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Chapter 2: Numerical integration for ship geometry

Version 1.0 · last revised 5 November 2026 · corrections

Prologue: Chapman’s quarter circle

In 1775 the Swedish shipbuilder Fredrik Henrik Chapman published a treatise on shipbuilding, the Tractat om Skepps-Byggeriet, which his French translator called the first work on the subject by a man of the trade¹. Its first chapter is about displacement. A ship, Chapman explains, sinks until it displaces its own weight of water. If the weight of the hull and rigging is known and the displacement is known too, the weight of the cargo is their difference. For warships and privateers, whose ballast and stores were fixed, a reliable displacement allowed a designer to give a ship good stability and good sailing together. The methods in use for calculating it were, in his judgement, imperfect, and he offered another, illustrated by examples, in the hope that shipbuilders would find it simple enough to stop being daunted by the amount of arithmetic the older methods demanded¹.

The usual practice, his French translator explained, joined the ends of neighbouring ordinates by straight lines; Chapman’s method treated the curve through three neighbouring ordinates as part of a parabola¹. The area under each such piece is then found by geometry: the trapezium under the chord, plus two-thirds of the parallelogram that encloses the parabolic segment above it. Added up along a curve divided into an even number of equal parts, these pieces give a rule that every naval architect still learns. The first and last ordinates are multiplied by 1, the others alternately by 4 and 2, and the sum is multiplied by one-third of the spacing¹.

To show that the rule was accurate enough for practice, Chapman applied it to a quarter of a circle of radius 8, whose area he could also find exactly, with five ordinates 1 apart, from the centre out to half the radius (Fig. 2.1). The rule gave 30.611312 for the area under those ordinates. Subtracting the triangle between the centre, the last ordinate and the arc left a sector of 30°, and three such sectors gave 50.2647 for the quadrant, against the exact $16\pi = 50.26548$. The calculation was, in our translation of his words, exact “to the fifth figure”; a footnote in the French edition adds that the ordinary way of calculating gave about a twentieth of a square foot less¹.

Figure 2.1 Chapman’s test of his rule: (a) a quarter circle of radius 8 with five ordinates 1 apart; (b) the gap between the arc and the straight chords, magnified 1000 times.

He then turned to a ship. For a privateer whose lines are drawn in the treatise, he measured the half-breadths of seventeen equally spaced sections at seven waterlines, 1.62 feet apart, put the seven ordinates of each section through the rule, and added the small triangle between the lowest waterline and the keel. The section areas were then treated as ordinates in their turn, 6.27 feet apart, and put through the rule again along the length, and the slivers beyond the end sections, towards the stem and the sternpost (the upright timber rising from the after end of the keel), were estimated separately. Doubled for both sides, the result was 16,210 cubic feet (about 420 to 560 m³, depending on the foot used), measured, he noted, to the outside of the frames and excluding the planking, the stem, the sternpost and the keel. In today’s terms this is a moulded volume. The moments of the section areas gave him the longitudinal position of the centre of the displacement, and, rearranging the ordinates by waterline, he found the areas of the seven waterplanes and from them its height¹.

Chapman, born in 1721, was the son of the superintendent of the naval dockyard at Gothenburg, and went to England in 1750 to improve his mathematics. In London he took lessons from Thomas Simpson, who taught mathematics at the Royal Military Academy at Woolwich, and Simpson reportedly taught him this rule². Simpson, a weaver who had taught himself mathematics, never claimed to have invented it; he acknowledged learning it from Newton³, and it had been published in 1730 by James Stirling⁴. After further study in France and Holland, Chapman returned to Sweden in 1757 and made it his habit to calculate the displacement of his ships, also as a function of draught, and their metacentre, with the parabolic rule². In Britain it kept Simpson’s name, and by the end of the nineteenth century Attwood⁴ could describe it as the rule “most extensively used” in ship calculations. Three-quarters of a century later, the International Convention for the Safety of Life at Sea, 1974 (SOLAS), as adopted, required the cubic capacity of a lifeboat to be found by “Simpson’s (Stirling’s) Rule” or by a method of equal accuracy, and the number of persons the boat could carry was based on that capacity⁵.

Chapman stopped his test at half the radius. Beyond that point the arc turns down ever more steeply and meets the axis at a right angle, and the hull of a ship has places like the end of the quadrant. Where a round bilge leaves the flat bottom along an arc of a circle, the half-breadth of the section, plotted against the height above the keel, starts with a vertical tangent, as Chapman’s arc does where it meets the axis. The forefoot, where the stem curves down into the keel, and the stern turn in profile as a round bilge turns in section. A table of offsets, however, gives no ordinates between its stations from which such places could be found. We return to Chapman’s quadrant in Sec. 2.7, where the error of a rule is estimated.

In this chapter

By the end of this chapter you will be able to:

• integrate a table of offsets by the trapezoidal rule and Simpson’s rules;
• compute the centre of flotation and the transverse and longitudinal second moments of a waterplane;
• find the volume and the centre of buoyancy on a displacement sheet, and the volume curve;
• estimate the error of a result by a test case and at another spacing, and find a mistyped area from second differences.

What it assumes. You need the integrals of Eqs. (1.6) to (1.9) and the table of offsets of Sec. 1.5.3. The Wigley hull of Eq. (1.5) is the test case with exact answers.

A first reading. Sections 2.1 to 2.8 carry the main thread, from offsets to areas, moments, volumes and their error; the rules of Sec. 2.4 for real tables of offsets are used throughout Secs. 2.5 to 2.8. On a first reading Sec. 2.4.5, on the ends of a curve, and Sec. 2.5.5, on the moment of the area between two ordinates, can be skimmed, and the alternatives to Simpson’s rules in Sec. 2.9 can wait for a second reading.

2.1 From offsets to integrals

At the end of Chapter 1 you had the hull as a table of half-breadths at stations and waterlines, or as a surface that can be cut wherever you choose, and the integrals for its volume, waterplane area and centres written down but, for the Series 60 hull, not yet evaluated. Every quantity that the later chapters need is an integral of this geometry. The area of a waterplane is an integral of its half-breadth along the length; the area of a section is an integral of its half-breadth up the height; the volume is an integral of section areas along the length, or of waterplane areas up the height; and the centres and second moments are integrals of half-breadths and areas multiplied by a distance or raised to a power (Sec. 2.5). They are listed in Table 2.1, with the chapters in which each is used.

Table 2.1 The integrals of ship geometry. $y_W(x)$ is the half-breadth of the waterline, $A(x)$ the immersed section area and $A_W(z)$ the waterplane area at height $z$.

Every integrand in the table is known only at a finite set of points. For a lines plan they are the stations and waterlines; for a surface model, they are wherever we choose to cut it; for the righting lever of Chapter 8, they are the angles at which it has been computed. We need a rule that turns a list of ordinates $y_0, y_1, \dots, y_n$ at abscissae $x_0, x_1, \dots, x_n$ into an approximation of the integral. All the rules of this chapter take the form of a weighted sum of the ordinates,

in which the weights $w_i$ depend only on the positions $x_i$, not on the ordinates. Written as small whole numbers with a common factor taken outside, the weights are called multipliers in naval architecture. A rule is a set of positions and weights, chosen once and applied to any curve. The products $w_i f(x_i)$ were traditionally called the functions of the ordinates, and a calculation laid out in a table of ordinates, multipliers and products is still the clearest way to see what a rule does.

There are three ways to arrive at the weights. One is geometric, as Chapman’s was: replace the curve between ordinates by simple pieces whose areas are known. A second replaces the curve by the polynomial that passes through the ordinates and integrates the polynomial exactly; this is the route that generalizes most easily, and it is used below for unequal spacing. A third expands the integrand in a Taylor series about each ordinate, which is how the error of a rule is found; all three lead to the same rules.

Two of the reference hulls of Chapter 1 serve as test cases throughout. The surface of the Wigley hull is given by a formula, so its integrals are known exactly and the error of any rule applied to its offsets can be measured. Its offsets in Table 1.2, eleven stations 10 m apart and five waterlines 1.5625 m apart, are the input. The Series 60 hull is the realistic case, and for it there is no exact answer. Its input is the table of offsets in which Todd⁶ published it, transcribed in Chapter 1: 25 stations, with half stations near both ends, and waterlines at the baseline and at 0.075, 0.25, 0.5, 0.75 and 1.0 times the design draught, the uneven spacing that a real table often has. Todd’s own tabulated values, and those of the surface that Chapter 1 built through Todd’s table (below called simply the smooth surface), serve for comparison with the results from the table. The three are compared in Sec. 2.6.3, and the difference from Todd’s displacement is discussed in Sec. 2.7.

