Race statistics and tine allowance comparisons

ALLOWANCE COMPARISONS

by

J. N. Newman and G. S. Hazen

February, 1975

This research was carried out under the North American

Yacht Racing Union Ocean Race Handicapping Project,

M.I.T.

OSP Project No. 81535.

_{The generous support of the individual}

donors to this program is gratefully acknowledged.

AB S TRACT

Race results from twenty-one major ocean races are analysed in

con-junction with eight different time allowance systems.

A two-parameter

family of speed curves is used to characterize the races and to derive a

new time-on-time system.

Numerical figures of merit are derived for

comparing the different time allowance systems.

Significant differences

are noted between American and British races, which appear to justify

the traditiönal choices of time-on-distance and time-on-time respectively.

Extensive computer listings of the race results are included in the

Page

L Introduction

1

Description of Race Data

5

Description of Time Allowance Systems

7

Least-Square Speed Curves---

13

Figures of Merit

- 17

Description of Computer Progr

23

Modified NAYRU Systems

25

Sununary and Conclusions

28

References

- --- -- 29

Appendix - Listings of Race Results

72

(see separate list on page iii)

Figures

Expected performance corresponding to the NAYRU

time-on-distance and NAYRIJ time=on_t:irné systems - 9

Least-square values of the parameters

C1

and

C2

for

twanty-one races

-

- - - 15

Summary of figure of merit for NAYRU time-on-distance

formula and variations of the coefficient 0.6

26

Summary of figure òf merit for NAYRU time-on-time formula

and variations of the coefficient 0.0567

27

Tables

1.

Listing of twenty-one races

--

----

- 4

2..

Time allowance formulae ---

6

Mean of magnitudes of four figures-of merit and all races - 20

TABLE OF CONTENTS

(continued)

Race Results

Race Name

Page

Appendix

Page of

Listing.

Figure

Number

5

Seine Bay

30

73 6

Harwich-Hook of Holland

32

76

7

LeHavre-Royal Sovereign

34 79 8

Morgan Cup

36 82 9

Cowes-Dinard

38 86 10

Channel

40

92 11 Fas tnet 42 97 12

Honolulu

44 104 13

Sydney-Hobart

46 107 14

Anclote Key

48

110

15

St.Pete.-Ft.Lauderdale

50 114 16

Ocean Triangle

52 117 17

Lipton Cup

54

121

18.

Miami-Nassau

56

125

19

Nassau Cup

58 129 20

Bermuda Race

60 133 21

Chicago-Mackinac

62 138 22

LeHavre-Royal Sovereign

64 144 23

Morgan Cup

₆₆

₁₄₇

24

Cowes-Dinard

68

150

25

Channel

70

155

NONENCLATIYRE

A

Factor of time in corrected time equation

B

Factor of distance in corrected time equation

C

Corrected time

C1,C2

Coefficients of speed curve, equation (10)

D

Race distance

E

Elapsed time

k

Arbitrary constant in time allowance system related to

scratch rating

N

Total number of boats 1n fleet

n

Subscript to denote n-th boat in fleet

R

Rating

Scratch rating

TOT

Time-on-time

RACE STATISTICS AND TIME ALLOWANCE COMPARISONS

by

J. N. Newman and G. S. Hazen

1.

INTRODUCTION

As part of an on-going effort to gather race data and use this to

judge different handicapping systems, and particúlar aspects thereof, the

results from twenty-one major ocean races have been collected from the United

States, Great Britain, and Australia, for the 1973-4 sailing seasons.

These

results, consisting of yacht names, sail numbers, ratings and elapsed times,

have been punched onto cards in a form suitable for computer analysis.

In

order to process this data, and future race results, an extensive computer

program has been developed.

The program accepts simultaneously eight

dif-ferent time allowance systems, fOr use with the prescribed rating of each

yacht, and comparisons are made of the race results under each time allowance

system.

The output from this program forms the bulk of the present report, and

will be described in more detail in the following sections.

The principal

results include for each race a listing of fleet positions and corrected

velocities under each time allowance, corrected velocities being defined

as the ratios of race distance to corrected time.

In addition to these

tabular listings which are contained in the Appendix, the results are also

displayed in graphical form on computer-generated plots of elapsed and corrected

velocities vs. ratings.

Additional steps have been taken to quantify these race results in

terms of least-square curve fits to the elapsed and corrected velocities of

each race.

The parameters defining these curves can then be utilized as

statistical descriptions of the races, and as quantitative measures of the

efficiency of the different time allowance systems.

For this purpose, the

elapsed velocities have been fit with a two-parameter "speed curve" of a

defining this speed curve are stu4ied to reveal differences and similaritiçs

between the various races, as well as to validate the NAYRTJ system for the,

different races.

