# Geometry

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Definitions

Length overall (LOA)

Length of water line (lwl)

Length between perpendiculars (LFF)

Rated length

he hull of a yacht is a complex three-dimensional shape, which cannot be defined by any simple mathematical expression. Gross features of the hull can be described by dimensional quantities such as length, beam and draft, or non-dimensional ones like prismatic coefficient or slenderness (length/displacement) ratio. For an accurate definition of the hull the traditional lines drawing; is still a common tool, although most professional yacht designers now take advantage of the rapid developments in CAD introduced in Chapter 1.

In this chapter we start by defining a number of quantities, frequently referred to in yachting literature, describing the general features of the yacht. Thereafter, we will explain the principles of the traditional drawing and the tools required to produce it. We recommend a certain work plan for the accurate production of the drawings and, finally, we show briefly how the hull lines are generated in a modern CAD program.

The list of definitions below includes the basic geometrical quantities used in defining a yacht hull. Many more quantities are used in general ship hydrodynamics, but they arc not usually referred to in the yachting field. A complete list may be found in the International Towing Tank Conference (ITTC) Dictionary of Ship Hydrodynamics.

The maximum length of the hull from the forwardmost point on the stem to the extreme after end (see Fig 3.1). According to common practice, spars or fittings, like bowsprits, pulpits etc are not included and neither is the rudder.

The length of the designed waterline (often referred to as the DWL).

This length is not much used in yachting but is quite important for ships. The forward perpendicular (FP) is the forward end of the designed waterline, while the aft perpendicular (AP) is the centre of the rudder stock.

The single most important parameter in any rating rule. Usually L is obtained by considering the fullness of the bow and stern sections in a more or less complex way.

The maximum beam of the hull excluding fittings, like rubbing strakes.

DWL

Fig 3.1 Definitions of the main dimensions

Beam of waterline (bwl)

Displacement

The maximum beam at the designed waterline.

The maximum draft of the yacht when floating on the designed waterline. Tc is the draft of the hull without the keel (the 'canoe' body).

The vertical distance from the deepest point of the keel to the sheer line (see below). Dc is without the keel.

Could be either mass displacement (m) ie the mass of the yacht, or volume displacement (V or V), the volume of the immersed part of the yacht. mc, Vc and Vc are the corresponding notations without the keel.

Midship section For ships, this section is located midway between the fore and aft perpendiculars. For yachts it is more common to put it midway between the fore and aft ends of the waterline. The area of the midship section (submerged part) is denoted AM, with an index 'c' indicating that the keel is not included.

Maximum area section For yachts the maximum area section is usually located behind the midship section. Its area is denoted Ax (AXc).

Prismatic coefficient This is the ratio of the volume displacement and the maximum section (CP) area multiplied by the waterline length, ie CP = V/(AX • Lwl). This value is very much influenced by the keel and in most yacht applications only the canoe body is considered: CPc = Vc(AXc • Lwl). See Fig 3.2. The prismatic coefficient is representative of the fullness of the yacht. The

Circumscribed cylinder volume = v = L^ Ay

Circumscribed cylinder volume = v = L^ Ay

Fig 3.2 The prismatic coefficient

Max. area

BOX WL WL c

Circumscribed box volume =

BOX WL WL c

Fig 3.3 The block coefficient

Block coefficient ( CB)

Centre of buoyancy (B)

Centre of gravity (G)

Sheer line

Freeboard fuller the ends, the larger the Cp. Its optimum value depends on the speed, as explained in Chapter 5.

Although quite important in general ship hydrodynamics this coefficient is not so commonly used in yacht design. The volume displacement is now divided by the volume of a circumscribed block (only the canoe body value is of any relevance) CBc = V J(Lwl • BWL • Tc). See Fig 3.3.

The centre of gravity of the displaced volume of water, its longitudinal and vertical positions are denoted by LCB and VCB respectively.

The centre of gravity of the yacht must be on the same vertical line as the centre of buoyancy. In drawings G is often marked with a special symbol created by a circle and a cross. This is used also for marking geometric centres of gravity. See. for instance, Figs 5.27 or 8.2.

The intersection between the deck and the topside. Traditionally, the projection of this line on the symmetry plane is concave, the 'sheer* is positive. Zero and negative sheer may be found on some extreme racing yachts and powerboats.

The vertical distance between the sheer line and the waterline.

### Tumble home

When the maximum beam is below the sheer line the upper part of the topsides will bend inwards (see Fig 3.4). To some extent this reduces the weight at deck level, but it also reduces the righting moment of the

Fig 3.4 Definition of tumble home and flare

Tumble home

DWL

Tumble home crew on the windward rail. Further, the hull becomes more vulnerable to outer skin damage in harbours.