2.2 The trapezoidal rule

The simplest rule, and the usual practice that Chapman set out to replace, joins the tops of neighbouring ordinates by straight lines and adds up the trapezia underneath. A strip of width $x_{i+1} - x_i$ between ordinates $y_i$ and $y_{i+1}$ has area $\tfrac{1}{2}(x_{i+1} - x_i)(y_i + y_{i+1})$, and the sum over all strips is

which for equal spacing $h$ becomes the trapezoidal rule,

Its multipliers are ½, 1, 1, …, 1, ½, because each strip gives half its width to each of its two ordinates, and every ordinate except the first and the last belongs to two strips. The rule is exact for straight lines but not, in general, for curves, and its error is the sum of the slivers between the chords and the curve (Figs. 2.1(b) and 2.2). Where the curve bulges away from the base, as a waterline does, each chord cuts inside it and the rule underestimates the area; where the curve sags towards the base, each chord passes outside it and the rule overestimates⁴. To first order, each sliver is the area between a chord of length $h$ and a parabola with the curvature of the curve, $-h^3 f''/12$, and adding the $n$ slivers over the interval $b - a = nh$ gives

where $Q_T$ is the value given by the rule and $x^*$ is some point in the interval⁷. The sign agrees with the picture. Where $f'' < 0$ the curve bulges away from the base, $E_T$ is positive and the rule is too small. An error quoted in per cent is that of the rule’s result relative to the exact value, so a rule that is too small has a negative percentage error. The value of $f''$ is not known in general, and the formula is more useful for what it says about spacing. The error is proportional to $h^2$, so halving the spacing reduces it to about a quarter. A rule whose error falls as $h^p$ is said to be of order $p$; the trapezoidal rule is of order 2.

Figure 2.2 The trapezoidal rule on half of the Wigley design waterline: (a) four strips, with the slivers between chords and curve; (b) the error against the number of strips.

Example 2.1. The trapezoidal rule on the Wigley waterplane. The Wigley hull is 100 m long and 10 m wide, and the half-breadths of its design waterline at stations 0 to 10, 10 m apart, are 0, 1.8, 3.2, 4.2, 4.8, 5.0, 4.8, 4.2, 3.2, 1.8 and 0 m. Find the waterplane area by the trapezoidal rule. Compare it with the exact value $\tfrac{2}{3}LB$, and predict the error with 20 strips instead of 10.

A second-order rule converges slowly; to bring the error of Example 2.1 down to 0.01 % would take about 100 strips, and a real table of offsets has 20 or so. The trapezoidal rule nevertheless remains in use, because it requires only that the curve be close to its chords. It works for any spacing and never overshoots between the ordinates. With the closely spaced ordinates that a computer can supply, its simplicity outweighs its low order. For some integrands the rule is much more accurate than its error formula suggests, for instance when the slope of the integrand is zero at both ends (Sec. 2.5.3).

2.3 Simpson’s first rule

Chapman’s rule replaces the chord by a parabola. His construction (Fig. 2.3) takes three equally spaced ordinates $y_0$, $y_1$, $y_2$, a distance $h$ apart, and passes a parabola through their tops. The area under the parabola is the trapezium under the chord from $y_0$ to $y_2$, of area $2h\cdot\tfrac{1}{2}(y_0 + y_2)$, plus the parabolic segment between the chord and the curve. That segment is two-thirds of the parallelogram that encloses it, a result known since Archimedes, and the parallelogram has width $2h$ and height $y_1 - \tfrac{1}{2}(y_0 + y_2)$, the amount by which the middle ordinate rises above the chord. Adding the two,

Figure 2.3 Simpson’s first rule by Chapman’s construction: the trapezium under the chord, plus two-thirds of the enclosing parallelogram.

Applied to successive pairs of strips along a curve divided into an even number $n$ of equal intervals, the rule for one pair gives Simpson’s first rule,

with multipliers 1, 4, 2, 4, …, 2, 4, 1. They are found by writing the multipliers 1, 4, 1 of each pair under its own three ordinates and adding each column⁴, as here for three pairs:

The ordinates at the junctions of the pairs carry 2, because each belongs to two pairs; the rule needs an even number of intervals, that is, an odd number of ordinates.

The rule was built to be exact for a parabola, and symmetry makes it exact for a cubic as well. Measured from the middle ordinate, a cubic term $c\,(x - x_1)^3$ is odd: it contributes equal and opposite amounts on either side, so its integral over the pair is zero, and the rule, whose weights are symmetric about $x_1$, also gives zero for it. The first term the rule cannot integrate exactly is the quartic, and repeating the reasoning that gave the error of the trapezoidal rule, with that term, gives the error of Simpson’s rule⁷

(the constant is found in Exercise 2.9). Simpson’s rule is of order 4, so halving the spacing reduces the error to about one-sixteenth. With the same ordinates as the trapezoidal rule it gains two orders, at the cost of a constraint on their number, and the gain depends on the curve being smooth (Sec. 2.7).

Simpson’s rule can also be obtained from the trapezoidal rule. Let $Q_h$ and $Q_{2h}$ be the trapezoidal results with spacing $h$ and with every other ordinate, spacing $2h$. Both errors are proportional to the square of the spacing, so the error of $Q_{2h}$ is about four times that of $Q_h$, and the combination $(4Q_h - Q_{2h})/3$ cancels the leading term. Written out, the combination is Simpson’s first rule, the trapezoidal rule with its main error removed. Combining two results to cancel the leading error term works for any rule whose order is known, and gives the error estimate of Sec. 2.7.

Example 2.2. Simpson’s rule on the Wigley waterplane. The design waterline of the Wigley hull has half-breadths of 0, 1.8, 3.2, 4.2, 4.8, 5.0, 4.8, 4.2, 3.2, 1.8 and 0 m at stations 0 to 10, 10 m apart. Find the waterplane area by Simpson’s rule.

In Example 2.2 the parabolic rule gives the waterplane area exactly, where the straight-line rule of Example 2.1 is 1 % out. The volume of the Wigley hull shows the difference again, because its sections are parabolas in $z$ and its sectional-area curve is a parabola in $x$. Simpson’s rule applied to the five waterlines of each section and then to the eleven section areas gives $\nabla = 2777.78$ m³ exactly. The trapezoidal rule, applied in both directions, gives 2707.03 m³, 2.55 % low.

2.4 Adapting the rules to real tables of offsets

The Wigley offsets used so far have eleven stations at one spacing and an even number of intervals, and a real table of offsets is seldom as tidy. The ends of a ship curve more sharply than its middle, so designers add half stations there, as Todd did at stations 0.5, 1.5, 18.5 and 19.5. Near the bottom the bilge turns through a right angle within a metre or two, so the lowest waterlines are placed closer together, as Todd’s are, at the baseline and 0.075 times the draught. Nor is the number of intervals always even. Each of these cases is handled by keeping in view the idea behind Simpson’s rules, a polynomial through a few neighbouring ordinates (a parabola through each successive pair of intervals, or a cubic through three), integrated exactly. The parabola of a pair also gives the area over a single interval. A curve that ends between ordinates needs, in addition, an assumption about its shape beyond the last ordinate.

2.4.1 Half stations

Where the spacing is halved, each pair of half-intervals is a Simpson pair with spacing $h/2$, so its multipliers 1, 4, 1 are halved when everything is written over the common factor $h/3$. Joining the halved pattern ½, 2, 1, 2, ½ at the ends to the normal pattern 1, 4, 2, …, 4, 1 between them, and adding the multipliers where the two meet, gives the sequence for Todd’s 25 stations (Fig. 2.4(b)):

The sum of the products is multiplied by $h/3$, where $h$ is the spacing of the whole stations. Halving the multipliers wherever the interval is halved also serves for quarter stations and for the second rule below^4,8. Some tables halve every multiplier, to ½, 2, 1, 2, …, with 1 and 2 instead of 2 and 4 at the whole stations, and multiply the sum by $2h/3$ instead of $h/3$. The result is unchanged, and these half multipliers save a little arithmetic^8,9. They are used, for instance, in a worked calculation of displacement from section areas⁸.