The corrected velocity distributions are f:it with straight lines,

which would be horizontal and independent of

rating if the handicap.png

process was free of bias toward the large or small boats.

The slopes of

these straight lines are therefore regarded as "figures of merit" by which

to compare the different time allowance systems in each race.

interesting

relationships emerge, not only with respect to the time allowance systems,

but also in comparing the British and American race results.

Initially this investigation was confined to a few existing

time-allowance systems .inclu4ing the NAYRU time-on-distance (TOD) formula, the

RORC performance factor system (PFS) used for British races during

1973-4,

two earlier RORC systems, and the Cruising Yacht Club of Australia

(CYCA)

system used during the samé period.

Inevitably other systems were included

f or comparison, particularly since the RORC was investigating existing and

proposed time allowance systems during the Pall of 1974 for use during the

1975 season.

As a result the above five existing time allowance systems

were supplemented by three proposed new

time-on-time formulae (TOT), including

a modified Australian system denoted as cYCA2, a.

modified RORC system denoted

as RORC2, and a new formula developed during

the course of this investigat.on

denoted as 'NAYRU TÔT.

The latter system, which will be described in some

detail in Section 3, has taken on added interest since it has been

adopted

by the RORC for use in British races during the 1975 season.

Certain individuals have been instrumental in supplying race

results

and time allowance formulae.

In this connection the authors wish to thank

Messrs. Roger Dobson of the RORC, Graham Newland of the CYCA, James

NcCurdi

of the CCA, Robert Potter of the LNYA, Jack Wirt of

thé SORC, and Gordon

Curtis of the TPYC.

Some additional race data has been gathered

from the

yachting press.

Corrections will be gratefully received and used to correct

and update the data bank of race results.

Work on this project was initiated during the first author's sahbatLcal

leave, and special thanks are due to his wife Kathy for programming assistance,

-3-to the University of Adelaide for use of their excellent computing facilities,

and to the Australian-American Educational Foundation and the John Simon

Guggenheim Memorial Foundation for fellwship support.

All of the

calcu-lations reported here have been performed using the facilities of the MIT

Information Processing Center.

Seine Bay

Harwich-Hook of Holland

LeHavre-Royal Sovereign

Morgan Cup

Cowes -Dinard

Channel

Fastnet

Honolulu

Sydney-Hobart

Anclote Key

St.Pete.-Ft. Lauderdale

Ocean Triangle

Lipton Cup

Miami-Nassau

Nassau Cup

Bermuda Race

Chicago-Mackinac

LeHavre-Royal Sovereign

Morgan Cup

Cowes-Di"nard

Channel

RACE NANE

DATE

SPONSORING ORGANIZATION

5/4/73

5/25/73

6/8/73

6/22/73

7/20/73

8/3/73

8/ii/73

7/4/73

12/26/73

2/i/74

2/6/74

2/14/74

2/23/74

2/25/74

3/1/74

6/21/74

7/24/74

5/17/74

6/21/74

7 /12/74

8/2/74

Royal Ocean Racing Club

ti t,

't

't i, t, t, f,

Transpacific Y.C.

Cruising Y. C. of Australia

Southern Ocean Racing Conference

Table 1 -- Listing of twenty-one races

t, t, it t, i, ii t, t, t, t, i, t,

Cruising Club of America

Chicago Yacht Club

Royal Ocean Racing

Club

'i

t, It

it

t, t' t, ti

t'

't

i

i

't

i

it

t

t

ii

'i

t'

't

't

't

't

t,

't

-5-2.

DESCRIPTION OF RACE DATA

The race data which has been analysed consists of the results for

twenty-one races held during the 1973-4 sailing seasons, in England, the

United States, and Australia.

These races are listed in Table 1.

Pertinent

data for each racé includes the race name and year, the length of the race

in nautical miles, and for each boat in the fleet the name, rating, and

elapsed time.

Elapsed times may be entered in hours, minutes and seconds,

or alternativly in decimal hours, and the program accepts which ever of

these two inputs Is given.

In those cases where the effective race, length

has been modified by the race cOmmittee this modified length is used.

Where

sail numbers are available for each boat these have been recorded, to facilitate

future identification of each boat on the computer tape ICR certificates.

The class or division in the race hs also been entered for each boat,

although this information is not used here.

No attempt is made to analyse boats separately within each class, or

to determine class finish positions, although this could be done simply by

running the program separately for each class.

Class performance may be

more significant than fleet performance from the racing standpoint, especially

in long races where separate weather systems may affect different portions

of the fleet.