Flare The opposite of tumble home. On the forebody in particular, the sections may bend outwards to reduce excessive pitching of the yacht and to keep it more dry when beating to windward.

Scale factor (a) This is not a geometrical parameter of the hull, but it is very important when designing a yacht. The scale factor is simply the ratio of a length (for instance the Lw,) at full scale to the corresponding length at model scale. Note that the ratio of corresponding areas (like the wetted area) is a2 and of corresponding volumes (like displacement) a3.

Lines drawing A complete lines drawing of the YD 40 is presented in Fig 3.5. The hull is shown in three views: the profile plan (top left), the body plan (top right) and half breadth plan (bottom). Note that the bow is to the right.

In principle, the hull can be defined by its intersection with two different families of planes, and these are usually taken as horizontal ones (waterlines) and vertical ones at right angles to the longitudinal axis of the hull (sections). While the number of waterlines is chosen rather arbitrarily, there are standard rules for the positioning of the sections. In yacht architecture the designed waterline is usually divided into ten equal parts and the corresponding sections are numbered from the forward perpendicular (section 0) backwards. At the ends, other equidistant sections, like # 11 and # 1 may be added, and to define rapid changes in the geometry, half or quarter sections may be introduced as well. In Fig 3.5 half sections are used throughout.

The profile is very important for the appearance of the yacht, showing the shapes of the bow and stern and the sheer line. When drawing the waterlines, displayed in the half breadth plan, it is most helpful if the lines end in a geometrically well defined way. Therefore a 'ghost" stem and a 'ghost' transom may be added. The ghost stem is the imagined sharp leading edge of the hull, which in practice often has a rounded stem, and the ghost transom is introduced because the real transom is often curved and inclined. If an imagined vertical transom is put near the real one at some convenient station, it will facilitate the fairing of the lines. The drawing of Fig 3.5 has been produced on a CAD system and no ghost stem is shown. However, a ghost transom is included.

In the body plan, the cross sections of the hull are displayed. Since the hull is usually symmetrical port and starboard, only one half needs to be shown, and this makes it possible to present the forebody to the right and the afterbody to the left. In this way mixing of the lines is avoided and the picture is clearer. Note that in the figure the half stations are drawn using thinner lines.

The above cuts through the hull are sufficient for defining the shape, but another two families of cuts are usually added, to aid in the visual perception of the body. Buttocks are introduced in the profile plan,

showing vertical, longitudinal cuts through the hull at positions indicated in the half breadth plan. The diagonals in the lower part of the half breadth plan are also quite important. They are obtained by cutting the hull longitudinally in different inclined planes, as indicated in the body plan. The planes should be as much as possible at right angles to the surface of the hull, thus representing its longitudinal smoothness. In practice, the flow tends to follow the diagonals, at least approximately, so that they are representative of the hull shape as "seen' by the water. Special attention should be paid to the after end of the diagonals, where knuckles, not noticcd in the other cuts, may be found, particularly on lOR yachts from the 1970s and the 1980s. Almost certainly, such unevenncss increases the resistance and reduces the speed of the yacht.

The other line in the lower part of the half breadth plan is the curve of sectional areas, representing the longitudinal distribution of the submerged volume of the yacht. The value at each section is proportional to the submerged area of that section, while the total area under the curve represents the displacement (volume). A more detailed description of the construction of the curve of sectional areas will be given in Chapter 4.

In order to define exactly the shape of the hull a table of offsets is usually provided by the designer. This is to enable the builder to lay out the lines at full size and produce his templates. Offsets are always provided for the waterlines, but the same information may be given for diagonals and/or buttocks also. Note that all measurements are to the outside of the shell.

Tools

The drawing should be made on a special plastic film, available in different thicknesses. The film is robust and will not be damaged by

Photo 3.6 Tools (triangle, plastic film, straight edge, brush, pens, pencil, erasing shield and eraser)

Photo 3.7 Tr¿\nster of measures from body plan (top) to half breadth plan (bottom) using a paper ribbon

Photo 3.7 Tr¿\nster of measures from body plan (top) to half breadth plan (bottom) using a paper ribbon

erasing. Furthermore, it is unaffected by the humidity of the air. which may shrink ordinary paper.

Since the film is transparent the grid for the lines drawing is drawn on the back so that it will remain, even after erasing the hull lines on the front many times. Great care must be exercised when drawing the grid, making sure that the alignment and spacing are correct and that all angles arc cxactly 90°. In Fig 3.5 the grid is shown as thin horizontal and vertical lines, representing waterlines, buttocks and stations.