Figure 2.4 Todd’s table as input to the rules: (a) the design waterline at 25 stations; (b) Simpson multipliers with half stations at both ends.

2.4.2 Simpson’s second rule

A cubic through four equally spaced ordinates, integrated over the three intervals between them, gives Simpson’s second rule,

with composite multipliers 1, 3, 3, 2, 3, 3, 2, …, 3, 3, 1 and the factor $3h/8$. Its weights follow from the condition that the rule integrate 1, $x$, $x^2$ and $x^3$ exactly. With the origin at the middle of the three intervals, the ordinates stand at $\pm h/2$ and $\pm 3h/2$; by symmetry the inner two carry one weight $\beta$ and the outer two another, $\alpha$, and the odd powers are then integrated exactly whatever $\alpha$ and $\beta$ are. The constant and $x^2$ give

so that $\alpha = 3h/8$ and $\beta = 9h/8$, the weights of the second rule⁸. Like the first rule, it is exact for cubics and its error is of order 4. The constant of its error is $1/80$ in place of the $1/180$ of the first rule, and at a given spacing its error is about twice as large. It is used for a number of intervals that is a multiple of three, or, combined with the first rule, for any odd number of intervals: three intervals by the second rule and the rest in pairs by the first⁸.

2.4.3 Unequal intervals

When the two intervals of a pair are not equal, the parabola through the three ordinates can still be integrated exactly. For ordinates $y_0$, $y_1$, $y_2$ with intervals $h_1$ and $h_2$ between them, the result is

which reduces to Simpson’s rule for one pair when $h_1 = h_2 = h$ (Exercise 2.8). The weight of $y_0$ becomes negative when the second interval is more than twice the first, $h_2 > 2h_1$, and that of $y_2$ in the opposite case. For Todd’s lowest waterlines, at 0, 0.075 and 0.25 times the draught, $h_2/h_1 = 7/3$, and the weight of the baseline ordinate is $-0.0139\,T$. The sign follows from the parabola (Fig. 2.5(a)). Raising $y_0$ while $y_1$ and $y_2$ stay fixed makes the parabola curve upwards more strongly. Over the long second interval it then sags below its chord, and the area lost there is more than the area gained over the short first interval. At the Series 60 draught of 6.5024 m the intervals are 0.488 m and 1.138 m, and a rise of 1 m in $y_0$ adds 0.219 m² under the parabola over the first interval and removes 0.310 m² over the second, a net change of −0.090 m², which is the weight of $y_0$ in metres. The negative weight is a property of the parabola, and it allows the rule to amplify errors in the ordinates, the more so the more unequal the intervals (Fig. 2.5(b)).

For any spacing, the intervals are paired from one end and this rule is applied to each pair; if one interval is left over, its area is taken from the parabola through the last three points. This is the construction behind the simpson function of the SciPy library¹⁰: for an odd number of equally spaced ordinates it is Simpson’s first rule, and for any other set it is the piecewise-parabolic rule just described. Version 1.18.1 was checked, and its results agree with this chapter’s scripts to rounding.

Figure 2.5 The negative weight of the rule for unequal intervals: (a) the parabola through Todd’s three lowest midship half-breadths, and with the lowest raised by 1 m (dashed); (b) the weights of the end ordinates against the ratio of the intervals.

2.4.4 The area between two ordinates

The parabola through three equally spaced ordinates can also be integrated over its first interval only, which gives the area between $y_0$ and $y_1$:

This is the five-eight rule, sometimes called Simpson’s third rule^4,11. The 5 belongs to the end ordinate of the pair that bounds the strip, the 8 to the middle ordinate and the −1 to the other end ordinate of the pair; for the other strip of the pair the order is reversed. Attwood checked it on a curve with ordinates of 8.5, 10.4 and 11.85, 2 apart: the five-eight areas of the two halves, 18.97 and 22.32 in his figures (18.975 and 22.325 before rounding), add up to the 41.3 that Simpson’s rule gives for the pair, as they must, since the two sets of weights add up to those of Simpson’s rule. With unequal intervals, $h_1$ the interval integrated and $h_2$ the next, the weights are

With the five-eight rule and its form for unequal intervals, a curve can be integrated up to an ordinate that is not at the end of a Simpson pair, as the integral with a variable upper limit requires (Sec. 2.8.1). Neither form is exact for cubics. Simpson’s rule owes its exactness for cubics to the symmetry about the middle ordinate, and that symmetry is lost when only one interval of the pair is integrated.

2.4.5 Curves ending between ordinates

A curve seldom ends at a station. The design waterline of the Series 60 hull carries on 2.06 m aft of the AP, to the cruiser stern⁶, and a curve of section areas taken to a low waterline may end between stations at both ends. There are four ways of including such a part, each resting on an assumption about the curve beyond the last ordinate, where the table gives nothing.

The first is to measure one extra ordinate, midway between the last station and the extremity, and to integrate the end as a Simpson pair of its own, with a zero ordinate at the extremity. Its multipliers are 1, 4 and 1 times the ratio of the half-length of the end to the station spacing, and they are added to those of the main integration where the two meet, so that one sum gives the whole area⁸. The US Navy’s Ship Hull Characteristics Program worked in this way: where a curve did not end at a station, it extrapolated the curve to the axis and placed one offset midway between the extremity and the first known offset⁸. The second is to replace the end by a shape whose area and centroid are known, a triangle or a parabola, as in Example 2.3. The third keeps the stations and changes the end ordinate instead: a reduced ordinate is chosen, by eye or by a construction, with which the rule gives the area under the real curve⁷. The fourth takes the shape of the end from the table itself. The extremity is taken as an ordinate of zero, and the parabola through it and the two nearest ordinates is integrated over the end by the five-eight rule for unequal intervals, Eq. (2.11). A related rule, for a curve that ends short of a station, also assumes a parabola, fixed by the two last ordinates and a fictitious, negative ordinate read at the station beyond the extremity from the curve continued past it⁸.

The endings that these assumptions give for the Series 60 waterline are drawn in Fig. 2.6. A parabola with its vertex at the AP leaves the AP without slope, and a parabola with its vertex at the extremity meets the centreline at a right angle. Both enclose two-thirds of the rectangle that bounds them, so each has an area of 1.83 m² for the two sides together, but their centroids lie at 3/8 and at 0.4 of the length of the end aft of the AP. The smooth surface carries the waterline almost straight from the AP to the extremity and encloses 1.39 m². The three curves cross the extra ordinate at 0.500 m, 0.471 m and 0.340 m. With these midway ordinates, Simpson’s rule on the extra pair gives 1.83 m² for the parabola with its vertex at the AP, which it integrates exactly, and 1.39 m² for the surface. For the parabola with its vertex at the extremity it gives 1.75 m², because that curve meets the centreline with a vertical tangent, as Chapman’s arc meets its axis, and the parabola of the rule cannot follow it. In the fourth way the ordinates are 0 at the extremity, 0.666 m at the AP, 2.06 m forward of it, and 1.569 m at station ½, a further 3.048 m forward. Equation (2.11) gives 0.694 m² for one side and 1.39 m² for both, and the first moment of the parabola puts its centroid 0.335 of the length of the end aft of the AP. These are the values of the surface’s ending to the precision printed. Both curves pass through the three points, and between them they hardly differ: the parabola crosses the extra ordinate at 0.339 m, against 0.340 m for the surface, and Fig. 2.6 does not draw it separately. The ordinate at the AP and the length of the end are known, but the shape between them is not, and the possible endings differ in area by about a quarter of the area of the end. That is 0.03 % of the waterplane; its effect on the centre of flotation is found in Example 2.5.

Figure 2.6 Three possible endings of the Series 60 design waterline aft of the AP, with the extra ordinate midway between the AP and the end of the waterline.

A volume also has ends beyond its ordinates, the part below the lowest waterline, where the keel rises aft or the hull narrows to a fin, and the parts beyond the end stations. Chapman allowed for both on his privateer, with the small triangle below the lowest waterline in each section and a separate estimate of the slivers beyond the end sections. Such parts are computed separately, each with its volume and centre, and combined with the main body by moments, as in Attwood’s worked calculation of displacement for the appendage below its lowest waterplane⁴. The volume of the shell and of true appendages, which lie outside the moulded surface, is also added separately (Sec. 3.4.6).