But for purposes of judging the overall nature of the race,

and the equitability of different handicap systems, subdivision by class

seems less desirable.

The division between classes is arbitrary, and this

division may be crucial if the handicap system is not equitable throughout

the fleet.

Thus, in a situation where small boats are favored in the fleet,

one may expect that the smallest boats in each class will also

generally

be favored within their respective classes, and vice versa.

Under this

circumstance an individual boat's perfôrmance will be vitally affected if

it is placed at the top of one class, or alternatively at the bottom of

the next class.

Thus it seems clear that for purposes of handicap system

development and evaluation, one should strive for equitable handicapping

of the entire fleet, as opposed to each separate class.

By the same token,

one may anticipate that in most cases a handicap system which works well for

the fleet as a whole will be equitable for each class, and vice versa.

RORC

PFS

NAYRU

TOT

k Rh/2

i + 0.O57R:I

k

l-A

1.5

-2.75R_h1'2

2.75

O

k

R1'2

[1.5 C/D +

k;l]R1/2

+ 1 - 2.75 C/D

k(R11'6

- 0.96)

C/D

k[(R-8)''1 - 0.75].

C/D

k(R1'2

+ 2.6)

C/D

k(.l999Rh/2

+ .34)

C/D

k

Rh/2

(1 + 0.057Rh/2)(C/D)

Table 2 -- Time allowance formulae.

Note that

R

is rating in feet, and

k

is an arbitrary constant

related to the scratch--boat-ratingj---R

,

suchthat

A

1.0

,

B = 0

for

For the present analysis of race data

R5 = 29.0 feet.

Rh/2

NAYRU

TOD

1.0

Ò.6(k

0..6 ± (cID

-k)Rh/2

1.43

Rh/2

+ 2.6

RORC

TOD

l.O

k-R1#'2

± 2.6

1.43 + (CID.

-k)(Rh/2

CYCA2

TOT

k(Rh/6

- 0.96)

o

CYCA

TOT

- 0.75]

O

RORC

TOT

k(R1'2

+ 2.6)

o

RORC2

TOT

k(.l999Rhi2

+ .34)

O

Formula

A

V

3.

DESCRIPTION OF TIME ALLOWANCE SYSTEMS

The eight time allowance systems evaluated here are listed in Table 2.

These include two time-on-distance formulae, the RORC performance factor

system, and five time-on-time formulae.

All eight systems can be expressed

in terms of the common equation

C=AE+BD

(1)

where

C

is corrected time,

E

is elapsed time,

D

is the race distance,

and

A

and

B

are coefficients depending on the particular time allowance

system and on the rating of each boat.

In the time-on-distance approach

A=l ;

for time-on-time systems B0 .

The complete formulae for these

coefficients are listed in Table 2 for all eight systems.

All time allowance systems are explicitly or implicitly related to

corresponding "speed curves", of hypothetical boat speed vs. rating.

Indeed,

-if al-1 boats f insh with idéntical corrected times.,

D ,

in an "ideal" race,

it 'follows by solving (1) for the elapsed times of each boat that the velocities

will be

-=D/E-

A

CID - B

Thus, for a given time allowance system, (2) gives the velocity for a given

rating, such that the boat will perform .as expected in a given race.

It

should be noted that this formula includes one descriptor

(CID)

of the

race conditions, which will be large or small depending on whether the race

is slow or fast, respect-ively.

In add-ition,

(2) contains two parameters

(A,B)

dependent upon the boat' rating, and thus its speed poténtial.

For

time-on-distance

ör

time-on-time

the

special choices of

A=l

or

B=O

respectively restrict the flexibility of (2).

Table 2 includes the expected

velocity based on (2) for each of the eight time allowance systems.

As an exemple, let us first consider the NAYRTJ TOD formula, where Al.O

and

.B = O.6(Ç'"2 -

R_1/2)

,

with

_Rs

the rating of the scratch boat.

Thus, from (1), the corrected time is

C = E +

.6açh/2

-

R'2)D

' (3)

and from (2) the velocity of each boat is related to its corrected time

by the formula

1 (4) =

C/D

0.61Ç1"2 + 0.6R1'2

Writing (4) in a slightly different form,

1.67 Rh/2

V

=

(l.67C/D -

çh/2)R1/2

(5)

(In the present discussion 1.67 is used as a convenient approximation to

the inverse of 0.6.)

The NAYRU TOD system is thus equivalent to the family of speed curves

given by

v=

-

12

1 + C2R

1.67 Rh/2

(6)

where. C2

depends on the.speed of the race.

