to return to the text. The one exception is Figure 26, which is in Adobe Acrobat (PDF) format. If you do not have a reader for this, one can be downloaded free from Adobe.

The words to follow are those of Thomas Moore, except for my notes inside [ ].

The Lines

The lines of a boat are intended to show the outside surface of its form. To accomplish this object on the flat surface of a piece of drawing paper, it is necessary to use a little imagination. It is necessary to imagine that our boat is made of a transparent substance that may be cut in the vertical and horizontal direction and even in a diagonal direction by planes and that wherever these planes cut the outside surface of our boat they are distinctly visible as a continuous curved line. The lines of a boat show one-half of the boat, the other half being exactly similar. There are three views, the profile, the half breadth or plan, and the body plan.
 

The Profile

The Half Breadth

The Body Plan

The curved lines formed by the intersection of the surface of our boat and the above mentioned planes are called sections.

It will be noted that in the case of water lines, buttocks and sections, lines appearing curved in one of the three views are straight in the other two views.

Laying Off

First, let us take the case of the prepared design, which will probably be chosen from one of the well known monthly magazines devoted to the sport of yachting. The design, as printed in the magazine, will no doubt be on a small scale, and the problem is to enlarge it to the size required for the model. There are two ways of doing this. If it is simply desired to enlarge the lines a given number of timesólet us say, sixóthe way to do it is to lay out the framework of water lines, sections and buttock lines with just six times the spacing shown in the original drawing. It is best to use the small bow-spring dividers for making this spacing.

The set of lines in Fig. 19 is given as an example.

It is now desired to enlarge the above set of lines six times. Fig. 20, printed on a greatly reduced scale, shows how this is done. We first construct our framework of water lines, buttock lines, and sections. As will be readily seen, the water lines in our enlarged drawing are spaced six times the distance apart that they are in the original small drawing. The same with the buttocks and sections. The next problem is to get curved lines of the profile, half-breadth plan and the body plan correctly superimposed on our large scale framework.

[The text that follows is not completely obsoleted by the ability to enlarge drawings using photocopying equipment. You must remember that old drawings and old publications will inevitably have shrunk or otherwise distorted over the years. It is not at all unusual to have an old full-scale blueprint that has shrunk 5% or more horizontally and be exactly to dimension vertically. A typical way of fixing this is to cut the drawing up at the section lines, lay the pieces out at the exact section spacings, and then redraw and refair the lines on tracing paper. Once you begin doing that you are just about where Moore is when he starts his discussion of scaling up. --EB]

We will start by obtaining points for the sheer line. Measure the distance on the small drawing, between the base line and the sheer line on Station No. 1 by means of the dividers. Set this distance up from the base line on No. 1 Station of the large drawing six times, and make a little dot. Repeat this process on the other four stations and at the extreme ends of the boat. We now have a series of dots through which we may draw a fair, continuous curve which forms the sheer line of our enlarged drawing.

It may be well to state here that in drawing these long curves we will need several long, limber strips of wood, of approximately square section, called splines or battens which may be held in place on the plan by means of lead weights called dogs.

[The weights are also called "ducks" or "whales."  They are somewhat hard to find these days. --EB]

These splines and dogs may be made at home or they can be purchased in a store where draughtsmenís supplies are sold. Splines may also be obtained made of hard rubber or celluloid in lieu of wood. These are useful on the more abrupt curves. If these regular draughtsmanís materials are not available, very good results may be obtained by ordinary home-made pine battens held in place by flat irons, pins, or willing hands. It is obvious that several battens, ranging in thickness from 1/4 inch to 1/16 inch will be useful, the thicker battens to be used on the flatter curves and the thin battens on the more abrupt ones.

[Masking tape also works well as a way of holding down splines or battens. --EB]

Now to tackle the profile. We measure up on each section on the small plan the distance above the base that the profile crosses the sections, and then transfer this distance six times above the base on our large plan and make a dot. A curve drawn through the dots gives us the profile on our large plan and we begin to see how big our boat is going to be. Now let us turn to the half-breadth plan and get our water lines drawn by measuring up on the sections in just the same way as we did for the sheer and profile. The actual terminations of the water lines may be obtained directly from the profile by dropping perpendiculars down to the center line of the half-breadth from the point where the profile crosses the particular water line with which we are dealing. Take the water line just above the load water line (Fig. 20). A Aí is the forward perpendicular and Aí is the forward termination of this water line. The after termination is found in the same manner.