Table 2.2 Waterplane sheet for the design waterline of the Series 60 hull, from Todd’s offsets: the area. $y$ is the half-breadth (m), printed to 1 mm; products are computed from the unrounded half-breadths. The multipliers are those of Fig. 2.4(b).

Example 2.3. The waterplane of the Series 60 hull from Todd’s table. Using the half-breadths of the design waterline at Todd’s 25 stations (Table 2.2), find the waterplane area by Simpson’s rule with half stations, add the small part of the waterplane aft of the AP, where the waterline ends 2.06 m behind it, and find $C_{WP}$. The hull has $L_{PP}$ = 121.92 m and $B$ = 16.256 m.

2.5 Moments, centroids and second moments

The area of the Series 60 waterplane came from 25 half-breadths and their multipliers in Table 2.2. Its centre of flotation needs one more column on the waterplane sheet, because a centre is a first moment divided by an area, and a first moment is the integral for an area with the integrand multiplied by a distance. The rule that gave the area therefore serves for the moment, applied to the products $x\,y$ instead of $y$. In a hand calculation the distance is written as a lever in units of the station spacing, here the station number $k$, with $x = k\,h$, and the factor $h$ is applied once at the end:

where $m_i$ are the multipliers. Any reference point for the levers will do, provided that all the products of one moment use it. Levers measured from amidships, positive forward and negative aft, keep the numbers small and were the traditional choice⁴. In this section the first moment is taken about the AP, the origin of every longitudinal position, and its levers are the station numbers; the second moment is taken about the transverse axis amidships, where the squares of the levers are smaller.

2.5.1 Moments of a strip and the transverse second moment

The integrands come from the moments of a strip (Fig. 2.7(a)). A strip of the waterplane, of width $dx$ and half-breadth $y$ on one side, has area $y\,dx$, and all of it lies at the distance $x$ from the AP, so its first and second moments about the AP are $x\,y\,dx$ and $x^2 y\,dx$. About the centreline the distance varies across the strip, from 0 at the centreline to $y$ at the waterline. The centroid of the strip is at $y/2$, which gives its first moment about the centreline, $\tfrac{1}{2}y \cdot y\,dx = \tfrac{1}{2}y^2\,dx$. For any plane area whose ordinates stand on a straight base, the integrand $\tfrac{1}{2}y^2$ gives the moment about that base (Exercise 2.6). The second moment of the strip about the centreline adds up the squares of the distances $\eta$ of its elements $d\eta\,dx$,

and for both sides of the waterplane it gives the transverse second moment of the waterplane, about the centreline^8,9,

It is called transverse because it governs heel, the inclination of the ship in the transverse plane. How far the centre of buoyancy moves sideways when the ship heels by a small angle depends on $I_T$ and on the volume, and that movement fixes the position of the metacentre. Second moments of area are also called moments of inertia, and $I_T$ is the transverse moment of inertia of the waterplane¹². It depends only on the geometry of the waterplane and is measured in m⁴. The mass moment of inertia of the ship in roll is a different quantity¹², the mass of the ship times the square of its radius of gyration about a longitudinal axis through its centre of gravity, measured in t·m². The integrand of $I_T$ is the cube of the half-breadth, so the breadth of the waterplane dominates its value.

The transverse inertia coefficient compares $I_T$ with $B^3L/12$, the second moment of the rectangle $L \times B$ about its centreline^12,13,

which is 1 for a rectangle, 48/105 = 0.457 for the Wigley waterplane (Example 2.4) and 0.544 for the Series 60 waterplane (Example 2.5). The second moment of the rectangle overstates $I_T$ by the factor $1/C_{IT}$, 2.19 for the Wigley waterplane and 1.84 for the Series 60, and $B^3L/12$ is no substitute for integrating the cubes of the half-breadths.

2.5.2 The longitudinal second moment

The longitudinal second moment $I_L$ is the second moment of the waterplane about the transverse axis through its centre of flotation F. It is taken first about a convenient transverse axis, through the AP or amidships, and then moved to F (Fig. 2.7(b)). Writing each distance from the AP as $x = (x - x_F) + x_F$ and squaring it,

The middle integral is the first moment of the waterplane about an axis through its own centroid, which is zero by the definition of the centroid, and the last term is $A_W\,x_F^2$. What remains is the parallel-axis theorem,

The second moment about an axis through the centroid is the smallest of all those about parallel axes: moving the axis a distance $d$ away from the centroid adds $A\,d^2$, where $A$ is the area, and moving it back to the centroid subtracts it^8,14. A sign error here is easy to make and hard to see, because both results look plausible.

Figure 2.7 Moments of the Series 60 design waterplane: (a) one strip; (b) parallel axes through the AP and through the centre of flotation F. Half-breadths are drawn at 5 times (a) and 2.5 times (b) the scale of lengths.

The longitudinal second moment governs trim, the inclination of the ship in the longitudinal plane, and it is taken about F because a ship at constant displacement trims about F. Only about an axis through the centroid of the waterplane are the wedges immersed and emerged in a small trim equal in volume⁹. The longitudinal inertia coefficient compares $I_L$ with $BL^3/12$, the second moment of the rectangle $L \times B$ about its transverse axis through the centre¹²,

The design waterline of the Wigley hull is $y_W = \tfrac{B}{2}(1 - \xi^2)$ with $\xi = 2x/L - 1$, and its centre of flotation is amidships, so that

and $C_{IL}$ = 0.4; the Series 60 waterplane has $C_{IL}$ = 0.428 (Example 2.5).

The two second moments take their value from different parts of the waterplane. In $I_L$ each strip is weighted by the square of its distance from F, and in $I_T$ by the square of its own half-breadth. The quarters of the length at the two ends of the Wigley waterplane, where $|\xi| \ge \tfrac{1}{2}$, hold 5/16 of its area but give 47/64 of $I_L$, about 73 %, and only 289/2048 of $I_T$, about 14 %:

Errors in the end offsets, half stations included, and in the shape assumed aft of the AP matter far more for $I_L$ than for $I_T$.

Second moments about other axes are built from strips and moved by the parallel-axis theorem. A waterplane that is not symmetric about the centreline, as at an angle of heel or with one side damaged, also has a product of inertia, and its principal axes are no longer the centreline and a transverse line¹⁴. The heeled waterplanes of Chapter 8 and the damaged waterplanes of Chapter 12 are asymmetric in this way. The product of inertia of a polygon is given in Sec. 2.9.3.

2.5.3 Accuracy of the rules for second moments

A rule that is exact for the half-breadth need not be exact for its cube, the integrand of $I_T$, or for $x^2 y_W$, the integrand of the second moment about a transverse axis. The accuracy of $I_T$ matters because in Chapter 4 it is divided by the volume to give the distance through which the centre of buoyancy moves sideways per radian of a small heel, and a relative error in $I_T$ passes unchanged into that distance.

Example 2.4. The transverse second moment of the Wigley waterplane. The design waterline of the Wigley hull, 100 m long and 10 m wide, has half-breadths of 0, 1.8, 3.2, 4.2, 4.8, 5.0, 4.8, 4.2, 3.2, 1.8 and 0 m at stations 0 to 10, 10 m apart. Find $I_T$ of the waterplane by Simpson’s rule and by the trapezoidal rule, and compare both with the exact value. Find $C_{IT}$.

Simpson’s rule integrated the parabolic Wigley waterline exactly, but the cube of the half-breadth has degree 6 and $x^2 y_W$ has degree 4. With ten intervals its error is −0.087 % in $I_T$ and −0.16 % in $I_L$ about amidships; Hamlin warns of such errors in general terms⁸. The order of a rule is also an incomplete guide to its accuracy. The error of the trapezoidal rule can be expanded as a series, the Euler–Maclaurin formula¹⁵, in which the term in $h^2$ is proportional to the difference of the slopes of the integrand at the two ends, $-\tfrac{h^2}{12}\left[f'(b) - f'(a)\right]$, and the next term, $+\tfrac{h^4}{720}\left[f'''(b) - f'''(a)\right]$, to the difference of the third derivatives. The cube of a half-breadth that falls to zero at both ends has zero slope there, because $d(y^3)/dx = 3y^2\,dy/dx$, so the $h^2$ term vanishes. The trapezoidal rule then behaves as a rule of order 4, with an error a quarter of Simpson’s and of the opposite sign; in Example 2.4 the trapezoidal rule is 0.88 m⁴ high and Simpson’s rule 3.30 m⁴ low. The effect occurs on fine waterlines with pointed ends and disappears for a waterline that ends in a transom, where $y$ is not zero.