From (5),

C2

and the correctèd

time are related by

C2 = l.67C/D - Ç"2

(7)

or

C

0.6(C

+

ÇhI'2)D

(8)

Therefore the parameter

C2

and the corrected time

C

are trniquely related

to each other, and either one is suitable basis for determining the winner

of a race, ¿lthough the correctéd time is a more readily understood parameter.

To pursue this viewpoint further, the family of speed curves defined by (6)

is plötted in Figure 1 (solid lines), for different values of

C2

ranging

f rom O to 1/4 .

These define performance curves along which boats of

different rating müst sail, in order to finish with the same corrected times

under the NAYRU TOD system.

Alternatively, if all boats in a given race

are plotted on this Figure, the boat lying on the

highest of the family of

sôlid curves will be the winner under the NAYRTJ TOD time allowance system.

It may be noted from (7) that the speed-length ratio

(or more precisely,

the speed-rating ratio) expected by the NAYRU TOD system is

V/R1

2

= 1+ C2R1/2

1.67

I0

9

5

(jtf

/

3--I

20

30

9

40

50

RATING (FEET)

Figure 1 -- Expected performance corresponding to the NAYRTJ

time-on-distance (solid line) and NAYRU time-on-time (dashed

line) systems.

This equation is displayed to emphasize the fact that the speed-length

ratio associated with the NAYRU TOD system is not a constant for all boats

in the fleet, or for any particular boat within the fleet.

Instead the

speed-length ratio is expected to vary both with the rating R ,

and with

the conditions of the race as represented by the parameter

C2 .

This

important feature is not sufficiently appreciated, and critics of the

NAYRTJ

TOD time allowance system often state incorrectly that it will only apply

when the boats involved are sailing at a hypothetical assume4 speed-length

ratio.

A more general speed curve is obtained if the coefficient

0.6

is

allowed to vary, corresponding to a change in the effective race distance

Under the NAYRU TOD system.

This is equivalent to changing the constant

1.67

in (6), and hence to the form'la

C1R112

=

1 + C2R112

(10)

where

C1

and

C2

are cOnstants defining the particular speed curve.

Equation

l0) wilibe used here not only to relate to the NAYRU TOD system,

but to derive the so-called NAYRU TOT time-on-tie system, and, also, to

fit on a post-race basis the elapsed velocity distributions of the fleet

to a least-squares curve which can then be used to describe the

character

of each race.

Before dong so, however, we note that in its most general

f àrm (10) can not be used as the basis for a (pre-determined) time allowance

system since the twO unknown coefficients

C1

and

C2

can not be determined

uniquely and independently from the single equation (10) for a given velocity

V

and rating

R

To overcome this difficulty it is necessary to restrict

one or the other of the two independent

coefficients, such as is done in

the NAYRU TOD system when

C1

is set equal to

1.67

An alternative assumption is to fix

C2

and allow

C1

to vary with

the conditions of the raòe.

This complementary approach lads to a

time-on-time system, with

k

R1'2

A

=

1

C2R1/2

(Il)

il

-C1

=

D/Ck

(13)

since, with these substitutions, (2) and (lo) are Identical.

Here

k

is

an arbitrary constant, related to

R5 ,

whióh caribe conveniently chosen

H

so that A

1.0 when

R

R

-Thus, to obtain a on-time formula analogous to the NAYRU

time-on-distance system,

and

denoted here as the

NAYRU

TOT formula, it is only

necessary to find a

uitable constant

C2

to use in the formula (1).

Initially we selected the value of C2

= .057

f-rom a preliminary graphical

fit. to race data.

Subsequently this formula has been adopted by the RORC

for üse during the 1975 season in British Races, but with the coefficient

0.057

changed to

0.0567

.

This minor modification with

k

= 0.2424

in,

the numerator, gives the time correction factor

0.2424

Rh/2

A

= 1

0.0567 R112

(14)

such that

A

- 1.0002

for a scratch rating

R5

29.0

.

In earlier work

(Ref. 1) results were presented based on our original value of 0.057, but

subsequently the program has been changed so that the handicap system

identified here as NAYRU TOT has the coefficient set equal to

0.0567

and thus agrees exactly with the system to be used in the 1975 RORC races.

The family of speed curves equivalent to the

NAYRU

TOT system are

given by

C1R/2

V

=

1 + 0.0567

(15)

and these are plotted as dashed lines in Figure 1, for comparison with the

NAYRU

TOD system.

In this case the corrected time could be replaced by

the parameter

C1

for purposes of determining race results, the boat with

the largest value of

C,1

being the winner of the race under the NAYRIJ TOT

system.

Figure 1 can be used to empare the expected results from

handi-capping under thè NAYU TOD and TOT systems, the two being coincident when

C1

=

1.67

and C2 =

0.0567

.