The sections should now be drawn in. It is no longer necessary to refer to our small plan unless we wish to do so as a check. We can take our plain dividers and, using Section No. 3 as a center line, we can set off the sections in the forebody to the right of this line and the sections in the afterbody to the left of the line. Measure directly from the half-breadth in the large plan the width of the deck (from the center line up to the curve) on Section No. 3. Set this distance out to the right from Section No. 3 in the profile. Now, as the height of the sheer above the base varies forward and aft we must also be sure to set this half-breadth measurement at such a height as to have it correspond with the height of the sheer line at Section No. 3. We will get a point at C which fulfills the above requirements. It is now simply necessary to set out on the several water lines in the profile, the widths of the corresponding water lines at Section No. 3 taken from the half breadth. This gives us a series of rather close spaced spots through which we may draw a fair curve freehand. There are, however, ship draughtsmenís curves for this work and it would help the accuracy and finish of our design if they were employed. It will be necessary to piece these curves together, as it is very rarely possible to find even among a whole set of these ship curves one that will lend itself to drawing an entire section, and have the line go through all spots. The other sections may now be drawn as described for Section No. 3, and we have then the body plan complete. Our drawing is apparently sufficiently complete to allow us to proceed with the construction of the model, but can we be sure of its accuracy! It is inevitable, in the process just described, that small errors will be made in transferring measurements, and these enlarged six times have made a few of our spots appear uneven We have endeavored, in drawing our lines through these spots, to strike a general average, or to adhere to the most spots which gave us a fair curve. It is now necessary, to insure the symmetry and accuracy of our model, to obtain complete agreement between the sections in the body plan and the water lines in the half-breadth plan. Any disagreements of this nature are best shown up by introducing a few buttock lines and three or four diagonals. The purpose of these lines is principally for fairing our hull, and their general nature has been previously discussed. By study of a set of published lines shown in Fig. 4 and from the above description of transference of ìspotsî it is believed that the reader can master the use of these lines without further explanation. It will be noted that the buttocks are really very similar to water lines, but are run vertically instead of horizontally. The diagonals are run at varying angles between the vertical buttocks and the horizontal water lines, but the general idea of a diagonal is to have it approximately perpendicular, so far as possible, to the surface it intersects. Beside the property of fairing the outside surface of our hull, the water lines, buttocks and diagonals all combine to tell us about the form of the boat. As soon as we have learned to interpret the form of the boat from these lines we can better understand the relation they have to speed, stability, balance and, last but not least, beauty.

The Offset Table

In large vessels the measurements are expressed in feet, inches and eighths of an inch, and an architectís scale is used in taking these measurements, which are expressed, for example, 31ó10ó5, which is 31 feet 10 5/8 inches. The following figure (Fig. 24) gives the mold loft offsets of the set of lines shown in Fig. 19.

It will be noted that the offset table is in two principal parts: the height above the base of lines appearing in the profile and the half-breadths, or distances out from the center line of lines appearing in the half-breadth plan. Take first the heights above base. Under the heading ìSheerî are given the height of this line at bow or fore perpendicular, at each section, and at stern or after perpendicular, above the base. If we have our framework accurately drawn up we can set these heights on the respective stations, draw a line through the spots and thus we have the sheer line in our new enlarged drawing. Now follow the same procedure with the buttock lines and profile.

Now turn to the half-breadth part of the offset table. Under the heading ìDeckî are given the distances out from the center line of the deck at side on each section. Set these distances on their respective stations on our new enlarged drawing, run a line through the spots and we have our deck line. Follow the same procedure with the water lines, each in turn.
The process of fairing up between the lines of the profile and half-breadth plans should now be gone through as previously described while we are drawing in the sections on the body plan, the latter being made from measurements taken directly from the profile and half-breadth of the large plan we have just drawn.
 