Black ink should be used when drawing the grid and preferably when finishing the hull lines also. However, when working on the lines a pencil and an eraser are needed. There are, in fact, special pencils and erasers for this type of work on plastic film. An erasing shield and a brush are also most useful (see Photo 3.6).

For creating the grid a long straight edge is required, together with a

Photo 3.8 Ducks and a spline used for drawing a water Iine

Photo 3.8 Ducks and a spline used for drawing a water Iine

Photo 3.9 Templates used for drawing lines with large curvature

Photo 3.9 Templates used for drawing lines with large curvature

large 90° set square. It is very convenient to have a bunch of paper ribbons, which can be used for transferring different measures from one plan to the other. For example, when drawing a waterline the offsets of this line may be marked on the ribbon directly from the body plan and moved to the half breadth plan (Photo 3.7).

To draw the hull lines it is necessary to have a set of splines and weights or ducks. Long, smooth arcs can be created when bending the splines and supporting them by the ducks at certain intervals. Photo 3.8 shows how these tools are used when drawing a waterline. The splines should be made of plastic, somewhat longer than the hull on the drawing, and with a cross-section of about 2.5 mm2. Many different types of ducks can be found, some of them home made. Preferably,

Photo 3.10 PI an i meter they should be made of lead, and the weight should be between 1.5 and 2.5 kg. To be able to support the spline, they should have a pointed nose, as shown in Photo 3.8.

The splines are needed when drawing the lines in the profile and half breadth plans. However, the lines of the body plan are usually too curved for the splines, so it is necessary to make use of a set of templates especially developed for this purpose. The most well known ones are the so called Copenhagen ship curves, the most frequently used of which are shown in Photo 3.9.

A very convenient instrument, well known in naval architecture, is the planimeter, used for measuring areas (see Photo 3.10). The pointer of the planimeter is moved around the area to be measured, and the change in the reading of the scale when returning to the point of departure gives the area enclosed by the path followed. Considering the difficulty in following exactly any given line the accuracy is surprisingly high, more than adequate for the present purposes. The need for measuring areas will be explained in the next chapter.

Since many calculations have to be carried out when preparing the drawings, and indeed in the whole design process, an electronic calculator is essential. A simple one would be sufficient in most cases, but a programmable calculator would simplify some of the calculations, particularly if a planimeter is not available. Most scientific calculators have programs for calculating areas with acceptable accuracy, and programs are available for most of the calculations described in the next chapter.

### WORK PLAN

Designing the hull is a complex process, and many requirements have to be considered. One difficulty is that important parameters, such as the displacement cannot be determined until the lines have been fixed. This calls for an iterative method. Such a method is also required in the fairing of the lines. The problem is to make the lines in one projection correspond to smooth lines in the other two projections. For an inexperienced draftsman this problem is a serious one, and many trials may be needed to produce a smooth hull.

While the preferred sequence of operations may differ slightly between yacht designers the main steps should be taken in a certain order. In the following, we propose a work plan, which has been found effective in many cases. It should be pointed out that the plan does not take into account any restrictions from measurement rules.

Step 1: Fix the main dimensions These should be based on the general considerations discussed in Chapter 2, using information on other yachts of a similar size, designed for similar purposes. This way of working is classical in naval architecture, where the development proceeds relatively slowly by evolution of previous designs. It is therefore very important, after deciding on the size of the yacht, to find as much information as possible on other similar designs. Drawings of new yachts may be found in many of the leading yachting magazines from all over the world.

The dimensions to fix at this stage are: length overall, length of the waterline, maximum beam, draft, displacement, sail area, ballast ratio, prismatic, coefficient and longitudinal centre of buoyancy. One of the aims of this book is to help in the choice of these parameters and to enable the reader to evaluate older designs when trying to find the optimum for his own special demands.

Step 2: Draw the profile As pointed out above, this step takes much consideration, since the aesthetics of the yacht are, to a large extent, determined by tBfe pi^ffle-

Step 3: Draw the midship section The midship section can be drawn at this stage, or, alternatively, the maximum section if it is supposed to be much different. This may occur if the centre of buoyancy is far aft. The shape of the first section drawn is important, since it determines the character of the other sections.

Step 4: Check the displacement To find the hull displacement calculate (or measure) the submerged area of the section just drawn and multiply by the waterline length and the prismatic coefficient chosen for the hull. From the ballast ratio, the keel mass can be computed and the volume can be found, dividing by the density of the material (about 7200 kg/m3 for iron and 11300 kg/m- for lead). Assume that the rudder displacement is 10% of that of the keel and add all three volumes. If the displacement thus obtained is different from the prescribed one, return to step 3 and change accordingly.