This condition corresponds to a rather fast

race, with boat speeds between

6 and

9.5

knots for ratings in the range

level of performance from the larger boats, and hence favors the small end

of the fleet., by comparison to the TOD system.

Thus, in all but the fastest

racing conditions, large boatS will perform the best under the NAYRU. TOD

system, and small boats will be favored by the NAYRU TOT system by comparison.

This difference is typical of all timS-ori-time. vs

timeondistance comparisons,

as noted earlier in. Reference 2.

'

The above discussion

applies only to. a relative comparison btween

the two time allowance systems, however, and it would be premature to conclue

that small boats are unduly favored in a race sailed under the TOT system,

or conversely that large boats are favored in 'a race sailed under the TOD

system.

An evaluation of this important subject

ust be based on the

exam-ination of actual race data, and In particular on the considerat-ion of what

constitutes in practice the most. relistic family of speed curves..

13

-4.

LEAST-SQUARE SPEED CURVES

In order to give a quant:itative descrptjon of the character of each

race, it is desirable to fit a suitable least-squares curve to the elapsed

velocities of all boats in the fleet

For this purpose the family of

speed-curves defined by (10) has been adopted, the procedure being to

determine the two coefficients

C1

and

C2

in a least-squares manner so

as to give an optimum fit to the elapsed velocities.

If the

N

boats in a given race are denoted by ratings

R,

and

(elapsed) velocities V,,

with n=l,2,..N,

the dtrect least-squares

approach would be to minimize the sum of the squarés of the errörs

C,R,/2

i +CR'12

2n

However this leads to nonlineâr equations for

C1

and

C2 ,

and to avoid

the difficulties associated with diret solutions an alternative approach

is adopted, based on minimizing the sum of thé squares of the time differençes

-i +

n

_CR''2

in

Ç' -

(l/C11Çh/2

E-.

This is a more conventiOnal

tOb1em involving the least-!squàrés detérmination

o

thé two àônstants

= (1/C)

and

(C/C,)

such that

-

rC1'2

)2

= minimUm:

-(16)

Differentiating (16) with respect to

o.

and

ad' setting these derivatives

é4Ual. to zero we obtain two simúltaneoùs eqüations for

o. aid

:-N

_i

-

)[ç1'/2]

= o

(17)

R_1/2

-

_N

_Ç1

R1h:

[

ç1l2j2

_N

1Ç'

where in all cases

denotes the sum over all

N

boats in the fleet.

Finally, substituting in the expressions for

C1

and

C2

ci

=

(

R'J2

-

N

_Ç1

R;1

-

N

_I

_Ç1

Ç'

p_1/2Vi R_1/2

-

_{I Ç1 I}

_Ç1'

I Ç'

Çhu/2

-NI

Ç1

Rhh'2

(18) (19) (20) (21)

These last two equations are the basis for the computations of

C and C2

which are plotted in Figure 2 and listed in the Appendix for each race with

typical values of

V

computed for ratings 21, 29, 40 and 60 feet,. as well

aS the slope

dv/dR

evaluated at

R=29

.

The computed values of

C1 and

C2

have also been used in (10) to construct the least-square speed-curves

for each race which are plotted in the elapsed velocity Figures 5a-25a.

The computed values of

Ç1

and

C2

plotted in Figure. 2.can be compared

with the assumed values of thesE coefficients in the two NAYRU time allowancé

Systems.

The NAYRU TOD assumed value

C1 = 1.67

is shown as a horizontal

line in this Figure, and the NAYRU TOT value

C2 ' 0.0567

as a vertical lin.

Races which lie on or near one of.. these twö lines will be

well handicapped

by the corresponding system.

In.geiera, the U.S. races are, scattered

more-or-less equally about the NAYRU TOD line, indicating that the

coefficient

H

0.6 is a reasonable mean valúe for american races.

British races and the

Hobart Race tend to lie substantially below this line,

indicating that for

these races

C1

should be. reduced or the coefficient

0.6

increas.

Regarding the NAYRU TOT system, Figure 2 indicates that while the va1ue

C2 = 0.0567 is a reasonab1e average, valúe

for all of the races, most of the

I

3.0

2.0

1.0

O

-0.I

/

-

15

-x

_{U.S. RACES}

O

BRITISH RACES.

HOBART RACE

/

X

CI.67

X

- -;

Ox

(NAYRUTOD)

C20.O567

(NAYRU

OT)

0.1

0.2

0.3

C2

0.4

0.5

Figure 2 -- Least-square values of the parameters C1 and C2

U.S. races are to the right of this line and most of the

other

races

are to the left.