Some Hints in Regard to Design

Before the urge to create a new design comes to us we generally have had a number of thoughts running through our heads along such lines as these: ìIf I make my next boat a little longer and narrower, I will get less resistance at high speed, hence she will be better in strong breezes where my present boat is a little weak,î or, ìMy present boat has such an excessive amount of displacement that she is handicapped when in competition with newer boats. I will design a new boat with less displacement to trim these fellows that have been beating me.î Then we get out a pencil and a piece of paper and make some tentative freehand sketches of the new creation. Eventually we arrive at the time when we are standing before our draughting table on which is a nice clean sheet of cheap but durable detail drawing paper, ready to elaborate upon our little pencil sketch and produce a design which shall be the last word so far as our experience goes.

It will be remembered that in connection with the determination of the form of our hull, the qualities of speed, stability, balance and beauty were mentioned. Of these qualities balance is undoubtedly the prime requisite, for no matter how speedy, or stable, or beautiful our craft is, unless she is able to sail consistently on a given course she will not be able to compete successfully with slower craft that can. And to sail consistently on a given course the boat must have balance in light airs, in moderate breezes and in heavy winds.

Balance

The weight of the boat W acts downward through the center of gravity CG of the model and is balanced by upward force of buoyancy B concentrated at the center of buoyancy CB.

Now suppose the model to be heeled over at an angle of, say, 35 degrees. It is obvious that there has been no change in weight of the hull, nor, with respect to the hull itself, in the position of center of gravity. There has, however, been a considerable change in the form of the immersed portion of the hull and it is entirely possible that the position of the center of buoyancy has moved.

Looking at the lower midship section of Fig. 25, we note that the center of buoyancy has made a lateral movement to Bí which places it in a vertical line with the center of gravity G. The model is at rest in an inclined position, held at the angle by an outside force. The point we now wish to bring out in the present discussion is the change in the fore and aft location of the center of buoyancy. If it has moved forward any considerable amount it means that our model has come up at the bow as she heeled over. If the center of buoyancy has moved aft it means that she has come up by the stern and that the bow has dropped. These changes of trim, as they are called, are very undesirable accompaniments of heeling as the model tends to lie at an angle to the direction of sailing which destroys the quality of balance, so essential in keeping a straight course.

To avoid these changes of trim in the heeling model it is necessary to design the ends of the boat to suit each other. As a general example, sharp V-shaped forward sections should be balanced by deep V-shaped sections aft. Full or U-shaped sections forward permit generally fuller sections aft, and flat sections forward call for flat sections aft.

Design

We may lay down our drawing on a scale of 1/2 inch to the inch, which will be a convenient size for the purpose of study. On the above scale the load water line measures 16 1/2 inches. At the lower part of the drawing, draw a horizontal line AB representing the center line of the boat in the Half-Breadth Plan. On this line lay off the length of the water line, 16 1/2 inches, and divide this distance into eight equal parts, 2 1/16 inches each. At each of the nine points of division thus made, erect perpendiculars to the line AB, and number these perpendiculars, beginning at the right, 1, 2, 3, etc. These perpendiculars represent the sections. Above the line AB, at a distance somewhat greater than half the breadth of the boat, draw the horizontal line CD as a base line at the bottom of the keel. The draught of water is 9 inches which on 1/2-inch scale is 41/2 inches, so draw at this distance above the base line another horizontal line EF for the load water line.

Displacement

It will be readily seen, then, that if we can start by drawing this curve of areas, we will have progressed a long distance in determining the shape of the underwater portion of our design.

In power propelled vessels it is usual to adopt a mathematical curve of areas. A method commonly used is to make the curve of areas for the forward half of the boat or entrance follow a curve of versed sines, while that for the after half, or the run, follows the trochoid. In sailing yachts, the keel is, of course, temporarily omitted in reckoning the areas of the cross-sections for determining the ordinates of the curve of areas, and only the fair body form is used. It is generally desirable to make various modifications to the above mathematical curve, such as placing the location of the generating circle from 2 per cent to 5 per cent of the water-line length aft of amidships and fining off considerably the form of the curve aft. In the after body the actual curve of areas adopted generally lies somewhere between a curve of versed sines and the above-mentioned trochoid. The relation of the area of this curve and the area of the enclosing rectangle should come somewhere between .50 and .60. This decimal is called the longitudinal coefficient. By reference to our preliminary sketch we can estimate that the portion of the displacement devoted to the keel, i.e., that portion below the curved dotted line GH in the profile view (Fig. 26) is about 35 cubic inches, leaving 345 cubic inches for the volume of the fair body of the boat. If we have selected a longitudinal coefficient of .50, the length of our water line being 33 inches, the total area of the under-water midship section, both sides should be

On the 1/2-inch scale this would be 5.2 square inches, and the area of the midship section, one side, would be 2.6 square inches.