2.5.4 The Series 60 waterplane

For the Series 60 waterplane the moments are added to the sheet of Table 2.2 as further columns, in Table 2.3.

Table 2.3 Waterplane sheet for the design waterline of the Series 60 hull, continued from Table 2.2: the moments. The lever for the moment is the station number $k$ (each lever times $h$ is the distance forward of the AP), and the product for the moment is the product for area times $k$. The product for $I$ about amidships is the product for area times $(k - 10)^2$, the square of the lever from amidships, and the product for $I_T$ is the multiplier times $y^3$. As in Table 2.2, products and sums are computed from the unrounded half-breadths; sums formed from the printed entries, or from the rounded half-breadths, differ by less than 0.003 %, which leaves the results of Example 2.5 for $x_F$, $I_T$ and $I_L$ unchanged at the precision given there.

Example 2.5. Centre of flotation and second moments of the Series 60 waterplane. From Tables 2.2 and 2.3, find the centre of flotation, $I_T$ and $C_{IT}$, and $I_L$ about the centre of flotation and $C_{IL}$. The stations are 6.096 m apart, $L_{PP}$ = 121.92 m and $B$ = 16.256 m. The waterplane area is 1402.12 m² between the perpendiculars and 1403.95 m² with the part aft of the AP, which has an area of 1.83 m² and ends 2.06 m aft of the AP. Take the centroid of that part three-eighths of its length from the AP, and neglect its second moments about its own axes.

2.5.5 The moment of the area between two ordinates

The first moment of a single strip was a familiar source of error in hand calculation. The five-eight rule gives the area between two ordinates, but multiplying its three products by their levers does not give the moment of that area, because the parabola was integrated over only part of its length. The correct moment about $y_0$ of the area between $y_0$ and $y_1$ comes from integrating $x$ times the parabola through the three ordinates over the first interval,

the three-ten rule. Attwood⁴ showed the error with a curve whose ordinates are 10, 9 and 7, 4 apart: weighting the five-eight products by levers puts the centroid 2.02 from the end ordinate, when the shape of the curve requires slightly less than 2, and the three-ten rule gives 1.965.

2.5.6 An integration sheet, and a waterplane of your own (Calculator 2.1, Explorer 2.1)

Calculator 2.1 sets out the integration of a plane area as a sheet, in the form of Tables 2.2 and 2.3: for each ordinate, the weight of the rule and the products that give the area, the first moment, the second moment and, for a waterplane, the cubes of the half-breadths. The rule is Simpson’s first rule when the ordinates are equally spaced and their number is odd, and otherwise the piecewise-parabolic rule; the trapezoidal rule is shown beside it for comparison. Below the sheet, the working gives the area, the centroid, the second moments and the moment of the area about the centreline or the base.

The calculator walks through the sheets of the worked examples, for the Wigley and Series 60 design waterlines, the waterplane of Exercise 2.5 and Todd’s midship section. The ordinates of each are fixed, so that the working can be followed line by line against the example. It checks each result against what any area whose ordinates are not negative must satisfy: a positive area, a centroid between the first and the last ordinate, and second moments that are not negative. When a result fails these checks, the spacing is too uneven for the rule, as in the problem “Very uneven spacing”, and the calculator says so instead of reporting the number as a property of the area. Passing the checks does not show that a result is accurate; its error has to be estimated separately (Sec. 2.7).

Explorer 2.1 is a waterplane to build: its length, the number of ordinates and each half-breadth are set by hand, the shape is drawn as the rule sees it, through the parabola of each pair of intervals, and the sheet, the area, the centre of flotation, the second moments and the coefficients change with it. It opens on the Wigley waterline of Example 2.2, so the first area shown is 666.67 m². Where the ordinates change sharply, a parabola can dip below zero between them or rise above the largest ordinate, and the Explorer reports it. It gives no results for a waterline that crosses the centreline, and it takes the breadth $B$ from the widest point of the drawn curve.

2.6 Volumes by integrating twice

The volume was written in Chapter 1 in two ways, as an integral of section areas along the length and as an integral of waterplane areas up the height. Either way needs two integrations, taken in one of the two orders of Fig. 2.8. In the first, each section is integrated up to the waterline to give its area $A(x)$. Plotted against $x$, these areas form the sectional-area curve: the area under it is the displacement volume, and the $x$-coordinate of its centroid is the longitudinal centre of buoyancy $x_B$, so integrating the curve along the length, with and without the lever $x$, gives both. In the second, each waterline is integrated along the length to give the waterplane area $A_W(z)$, and these areas are integrated up the height. The area between the curve of waterplane areas and the height axis is again the volume, and the height of its centroid is $z_B$. Chapman used both, the first for the volume and its longitudinal centre, the second for its vertical centre.

Figure 2.8 Two routes from Todd’s table to the displacement volume of the Series 60 hull: (a) by sections; (b) by waterplanes. Points are areas from the table, curves smooth guides through them. Heights are drawn 2.5 times (a) and 9 times (b) as large as lengths.

2.6.1 The displacement sheet

When both integrations were done by hand, they were set out together on one large form, the displacement sheet^4,8. Its rows are the stations and its columns the waterlines (Fig. 2.9), and every offset of the table is entered once. Each row is multiplied, ordinate by ordinate, by the multipliers of the rule up the height and summed, to give a function of the area of that section. Each column is multiplied by the multipliers along the length and summed, to give a function of the area of that waterplane. The functions of the section areas, multiplied by the multipliers along the length, and the functions of the waterplane areas, multiplied by those up the height, must give equal grand totals, and this equality checks the arithmetic of the sheet. Levers in the margins give the moments: station numbers for the longitudinal centre of buoyancy and waterline numbers for the vertical centre.

Figure 2.9 The layout of a displacement sheet for eleven stations and five waterlines, with Simpson’s multipliers. Each square is one offset; one section (solid arrow) and one waterplane (dashed arrow) are followed to the grand total.

For the Wigley hull the sheet is set out in Table 2.4. Its sections are parabolas in $z$ and its waterlines parabolas in $x$, so Simpson’s rule is exact in both directions and every entry can be checked against the formula of the hull. The displacement sheet of the Series 60 hull, from Todd’s table and with the weights of the next subsection up the height, is given with the chapter’s data (Notes on the data).

Table 2.4 Displacement sheet for the Wigley hull. The half-breadths are in metres; the column headings give the height of each waterline above the keel and its multiplier. The station multipliers are Simpson’s, with $h$ = 10 m, and the waterline multipliers Simpson’s, with $h_z$ = 1.5625 m. The lever for the moment along the length is the station number; up the height it is the waterline number, 0 to 4 from the keel or 4 to 0 from the waterline.

Example 2.6. The displacement sheet of the Wigley hull. The Wigley hull is 100 m long and 10 m wide and floats at a draught of 6.25 m. From Table 2.4, find the displacement volume by sections and by waterplanes, the longitudinal centre of buoyancy, and the height of the centre of buoyancy above the keel, with a check by moments about the waterline. Compare with the exact values $\nabla = \tfrac{4}{9}LBT$, $x_B = L/2$ and $z_B = \tfrac{5}{8}T$.

2.6.2 Sections with uneven waterlines

Todd’s waterlines are at 0, 0.075, 0.25, 0.5, 0.75 and 1.0 times the design draught: five intervals, the first two unequal. A suitable rule for them combines two of the rules above. The first two intervals, 0 to 0.075 and 0.075 to 0.25 of the draught, form a pair for the rule for unequal intervals; the remaining three, of 0.25 each, are a case for Simpson’s second rule. In metres, for $T = 6.5024$ m, the weights of the six ordinates are −0.0903, 1.2902, 1.0354, 1.8288, 1.8288 and 0.6096 (Fig. 2.10(a)); they add up to the draught, as the weights of any rule must add up to the length of the interval, since the rule has to integrate a constant exactly.