Thdeed, for five Brit±sh races and the

73 Hobart Race,

C2

_{is negative, indicating that thè speed length ratio of the larger boats}

in these races exceeded that of the Sma-ller bòats.

Finally, it is Obvious from Figure 2 that the most significant

trend is for ail of the races to be distributed not along the horizontal

NAYRU TOD line hor along the verticàl NAYRU TOT 1ne, büt instead along

a diagonal line where

C1

and

C2

increase simultaneously.

A fair

appröximation is

C1 = 1 + 6.25 C2 as shown, and this particular straight

line gives a relatively good fit simultaneously to all 21 races.

Indeed,

it can be confirmed from Figure 2 that this relation between

C1

and

C2

gives a reasonable "colläpse" simultaneously for the British and American

races, i.e., the scatter of all points about this one unified line is

less than the individual scatter of American races about the NAYRU TOD

line and British races about the NAYRU TOT line.

Noreover, the diagonal

line shown is also able to account for the very unusual conditions of the

1974 Lipton Cup Race, (shown by the point at the upper right corner of

Figure 2), where differences in elapsed time throughout the fleet were

minimal due to a wind shift, with small boats favored overwhelmingly., and,

on the other exterme, the 1974 Channel and Le Havre Races (lower left cÖrner)

where large boats were heavily favored.

Unfortunately it is not possible to use the diagonal lIne in Figure 2

as the basis for a time allowance system based on the speed fórmula (IO),

sincé the resulting family of speed curves is singular at

R = (6.25)2= 39 feet.

Nevertheless the correlatión in Figure 2 is quite striking, and should be

17

-5.

FIGURES OF MERIT

in order to provide quantitative measures of the equitability of each

time allowance system, in each race, and ultimately a summary of these for

all races, figures of merit have been developed which will be described here.

In an ideal situation, the distribution of performance throughout the

fleet should be independent of rating, with large and small boats performing

equally.

Thus if the results are expressed in terms of the corrected

velocities, equal to distance divided by corrected time, and these corrected

velocities are plotted on a graph vs. rating for each. boat, as in Figures

5b-25b, a measure of the quality of the time allowance system is the extent

to which the points on this graph are distributed horizontally.

In order

to express this measure quantitatively we fit to all the points on the graph

a least-squares straight line., that is a straight line drawn on the

graph

which has a minimum mean-square error with respect to all the points on the

graph.

The slope of this straight line can then be used as a measure of

equitability between large and small boats.

If the slope is positive it

indicates that large boats are favored in the race, and vice versa.

If

the slope is. zero and the straight line is horizontal one may conclude that

there is no bias in either direction.

This is the basis for figure of merit

number one.

The mathematical procedure for determining figure ofmerit one is as

follows.

Given a set of cOrrected velocities

V

and ratings

R

, we

n

.

n

wish to fit to these a straight line

V=ct-I-R

(22)

so as to minimize the mean-squared error

(Vn - -

R)2

As before, differentiating with respect to

and

and equating these

derivatives to zero gives the simultaneous equations

N1

(V -cx-R)=O

R

n

n

11=1

fl

J

(23)

Solving

for the unknowns

ci.

and

from Cramer's rule,

V

R2-.1VRR

n

n

nfl

._n

N

R2

-

(

R)2

N

VR -.

V1

R

'(25)

N1R2- (R)2

Figure of merit one is the slope of (22), equal to

,

and indicates the'

trend towards favoring large

(

positive)

or small

(

negative)

boats.

One possiblé objection to the above is.that there :Ls no attempt to.

discriminate..between "good" and "bad" boats in the least-squares fit to

the corrected velocities.

It is generally felt that the well-sailed boats

are more important in judging a handicapping system, and this suggests a

weighting of the least-squares fit to favor the good performers.

In order

.to provide alternative types of weighting, three substantiallydifferent

approaches have been adopted, which are identif led here as figures of merit

two, three, and four.

In figure of merit two the. weight

actors are related

linearly to the corrected fleet positions, i.e., each boat is given a weight

factor equal to one plus the number of boats it has. beaten.

The mathematical scheme for accomplishing this. requires that each. of

the

N

boats be ranked according to its fleet positIon (denoted as

IP

in

the computer listings).

Assuming this has been done, with

n=i

the race

winner, etc., and

n=N

the last boat, the appropriate

weighted errors are

1

N

T(N-n+l) (V-ct-R)2

We then proceed as before to find

and

so that this sum Is minimized.

The resulting value of

defines figure of merit two.

Note that the last

boat to finish the race has a small but non-zero influence on this figure

of merit.

(Here, as in all figures of merit, credit is given for beating

a DNF boat, but the DNF boat is iiot included

in

the least-squares fit.)