In the design shown in Fig. 26 we have placed the generating circle for our theoretical curve of displacement about 3 per cent aft of Station No. 5. This percentage times the water-line length of 16 1/2 inches on the 1/2-inch scale gives about 1/2 inch. On the line AB set this distance off aft of Station No. 5 and erect a perpendicular below the line AB. On this perpendicular set off JK equal to 2.6 on a scale where 3/4 inch represents one square inch. On AB divide J9 and J1 into six equal parts each. On JK draw the generating circle with a diameter 2.6. Divide the circumference into twelve equal parts. In the left half of the circle draw JM, JN, JO, JP and JQ, as shown. Parallel to these lines and equal in length to them draw lines from the division points in J9 to Mí, Ní, Oí, Pí and Qí. The points 9, Qí, Oí, Ní, Mí and K now denote the trochoid curve of displacement for our after body. On the right half of the circle draw radii from the center to the five divisional points on the circumference. Produce horizontal lines from these points on the circumference out into the fore body. Erect perpendiculars to AB at the five divisional points. Where these perpendiculars meet the horizontal lines through corresponding points on the circumference of the generating circle, will give a series of points K, R, S, T, U, V and 1 through which the curve of versed sines forming the curve of areas of the fore body will pass. Draw in this trial curve of areas.

The next step is to sketch in Section No. 5, getting the actual area of the fair body (minus the keel below the dotted line GH) from our theoretical curve of areas. It is about 2.58. Use cross-section paper to check up on the area, slipping a piece of this transparent paper over the section just drawn and calculating the area by counting the squares enclosed by the curve of the section and the center line and the load water line. We have also as a guide in drawing this section a half breadth of beam of 4 1/2 inches. This will be somewhat less at the water line, say about 4 1/8 inches, measuring with our 1/2-inch scale. As the character of the midship section will exert a very strong influence on the whole shape of our boat it would be well to study the midship sections of published designs, with a view to selecting the characteristics most desired. In our present model we are designing a rather narrow, fine-lined boat, one that shall be easily driven with moderate sail area and, of course, are depending for stability principally on a comparatively heavy lead keel. Hence considerable depth of body is necessary.

Now draw Section No. 3 and Section No. 7 similarly, being careful to give these sections the character they should have as previously discussed earlier in this chapter under the heading of "Balance." No. 3 should be about 1.07 square inches and No. 7 should be about 1.80 square inches, according to our curve, but in the present design is fined down to 1.60 square inches, considering only the fair body and omitting the keel below CIH. These sections should be drawn in connection with the load water line, the shape of the latter having a controlling influence on the width of the top of the underwater portion of the sections. The shape of the load water line, together with that of the midship section, really determines the form of the hull. The half breadth and length of the water line are given, and we are guided to some extent by the requirements of our curve of areas, but it will be necessary for us to be guided by experience as to the exact form of the curve itself. Broad, shallow boats are apt to have load water lines that are fuller at the ends than those of boats of deep, narrow body. The design we have before us shows a deep and rather narrow boat. The water line is kept rather fine at the ends to harmonize with the general idea of the form.

We should now draw in our sheer line in the profile. This should be a fair, flowing curve, but the exact form of the curve is a matter to be dictated by the experience and artistic sense of the designer. A sheer that would look well on a boat with clipper bow and long, overhanging stern might not look well on a vessel with short, full overhangs. The following rule is given in Yacht Architecture, by Dixon Kemp:

The length from the forward end of the water line to near the end of the stern overhang is divided into four equal parts. The height of freeboard at the forward end of the water line being taken as 1.00, the comparative heights of other given points in the length of the vessel are as follows:
 

It will be noted in the above that the point at the stern end of the water line is put in as an extra point. If a rail is added above the sheer line, it should follow the sheer parallel until near the 3/4 distance, when it should become a little straighter, that is, the height of the rail is slightly less until the stern is reached. This results in a graceful taper between the two lines. The sheer line on our modern racing craft is considerably straighter than the curve obtained from the above values, but the latter are considered to be a useful guide.