Amidships the half-breadths are 5.771 m at the baseline (the half-width of the flat bottom, or flat of keel), 7.039 m at 0.075$T$ and 8.006 m at 0.25$T$, and 8.128 m, the full half-breadth, above. The rule gives a half-area of 51.53 m², so $A_M = 103.07$ m² and $C_M = 0.975$. What the rule assumes between the ordinates is drawn in Fig. 2.10(b). Above half the draught the section is wall-sided (its side is vertical), and the rule integrates that part exactly. Below half the draught lies the whole turn of the bilge, where the half-breadth grows from 5.77 m to 8.13 m. The table gives four ordinates there, three of them up to a quarter of the draught, and the area depends on the curves that the rule draws through them: a parabola through the lowest three and a cubic from a quarter of the draught upwards.

The rule is also least reliable at the bilge. A round bilge leaves the flat of keel along an arc of a circle. Measured upwards from the keel, the half-breadth grows at first as the square root of the height, as the ordinate of Chapman’s arc does with the distance from the end of the quadrant, and no parabola can follow it. Todd gives the bilge radius as $R = 0.229\sqrt{BT} = 2.35$ m. If all six ordinates are taken from a circular bilge of that radius, whose section has an exact area of 103.32 m², the rule gives 103.52 m², 0.19 % too much, and the trapezoidal rule 102.74 m², 0.56 % too little. The table adds an uncertainty of its own. The circle passes 0.075$T$ at 7.208 m, where the table gives 7.039 m, and the ordinates give no means of deciding which is correct. Both effects amount to a few tenths of a per cent of the section area.

Figure 2.10 The rule for Todd’s waterlines: (a) the weights of the six ordinates; (b) the curves it draws through the midship ordinates of the Series 60 hull, with the circular bilge of Todd’s tabulated radius (dashed).

2.6.3 The volume, both ways

Applying the rule for Todd’s waterlines at all 25 stations gives the sectional-area curve, which agrees with Todd’s tabulated area fractions to within 0.004 of $A_M$ at every station. Integrated along the length with Simpson’s multipliers, half stations included, it gives $\nabla = 7719.9$ m³. By the other route, the six waterplane areas from the baseline to the design waterline are 610.02, 885.48, 1115.05, 1228.47, 1299.65 and 1402.12 m² between the perpendiculars, and integrating them up the height with the rule for Todd’s waterlines also gives 7719.9 m³.

That the two routes agree should not be taken as a check of accuracy. Both use every ordinate of the table, each multiplied by a weight $w_i$ in $x$ and a weight $v_j$ in $z$; the volume is the double sum $2\sum_i \sum_j w_i\,v_j\,y_{ij}$, and the order of summation does not change it. The agreement checks the arithmetic, and that was its use when the arithmetic was done by hand^8,9; it does not show whether the rules suit the hull. The comparison of the two routes becomes informative only when they use different information, for instance when the ends beyond the end stations are added in one route and not in the other. The cruiser stern adds a part of the volume aft of the AP, above about 0.9$T$, that is in neither sum; the smooth surface puts it at 0.42 m³, which is negligible.

Nor does the agreement show whether the offsets are right. An offset misread from the drawing or mistyped enters both routes alike and leaves the totals in agreement. Such an error shows instead as a point off a fair curve when the section areas or the waterplane areas are plotted^4,8. Where the values are equally spaced, it also shows in their second differences, the differences between successive differences, as one value out of line, with its neighbours moved half as far the other way.

Taken with levers, the sums of the two routes give the centre of buoyancy: $x_B = 59.17$ m, 1.47 % of the length aft of amidships, and $z_B = 3.48$ m.

A third route to $z_B$ uses the vertical moment of each section area about the base, $2\int z\,y\,dz$ up to the waterline; integrating these moments along the length gives $\nabla z_B$. With Simpson’s multipliers along the length and the rule for Todd’s waterlines up the height, it is again the double sum of the table, now with the lever $z_j$ as well, and it also gives 3.48 m. This route is useful because each section is integrated up to the point where the waterline cuts it. It works equally for a waterline that is not parallel to the base, when the ship is trimmed, and for a ship flooded over part of its length⁸ (Chapters 7 and 12).

Todd’s values and those of the smooth surface complete the comparison:

Example 2.7. Midship section and displacement from Todd’s table. The Series 60 hull has $L_{PP}$ = 121.92 m, $B$ = 16.256 m and a design draught $T$ = 6.5024 m. (a) At Todd’s six waterlines, from the baseline to the design waterline, the half-breadths of the midship section are 5.771, 7.039, 8.006, 8.128, 8.128 and 8.128 m, and the weights of the rule for these waterlines are −0.0903, 1.2902, 1.0354, 1.8288, 1.8288 and 0.6096 m. Find the area of the midship section and its $C_M$. (b) The sum of the products of the 25 section areas and their multipliers, with stations 6.096 m apart, is 3799.18 m². Find $\nabla$, $C_B$ and the displacement mass in sea water.

2.7 Estimating the error

For the Wigley hull the error of every result so far could be read off against the exact value. For the Series 60 results there is no exact value, and the error has to be estimated. The error formulae of the trapezoidal rule and Simpson’s rule show how an error changes with the spacing, but not how large it is, because the derivatives they contain are not known. In practice the error is assessed by separate checks, each catching a different source of error. The error that arises where the table does not define the hull closely enough cannot be revealed by comparing rules and needs a check of its own.

Whether the rules have been applied correctly is settled by a test case with a known answer. The Wigley hull is such a case; a scheme that does not return $\nabla = \tfrac{4}{9}LBT$ and $A_W = \tfrac{2}{3}LB$ from its offsets by Simpson’s rule is wrong. The size of the rule’s error for a given hull and spacing is estimated by repeating the calculation with a different spacing. If $Q_h$ and $Q_{2h}$ are results with spacing $h$ and $2h$ from a rule of order $p$, their errors are in the ratio $1 : 2^p$, and eliminating the error between them gives an estimate both of the error and of a better value,

This is Richardson extrapolation, and for the trapezoidal rule ($p = 2$) it is the combination that gave Simpson’s rule. For Simpson’s rule ($p = 4$) the correction is one-fifteenth of the difference between the two results.

On Todd’s table the test can be made by dropping stations. With the eleven even-numbered stations, spacing $2h$, Simpson’s rule gives $\nabla$ = 7729.6 m³; with all 21 whole stations, 7722.2 m³; with the half stations as well, 7719.9 m³. By Richardson extrapolation, the error of the 21-station result is about $(7722.2 - 7729.6)/15 = -0.5$ m³, which would put the integral at 7721.7 m³. The half stations give 7719.9 m³, 2.3 m³ below the 21-station result and 1.8 m³ below the extrapolated value. They add information near the ends, where the sectional-area curve changes too quickly for its error to fall as $h^4$ at this spacing, and the halving test cannot detect that part of the error. The error of the integration along the length is estimated at a few hundredths of a per cent. The integration up the height is less certain: replacing the second rule on the upper three intervals by parabolas through pairs, with the last interval taken from the parabola through the last three ordinates, which is equally a rule of order 4, changes $\nabla$ to 7724.9 m³, a difference of 0.06 %. Replacing the rules of order 4 by the trapezoidal rule in one direction or in both shows which direction carries the error:

The trapezoidal rule in both directions is nearly 1 % low, and most of that comes from the vertical integration, where the straight lines cut the corner at the bilge. Todd’s own displacement, from his faired lines, is 7737.4 m³, 0.23 % above our result. That difference is of the order of the two effects found at the bilge, the rule’s error where the half-breadth grows as the square root of the height and the table’s own uncertainty there. Between them they account for most of it. With a rule of order 4 and stations as close as Todd’s, the accuracy of the volume is limited by the bilge rather than by the integration along the length.

The error of a good integration is small beside the other uncertainties of a displacement. The density of sea water varies with temperature and salinity, and the 1025 kg/m³ used in design is a convention⁷; the moulded volume differs from the true displacement by the shell and appendages, typically by 0.5 % to 1 %. A ship may also carry a stability instrument, an approved onboard computer that calculates the ship’s hydrostatics and stability for the loading carried. When such an instrument computes from a model of the hull, the IMO Guidelines for the approval of stability instruments, MSC.1/Circ.1229, accept a displacement that differs by up to 2 % from the ship’s approved data¹⁶ (Sec. 2.10.2). An error of 0.01 % from an integration rule is negligible on either scale, and an error of about 1 %, such as the trapezoidal rule made on Todd’s table, would be within that tolerance. It is not small in other terms: 1 % of the Series 60 volume is 77 m³, or 5.5 cm of mean draught, a large amount in a lightweight survey or an inclining experiment (Chapter 10).