In

figure of merit three a more brutal approach is taken, in that all

boats lying below the middle of the fleet are eliminated and the remaining

(24)

19

-half of the fleet is fit as in figure of merit one, the slope of this new

straight line being defined as figure óf merit three.

Finally, as an

alternative but equally arbitrary approach, all of the boats below the

straight line corresponding to figure of merit one are eliminated and the

remaining half of the fleet is fit to give figure of merit four.

Note

that figures of merit three and four both eliminate approximately one. half of

the fleet, but the elimination procedures differ unless figure of merit one

is zero.

Haviig introduced f our separate figures of merit, it remains to decide

which of the four is most significant for á comparison of the different

time allowance systems. Fortunately, while significant dif:ferences exist

between the individual figures of merit for each race, as shown in the

separate tabulations in the Appendix,

these differences do not persist

when the figures of merit are first averaged over all the races and then

cómpared.

For this purpose Tàble 3 shows the mean, for all twenty-one races,

of the magnitudes of each figure of mer:it for each time allowance system.

From Tble 3 the eight time allowance systems can be ränked, with the smallest

figure of merit corresponding to the best. time 'allowance system, etc., and

in general similar conclusions follow regardless of which figure of merit

is used.

In particular, regardless. of which of the four figures of merit is

used, the Performance Factor System is the worst (i.e., largest figure of

merit) of the eight time allowance systems, and with only one exception the

time-on-time systems are superior to the time-on-distance.

The exceptional

case is figure of merit three, where the NÁYRIJ TOD system is slightly better

than the RORC2 TOT system.

In view of the arbitrary nature of figures of merit three and four,

and the relatively continuous weighting of figure of merit two with respect

to fleet performance, the latter has been adopted for comparative purposes in

the results tO be described later, but all four figures of merit are tabulated

-in the Appendix for each race and time allOwance system.

Restricting our attention to figure of merit two, Table 4 presents the

values of this parameter for all eight time allowances and twenty-one races.

Also shown are the mean values of the magnitude. of, this figure of merit,

Figure

of

Merit

NAYRIJ:

RORC

RORC

CYCA2

CYCA

RORC

RORC2

TOD

TOD

PFS

TOT

TOT

TOT

TOT

Table 3 -- Mean of magnitudes of four figures of merit and all races

0.0226

0.0219

.0.0288

0.0186

0.0173

0.0208

0.0188

0.6203

0.0195

0.02(46

0.0172

0.0162

0.0186

0.0175

3

0.0095

0.0098

0.0125

3.0.085

0.0092

0.0085

0.0096

4

0.0209

0.0211

0.0271

0.0165

0.0160

0.0185

0.3169

Table 4 -- Summary of figures of merit

NAYN'J

TOD -RONC

Tul)

NONC

CYCA

PFS

TOT

CYCA

TOE NOIÈC TOT

ROC2

îoî

\1AYRU

Tot

I 9,3

SE1E iAY NACE

0.0284

0.0243

0.0361

0.0i3i

0.015M

0.0225

0.0165

0.0164

HANW

ICH-HOOK

0.0313

0.0279

0.0424

0.0i(s

0.0147

0.0217

0,01b4

0.019

LE

HAVNE - NOVAL

0.0200

0.0161

0.0295

0.0069

0.0039

0.0116

0.005.1

0.0047

i1)N(,AN CUP

0.0214

0.0118

0.0310

0.0081

0.0059

0.012?

0.0060

0.0059

COWt.S-U I NANI)

0.079

0.0219

0.0300

0.02(3

0.0244

0.0328

0.0246

0.0244

CHANNEL NACE

- 0.01,7 -0.0221 -0.OIYÓ -0.-008f -0.uIlO -0.0042 -0,0116 -0.011s

FASTNET NACE

0.0229

0.0194

0.0301

0.0130

0.0112

0.0158

0.0115

0.0115

rIONULULU NACE

0.0036 «0.0119 -0.0023 -0.0043 -0.001M

-0.0067 -0.0118 -0.0070

SYL)NE Y-HO8AN T

0.017.

0.0129

0.0177

0.0193

0.0.189

0.0216

0.0166

0.0177

1i4

ANCLOTE

-0.0011 -0.0051

0.0042 -0,0138 -0.0131 -0.0130 -0.0179 -0.0161

T. LAU[)LNUALE

0.0040 -0.0008

0.008/ -0.0029

-0.0021

-0.0027 -0.0071 -0.0050

OCtAN TNIANGLE

0.0130

0.0066

0.0166

0.0082

0.0096

0.0081

0.0037

0.0060

L1tUN CUP

-0.0311 -0.03/9 -0.025s -0.0389 -0.0381 -0.0317 -0.0437 -0.0416

MIAMI-NASSAU

- 0.0143 -0.0258 -0.0204

-0.0026

-0.0016 -0.0023 -0.0087 -0.005'.