It would now be well to draw the profile line of the boat in the profile plan. The shape of the stem, forefoot and keel is very much a matter of taste, but the line should be so drawn that the fair body form of the boat is included within the profile and no undue bluntness to the forward ends of the water lines occur. In models it is well for the forward profile line to form a smooth, continuous curve from stem to keel without any sudden radical change in direction at any point.

A sharp, well-defined ridge should be formed along the center line of the model at the keel and forefoot, as this sharpness is said to help the boat in going to windward. Likewise, the forward end of the lead keel should not be blunt, as in the case of many large yachts, although the center of gravity of the lead itself should be well forward of the middle point of its length.

The rake of the stern post is a matter of importance with models. If the desire to cut away all useless deadwood were followed, as in the case of extreme freaks among the large racing craft, one could start at the keel at Station No. 7, run forward and up to a point halfway between Stations Nos. 6 and 7 and thence back to Station No. 7 at the point where the line GH crosses it. In this case, the rudder would be a balanced rudder with a vertical stock, located at about Station No. 81/2. In models designed and built in England, the foregoing form of profile aft is common practice. The short fin is said to give relief to pressures which form on the lee side, which pressures, in the case of a longer fin, have greater opportunity to exert an erratic influence, thus affecting adversely the steering of the model. The longer keel, as shown on the drawing, has, however, been used on many boats with good results. A moderate rake of stern post is desirable as with the rudder turned the current of water passing aft under the hull is directed upward, instead of being driven downward against its natural flow. On the other hand, an excessive rake of sternpost makes it very difficult to fit a suitable steering gear on the head of the rudder stock, as the angle of the latter with the deck is then very great. A moderate rake of sternpost is therefore the best.

The after overhang is somewhat a matter of fancy, but should balance the overhang of the bow to give a good appearance and should be so drawn as to give good sailing lines with the model heeled down under sail.

It should now be practicable to sketch in the sections at Stations Nos. 1 and 9 in the body. Next, carry up the underwater sections that we have drawn in slightly rounded topsides to the deck. Take a few half breadths from the sections on the several stations in the plan view. We already have the exact maximum half breadth of the deck amidships, which is 4 1/2 inches on the 1/2-inch scale. Now spring a batten around the points thus formed and draw a fair curve for the deck line, giving a little width at B for the sake of strength of construction and any width we like at A across the stern. We can now transfer the half breadths of the deck to the body plan and draw more exactly the upper part of our sections, giving them the exact height they should have above the water line as taken from the sheer line.
Now run in a diagonal across the bilge in the body plan, as shown in Fig. 26. Pick off the half breadth of these diagonals for the sections already sketched in and through the spots run a fair, free, flowing curve. Now using this diagonal and the load water line and deck line pick off the half breadths on Sections 2, 4, 6 and 8 and transfer them to the diagonal, the load water line and the deck line respectively in the body plan, making little marks on the diagonals, water lines and deck line. Now sketch in the new sections going through these marks and you have the body plan complete. In the profile draw a few water lines above and below the load water line and transfer the half breadths on each water line in the body plan down to the proper station in the half-breadth plan. Make spots on the sections and through the spots draw water lines. Do the same way with the buttocks. These should all come fair, but slight discrepancies may be expected when it is remembered that our sections are sketched in freehand and have not been accurately drawn by means of ship curves.

Fairing the Lines.

Calculating Displacement and Center of Buoyancy

Now to complete our calculation we wish to find the fore and aft position of the center of buoyancy. This information is useful as it enables us to determine where to locate the lead ballast.

The sum of the moments about the Station No. 1 is 150.28. Dividing the sum of the moments by the sum of the areas and multiplying by the common interval gives

Multiply by 2 for our scale and get 17.772. The Center of Buoyancy is therefore 1.272 inches aft of Station 5.

Center of Gravity

It will be noted from the foregoing that a position of center of gravity of lead has been assumed such that the total of the moments of the weights of the several parts of the boat about Station No. 5 is equal to the moment of the weight of the whole boat acting through the center of buoyancy about Station No. 5. This is done to bring the center of gravity of the complete boat and the center of buoyancy in the same vertical line. As explained previously, this produces equilibrium at the designed trim.