Richardson extrapolation assumes that the error falls as $h^p$, and the assumption should be checked, for instance on Chapman’s quadrant of the prologue. On his five ordinates, 1 apart from the centre to half the radius, Simpson’s rule missed the exact area, 30.6116, by 0.00025, less than a thousandth of a per cent. Had he carried ordinates 1 apart to the end of the quadrant, where the arc meets the axis at a right angle, nine ordinates would have given 49.939 against the exact 50.265, an error of 0.65 %, nearly eight hundred times the relative error of his own test. Over half the radius Chapman’s curve is smooth and Simpson’s rule converges as $h^4$; over the whole quadrant every derivative of the integrand is unbounded at the end of the arc, and the error falls only as $h^{1.5}$ (Fig. 2.11). With ordinates 4, 2 and 1 apart, Simpson’s rule gives 47.617, 49.338 and 49.939 for the quadrant. Extrapolating the last two with $p = 4$ gives 49.979, still 0.57 % low, and the estimate of the error is too small by a factor of eight. The order can be found from the three results. The ratio of successive differences, $(49.338 - 47.617)/(49.939 - 49.338) = 2.86$, is $2^p$ with $p = 1.52$, and extrapolating with that order gives 50.262, within 0.006 %. Three results at halving spacings allow the assumption to be tested before it is used; two results do not.

Figure 2.11 How the error falls as the spacing is reduced, for five pairs of rule and integrand. The slope of each line is the order of the rule for that integrand.

Explorer 2.2 applies the trapezoidal rule or Simpson’s first rule, with any even number of intervals up to 64, to the five integrands of Fig. 2.11, and shows the pieces that the rule integrates, its true error beside the error estimated by halving the number of intervals, and the error of both rules against the number of intervals.

Three faults spoil the result of a rule. The first is a point where the curve is not smooth: a jump in the slope, at a knuckle or chine; a jump in the ordinate, at the end of a transom; or a derivative that becomes infinite, as at the end of the quadrant or where a round bilge leaves the flat of keel. The edge of a flat of side, a part of the side that is flat, need not be one of these, since a fair transition keeps its slope and changes only its curvature, which lowers the order less. Attwood⁴ gave the remedy for a corner: “If the curvature changes abruptly at any point, this point must be at a dividing ordinate”, that is, at the end of a Simpson pair, and then no parabola spans the corner. An infinite derivative can be removed by a change of coordinates: measured from the centre of the bilge radius, a circular bilge has a constant radius, which polar integration (Sec. 2.9.2) integrates exactly. A parametric description of the section (Sec. 1.5.4) removes the difficulty in general. Both coordinates are then smooth functions of a parameter along the curve, and their derivatives stay finite where the section has a vertical tangent, so the area becomes an integral along the curve with the parameter as the variable¹⁴.

A second fault is a stretch where the curve changes too quickly for the spacing, which closer ordinates cure; half stations and Todd’s 0.075$T$ waterline serve this purpose, and in the quarter circle, halving only the last two intervals reduces the error of the nine-ordinate result from 0.65 % to 0.23 %. The third fault lies in the domain of integration rather than in the rule. A part of the hull beyond the end ordinates, such as a cruiser stern or a bulb, is not included by any rule applied between the perpendiculars. Such a part is added separately, as Chapman did and as we did for the cruiser stern, and the result depends on the shape assumed for it (Fig. 2.6).

2.8 Integrals with a variable upper limit

Many of the quantities of later chapters are curves. The volume of the hull to every draught, from which the draught at which a loaded ship floats is read (Chapter 3) and the hydrostatic curves are drawn (Chapter 7), is $\nabla(z) = \int_0^z A_W\,d\eta$; Chapman already calculated displacement as a function of draught. The Bonjean curves are the section areas to every height. The work done in heeling a ship to a given angle (Chapter 9) is its weight times the area under the curve of righting levers up to that angle. Each of these is an integral with a variable upper limit, wanted at every ordinate and not only at the last.

2.8.1 Running totals and the volume curve

The trapezoidal rule gives such a running total directly: each new strip adds $\tfrac{1}{2}(x_{i} - x_{i-1})(y_{i-1} + y_i)$ to the total so far⁷. With $Y_i$ for the approximation to $\int_{x_0}^{x_i} f\,dx$ and $Y_0 = 0$,

With Simpson’s rule the total is available directly only at the end of each pair. The value at the ordinate inside a pair comes from the five-eight rule, or its form for unequal spacing: the first strip of the pair is integrated under the parabola of that pair and added to the total at the end of the previous pair⁸. The pair is then completed by the rule for the whole pair. The running total is the integral of one piecewise parabola at every ordinate, and at the end of each pair it is the result of the rule for unequal intervals, or of Simpson’s rule when the intervals are equal. When the number of intervals is odd, the last strip has no pair of its own and is taken from the parabola through the last three ordinates. SciPy’s cumulative_simpson¹⁰, checked in version 1.18.1, follows this scheme and returns running totals identical to those of the chapter’s scripts.

Example 2.8. The volume curve of the Series 60 hull from Todd’s waterplanes. The waterplane areas of the Series 60 hull between the perpendiculars at Todd’s six heights, 0, 0.075, 0.25, 0.5, 0.75 and 1.0 times the design draught of 6.5024 m, are 610.02, 885.48, 1115.05, 1228.47, 1299.65 and 1402.12 m². Find the volume of the hull up to each of those heights, and compare it with the smooth surface, which gives 1544.6, 3461.9 and 5516.8 m³ at a quarter, half and three-quarters of the design draught.

In Animation 2.1 the route by waterplanes is followed as a running integral. As the waterline rises, each waterplane adds its area to the curve, and the area between the curve and the height axis, the volume below the waterline, grows with it.

Animation 2.1 The waterline of the Series 60 hull rising from the keel to the design waterline. Above, the profile with the rising waterline; below it, the waterplane at that height in plan, with its height and its area A_W; on the right, the curve of waterplane areas traced against height, with the area between it and the height axis shaded and stated as the volume below the waterline. The waterplane area grows from 610 m² on the flat of keel to 1403 m² at the design waterline, and the volume reaches 7709 m³, with its centroid at z_B = 3.49 m. These are values of the smooth surface built through Todd’s offsets; Todd’s table itself gives 7719.9 m³ and 3.48 m. Heights in the profile are drawn 2.5 times, and breadths in the plan twice, as large as lengths. No sound; 12.5 s.

The volume curve is least reliable near the keel^7,8, and for this hull the reason can be seen in Fig. 2.12(a). The waterplane area rises out of the flat of keel as the half-breadth does at a round bilge, at first roughly as the square root of the height, and the parabola through the three lowest waterplanes cannot follow that rise. Below about 0.5 m it lies under the curve of the surface, and above that height it lies over it, peaking at about 1118 m² just below a quarter of the draught, above the tabulated 1115 m². Up to 0.075$T$ the table-based volume is 14 m³ less than that of the surface, and from there to a quarter of the draught 31 m³ more, which leaves the difference of 1.1 % found in Example 2.8. The remedy, as for the bilge, is to place the waterlines closer together where the area changes fastest, and Chapter 7 applies it to the hydrostatic curves.

Figure 2.12 The Series 60 hull from Todd’s table and from the smooth surface (dashed): (a) waterplane area against height; (b) volume to each height, with the two areas that give the centre of buoyancy shaded.

2.8.2 The centre of buoyancy from the volume curve

The volume curve also gives the height of the centre of buoyancy. The vertical moment of the volume below a waterline at height $T$ is $\int_0^T z\,A_W\,dz$, and since $A_W\,dz$ is the volume $d\nabla$ of the layer between $z$ and $z + dz$, integration by parts turns it into an integral of the volume curve:

where $\nabla(z)$ is the volume up to height $z$ and $\nabla$ its value at $T$. In Fig. 2.12(b) the integral is the area between the volume curve and the height axis. The curve divides the rectangle of height $T$ and width $\nabla$ into that area, $\nabla(T - z_B)$, and the area to its right, $\nabla z_B$, so the depth of the centre of buoyancy below the waterline is the first area divided by the volume^4,8. Once the volume curve has been computed as a running integral, this relation gives $z_B$ with one more integration and no moments (Exercise 2.18); it needs the curve to be well defined at its lower end, where it is least reliable.