The procedure described is for a fin-keel yacht. For a hull with an integrated keel, as on more traditional yachts, the prismatic coefficient usually includes both the keel and the rudder.

Step 5: Draw the designed waterline One point at or near the midship station is now known, together with the two end points from the profile, so now a first attempt can be made to draw the designed waterline.

Step 6: Draw stations 3, 7 and the transom The waterline breadth is now known, as well as the hull draft, and the sections should have a family

resemblance to the midship section. Often it is helpful to draw a ghost transom behind the hull.

Step 7: Draw new waterlines Two or three waterlines can now be drawn above and below the DWL. If the appearance is not satisfactory, go back to step 6 and change.

Steps 8 and 9: Add new sections and waterlines

Once this is done, sections I-9 should be completed as well as 7-10 waterlines. Constant adjustments, have to be made in order to create smooth lines in the body plan, as well as in the half breadth plan.

Step 10: Recheck the displacement and the longitudinal centre of buoyancy The curve of sec tional areas can now be constructed. Its area gives the displacement (excluding that of keel and rudder) and its centre of gravity corresponds to the longitudinal position of the centre of buoyancy. If not correct, adjustments have to be made from steps 5 or 6,

Step 11: Draw diagonals Inspect the smoothness, particularly near the stern. Adjust if necessary.

Step 12: Draw buttocks This is the final check on the smoothness. Usually only very minor corrections have to be made at this stage.

### Computer aided design of hulls

As mentioned in Chapter 1, most CAD programs use master curves for generating the hull surface. Each curve is defined by a number of points, called vertices. Photo 3.11 shows, in a plan view, the grid of master curves used for generating the YD-40 hull. One of the transverse curves has been selected in Photo 3.12 and it can be seen how the smooth hull surface is generated inside the curve, which is shown as piece-wise linear between the vertices.

Photo 3.11 Grid of master curves used for the YD-40 (the vertical line to the right marks the origin of the coordinate system)

Photo 3.12 A section with vertices (crosses), master curve (between the crosses), hull surface and cuwature (outermost line)

Photo 3.11 Grid of master curves used for the YD-40 (the vertical line to the right marks the origin of the coordinate system)

Photo 3.12 A section with vertices (crosses), master curve (between the crosses), hull surface and cuwature (outermost line)

The task of the designer is to specify the vertices in such a way that the desired hull shape is created.There are different ways of achieving this. Some programs start from a long cylindrical body or a box, while others start from a flat rectangular patch, defined by an orthogonal grid. These original shapes are then distorted by moving the vertices around, and it is relatively easy to produce a yacht-like body. However, it takes experience and experimentation to obtain a shape that satisfies criteria set up beforehand. In practice, designers very seldom start from scratch, but work from earlier designs, which already have a desirable shape and a known grid of master curves surrounding it. Since most new designs are evolutions of previous ones this approach is very natural.

A problem encountered when the first CAD programs for yachts appeared was that the scale on the screen was too small, and the resolution too low to enable the designer to create fair lines. Small bumps on the surface could not be detected 011 the screen, and it sometimes happened that the bumps were noticed only after the start of the hull construction. Therefore the CAD program developers introduced plots of the curvature of lines on the hull. Such a plot is shown.in Photo 3.12. The curvature of the line, which essentially corresponds to a section, is almost constant, except at the ends where it goes to zero.

Photo 3.13 illustrates the sensitivity of the curvature to small changes of the surface. The sheer line is shown in a plan view. In the top photo (the real design) the curvature is smooth and relatively constant along the hull. In the bottom photo one vertex point has been moved 10 mm at full scale perpendicular to the surface. The resulting change in the sheer line is so small that it cannot be detected by eye, but the curvature exhibits a considerable bump and some smaller fluctuations, showing that the line is not smooth. By looking at the curvature, lines may thus be generated that look fair even at full scale.

Photo 3.13 Sheer line with vertices and curvature. (top) Real design. (bottom) One vertex point moved 10 mm

Photo 3,14 Perspective view A great advantage of most CAD programs is that the hull may be of the YD-40 shown in perspective. As pointed out in Chapter 1 it is important to study the sheer line in particular from different angles, since the impression of the hull contour in reality is also influenced by the beam distribution, which is not visible if only the profile view is studied. Fig 3.14 shows the YD-40 in perspective, and a good impression can be obtained of the shape. "

By using a CAD program a fair hull can be produced rapidly and different requirements may be satisfied without too much work, such as a given prismatic coefficient or longitudinal centre of buoyancy. Meeting such requirements accurately in a manual process is extremely time consuming, so it is understandable that CAD techniques are always used nowadays by professional designers. However, due to the considerable cost of a CAD system, most amateur designers will still have to use the manual approach described above.