The Sail Plan

Sail Area and Center of Effort

 If the mast is raked aft, it is best to take an axis perpendicular to the water line. The sail plan of Fig. 15 will give the method employed which, with the foregoing text, will enable the prospective builder to readily follow the several steps. There is a point to remember. If an axis or reference line is chosen which lies between the jib and the mainsail, the arm and consequently the moment of one sail or the other will be minus, while the other is plus.

[A more detailed set of instructions for finding the geometric center of effort can be found here. --EB]

Center of Lateral Resistance of Hull

The calculation for the center of lateral resistance for our boat is given in the following:
 

[A simpler method for finding the CLR is to use a piece of thick cardboard or thin ply in the shape of the underwater profile of the boat. Lay this horizontally on a knife edge vertical to the LWL, and slide it back and forth until it balances. That point of balance is the location of the CLR. --EB).

Now go back to the sail plan and draw in the hull definitely so that the CLR is slightly aft of the CE (center of effort). The amount of the horizontal distance between these two centers is determined by experience. Generally speaking, the distance should be greater on broad, full-bodied boats than on keen, sharp-lined craft. As our boat is distinctly in the latter category, it is believed that a ìleadîóas it is calledóof 1 1/4 inches of CE forward of CLR is about right. The general practice is to make this distance from 2 1/2 per cent to 5 per cent of the water-line length with an average say, of 4 per cent.
We can show a spinnaker on our sail plan, which should be proportioned as follows:

The hoist of the spinnaker is the distance measured from the deck on the foreside of the mast where the line of the luff of the jib when extended cuts the mast. The base of the spinnaker shall be from the forward side of the mast to the bowsprit end or point where the forestay meets the deck.

We have now performed all the ordinary operations of designing a model sailing yacht, such as lie within the scope of this work. A great deal of additional work may be performed, such as investigating the balance of the hull when heeled and investigating the conditions of stability.

Curve of Areas, Versed Sines, Trochoids, and the Wave Theory

The curve of areas for a hull is a graph which displays how the displacement, or underwater volume, of the hull is distributed along its length. The horizontal axis of the plot represents the load water line. The vertical axis represents the area of an underwater section of the hull at that particular point.  The areas are taken off the section plans.  In actual practice, the areas are sampled at fixed intervals (e.g., the stations) and a smooth curve is fitted to the points. In the drawing by Thomas Moore, the curve of areas is (as is common) overlaid on the half breadth plan.
The curve of areas clearly has a strong relationship to the wave making and resistance of a hull.  If the curve rises too sharply at the bow, the hull will be ìbluff bowedî and push water before it. Likewise, if it drops too sharply at the stern, there will be significant turbulence aft, as when a square sterned  boat ìsquatsî over the limit of its design speed. Conversely, a curve of areas that rises too slowly at the bow will lead to ìplungingî and excessive pitching.

Archer asserted (based on some fairly imaginative mathematical reasoning) that the curve of areas should itself resemble a wave line. It had already been learned by trial and error that the section of a yacht with the greatest area (the midbody) should be at about 6/10 of its length.  Archer further argued that the curve of areas ahead of the midbody should follow the mathematical curve called a curve of versed sines and that after it should be what mathematicians call a trochoid.
 
 

These curves are generated geometrically, as shown in the diagrams. They are both constructed from a circle and a line. The line is the length of the load water line and the circle is placed at the location of the midbody. The line is divided into equal parts (usually corresponding to the position of the stations; aí, bí, and cí in the diagrams) and the circumference of the circle is divided equally into the same number of parts (a, b, and c) as the line was. The upper diagram shows how to develop the curve of  versed sines. A series of rectangles is drawn (e.g. a to aî to aí) and the upper corners are connected as shown.  The trochoid in the lower diagram is a little more complicated. Lines are drawn from the point A where the circle touches the line , such as A to a.  A second line, of the same length and parallel to it (aí to aî) is constructed.  As before, the corners are connected with a smooth curve.

The wave theory  was attractive to Victorian naval architects because it was fundamentally geometric and therefore could be implemented using the drafting instruments of the time. It also had a nice mystical ring to it, and, most importantly, produced fast, steady, and good-looking boats. It fell out of favor in the early 20th century as more scientific evidence was gathered from towing model hulls in test tanks.

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