2.9 Other rules and methods

Everything so far has taken the ordinates where the table of offsets gives them, at fixed stations and waterlines. The rules below either use more of those ordinates at once, or place the ordinates where a surface that can be cut anywhere allows, or do without ordinates, by describing the section as a polygon or tracing it with an instrument. The rules used so far belong to a family, the Newton–Cotes rules, which integrate the polynomial through equally spaced ordinates. With five ordinates the family gives Boole’s rule, exact for polynomials of degree 5, with weights proportional to 7, 32, 12, 32 and 7; Hamlin⁸ tabulates the family up to nine ordinates. Adding ordinates does not improve the family indefinitely. The nine-ordinate rule, and every rule with eleven or more, has negative weights. The absolute values of its weights then add up to more than the length of the base, and errors in the ordinates can be amplified rather than averaged. For tables of offsets, composite Simpson rules on closely spaced stations are the better choice.

2.9.1 Tchebycheff and Gauss rules

Two other families drop the equal spacing. The Chebyshev rules, which naval architecture traditionally spells Tchebycheff, give all the ordinates equal weights and place them where equal weighting is most accurate. A hand calculation with them needs no multipliers, only a sum, and this was their attraction when ordinates were measured from a drawing^8,9. For two ordinates, measure $x$ from the middle of a base of half-length $s$, and place the ordinates at $\pm\lambda$, each with the weight $\omega$. Odd powers of $x$ are then integrated exactly by symmetry, and the rule must also integrate 1 and $x^2$ exactly:

so that $\omega = s$ and $\lambda = s/\sqrt{3} = 0.57735\,s$. The area is the mean of the two ordinates times the length of the base, and since $x^3$ is integrated exactly as well, two ordinates are exact for cubics, as the three of Simpson’s first rule are. With more ordinates the conditions become equations of higher degree in the positions, whose solutions are tabulated (Table 2.5). Classical Tchebycheff rules, in which $n$ equally weighted ordinates integrate every polynomial up to degree $n$ exactly, have all their ordinates real for up to seven ordinates and for nine, and for no other number¹⁵. The eight- and ten-ordinate Tchebycheff spacings in the tables of Hamlin and Tupper are two four- or five-ordinate rules placed side by side on the two halves of the base. The Gauss rules choose both the positions and the weights, and $n$ ordinates then integrate a polynomial of degree $2n - 1$ exactly, the highest degree that $n$ ordinates can reach; the positions are the roots of the degree-$n$ Legendre polynomial^9,15. For two ordinates the Gauss and Tchebycheff rules coincide, and three Gauss ordinates are as exact as the five of Boole’s rule.

Table 2.5 Positions of the ordinates of the Tchebycheff and Gauss rules, as fractions of the half-length of the base measured from its middle, and the weights of the Gauss rules as fractions of the length of the base. Each Tchebycheff ordinate has the weight $1/n$; the integral is the length of the base times the weighted sum of the ordinates.

Positions that are not equally spaced are of no use for a table of offsets measured at fixed stations, but they are useful when the hull is a surface that can be cut anywhere, as it is in every modern design system. The rules, each with five ordinates or fewer, are compared on two integrands in Table 2.6. On Chapman’s smooth arc the errors fall as the degree of exactness rises, and the Gauss rule with five ordinates is better than Simpson’s rule by a factor of twenty thousand. On the cube of the Wigley half-breadth that ranking breaks down. Of the five-ordinate rules, only the Gauss rule is better than the trapezoidal rule, which gains here for the reason found earlier for $I_T$, and Boole’s rule is the worst. Four Gauss ordinates give the exact answer, since the integrand has degree 6 and four Gauss ordinates are exact up to degree 7. The degree of exactness is a reliable guide on the arc, but not on an integrand whose slope vanishes at both ends, as that of the cubed Wigley half-breadth does; there only a test on that integrand shows which rule is better (Exercise 2.19).

Table 2.6 Errors of five rules on two integrands: Chapman’s arc, $\int_0^4 \sqrt{64 - x^2}\,dx = 30.6116$, and the cube of the Wigley half-breadth, $\int_{-1}^{1} (1 - \xi^2)^3\,d\xi = 32/35$.

Example 2.9. Tchebycheff and Gauss by hand. Find the area under Chapman’s arc, $y = \sqrt{64 - x^2}$ from $x = 0$ to 4, with the two-ordinate Tchebycheff rule and with the three-ordinate Gauss rule, and compare their errors with those of the trapezoidal rule and Simpson’s rule with five ordinates, −0.048 and −0.00025.

2.9.2 Polar integration

A boundary that is closer to a circle about some point than to a curve over a straight base is integrated more accurately in polar coordinates. Take a pole O and measure the radius $r$ to the boundary at angles $\psi$, in radians, measured downwards from the horizontal through O. The thin triangle between $\psi$ and $\psi + d\psi$ has the area $\tfrac{1}{2}r^2\,d\psi$, and its centroid lies two-thirds of the way out, at $\tfrac{2}{3}r\cos\psi$ across from O and $\tfrac{2}{3}r\sin\psi$ below it. Adding the triangles,

where $\bar{y}$ is the distance of the centroid across from O and $\bar{d}$ its depth below O^4,8. Any rule of this chapter evaluates the integrals, with the angular spacing as the interval and $\tfrac{1}{2}r^2$, $\tfrac{1}{3}r^3\cos\psi$ and $\tfrac{1}{3}r^3\sin\psi$ as the ordinates. For a transverse section the pole is usually placed on the centreline, and a radius is passed through every point where the outline turns sharply, such as the deck edge⁹.

The method also meets the difficulty found at the bilge of the Series 60 midship section. With the pole at the centre of the bilge radius, a circular bilge has a constant radius, which any rule integrates exactly. Seen from that centre (Fig. 2.13(b)), two of Todd’s three lowest midship offsets lie within 6 mm of the circle of Todd’s radius, and the third, at 0.075$T$, lies 0.10 m inside it; this is the uncertainty of the table at the bilge. Radial ordinates were also used for ships heeled to large angles: Barnes’s method, introduced in 1861, found the volumes of the wedges between the upright and the inclined waterplanes by integrating over radial planes at intervals of angle⁴.

Figure 2.13 Polar integration: (a) the Wigley midship section, one side, with radii 15° apart from a pole O on the waterline; (b) the Series 60 bilge seen from the centre C of the bilge radius.

Example 2.10. The Wigley midship section in polar coordinates. From a pole on the centreline at the design waterline, the radii to the midship section of the Wigley hull ($B$ = 10 m, $T$ = 6.25 m) at 15° intervals, from the waterline ($\psi = 0$) down to the keel ($\psi$ = 90°), are 5.000, 4.958, 4.890, 4.899, 5.068, 5.477 and 6.250 m (Fig. 2.13(a)). Find the area of the half-section, the depth of its centroid below the waterline and its distance from the centreline by polar integration, and compare them with the exact values $\tfrac{2}{3}\cdot\tfrac{B}{2}T$, $\tfrac{3}{8}T$ and $\tfrac{2}{5}\cdot\tfrac{B}{2}$.

2.9.3 Polygons and Green’s theorem

A section, or any other plane figure, can be described by a polygon instead of by ordinates, and the area and moments of the polygon are then found exactly. Take a closed polygon with vertices $(y_j, z_j)$, $j = 1, \dots, N$, in order round the boundary, with vertex $N + 1$ taken as vertex 1, and write $c_j = y_j z_{j+1} - y_{j+1} z_j$. The triangle with its corners at the origin and at vertices $j$ and $j + 1$ has the area $\tfrac{1}{2}c_j$, positive when its corners run counter-clockwise and negative otherwise, and its centroid is at a third of the sum of the two vertices. Added over all the edges, the triangles cover the polygon once; where they reach outside it, triangles of opposite signs cancel. The area and its first moments are therefore

which is Green’s theorem, turning an integral over an area into an integral round its boundary, applied to straight edges¹⁴. The triangles also give the second moments. A point of the triangle can be written $u\,(y_j, z_j) + v\,(y_{j+1}, z_{j+1})$ with $u, v \ge 0$ and $u + v \le 1$, an element $du\,dv$ then has the area $c_j\,du\,dv$, and the integrals of $u^2$, $uv$ and $v^2$ over that unit triangle are 1/12, 1/24 and 1/12. For one triangle, therefore,

and, over all the edges,