NASSAU CUP

- 0.0008 -0.00s5

0.0044 -0.0101 -0.0095 -0.0102 -0.0160 -0.0133

8ENMiJDA NACE

0.01d9

0.0113

0.0203

0.0171

u.00n

0.0163

0.0114

0.OlsI

CH1C4bO-MACr INAC

u.0029 -0.0010

0.0066 -0,0040 -0.0057 -0.0004 -0.00o8

-0.0064

LE

HAVNE - NUYAL

0.05/9

0.0549

0.0116

U.0U5

0.0381

0.0449

0.0387

0.0385

MONGAN CUP

0.0208

0.0145

0.019/

0.0:241

0.0185

0.0337

0.0228

0.0213

CUWES/DINARI)

-

0.00b7 -U.u0b

0.0043

-0.0208-0.0264

-0.0134

-0,0208 -0.0228

CHANNEL NACE

0.0651

0.0615

0.OTs2

0.Osid

0.0503

0.0.58?

0.0510

0.0508

MEAN 0f

MAuNITUL)E

81TISti RACES

O.02U

0.0264

0.0354

0,0211

0.02Do

0.0246

.0.0205

0.0203

US

ACES

0.0100

0.0114

0.01210.0115

0.0113

0,0108

0.0142

0.0129

ALL kACES

0.0203

0.0195

0.0246

0.01(2

0.0162

0.0186

0.0175

0.0170

For the average over ali races, the five time-on-time systems are

signi-ficantly better than the two time-on-distance formulae.

Among the five

time-on-tim

Systems the CYCA f orula is best, followed in òr4er by NAYRU TOT,

CYCA2, RORC2, and RORC.

When the méans are computed separately for the

British and U.S. races, however, significant differences emerge.

For the

British races time-on-time is superior, with YCA and NAYRU TOT the best.

But for the American races the time-on-distance f örmulae aré tore efféctive,

with NAYRU TOD the best of the eight.

Thus ve again see that American and

British races must be analysed separately.

Small differences between the figures of merit shown in Table 4 should

not be regarded as significant, since the definition of this figure oferi

introduces some degree of arbitrariness.

Moreover, one can envisage

small-modifications of the coefficients in the various time allowance systems

which would chaige the figures of merit and, if judged solely by this or

another measure, woùld lead to changes in the relative iâerit of the time

allowance systems.

But. large differences between the figures of merit shown

in Table 4 are thought to bé significant, and relevant to the selection of

equitable time allowance systems.

23

-6.

DESCRIPTION OF COMPUTER PROGRAM

In order to process the race results outlined in Section 2, using

the different time allowance systems listed in Table 2, a program has

been developed using the programming language PL/l and the computing

f acilities of the MIT Information Processing Center, notably an IBM 370-168

digital computer and a CALCOMP plotter.

As the CALCOMP and MIT-supplied

plotting subroutines are written in FORTRAN, the optimizing, PL/l compiler

was chosen in order that its interlanguage interpreting capabilities could

be utilized.

In the absence of the optimizing compiler, however, the same

language interfacing capabilities could be accomplished using the standard

PL1F compiler.

Brief ly, the program accepts as input on computer cards of specified

format the race data, including two cards to describe the race ánd one

card for each boat.

The eight time allowance systems are treated as an

external subroutine which can be changed readily without re-compiling the

main program.

Considerable effort was expended in order to make the program as

versatile and simple to use as possible.

Consequently, the first card in

the input data deck is for describing the execution options desired.

These

are entered using the PL/l data format, and all or any of the options may

be specified.

In the absence of explicit specifications, each option's

default value will be used.

On a primary level there are options to include

printed, punched and plotted output.

If plotted output is specified two

additional options are availáble controlling labeling

and size of plots.

As has already been mentioned, the time allowance systems can be changed

by means of changing a single subroutine.

In conjunction with this change

the second card of the input data deck contains the eight titles to be used

to describe the time allowance systems.

Following thede two control cards the data is entered race by race.

The first card in each race deck contains its title and date, followed by

a card specifying the race length and scratch rating used.

Next each boat

is entered on one card with fields for the boat's name, sail number, class,

rating and elapsed time (decimal hours or hours, minutes and seconds).

DNF boats are indicated on the input cards by an elapsed time of zero

The

end of a raca is designated. by a card with

a blank chàiacter in

o1umn one,

and a non-blank character in column. two.

Similarly, the eùd of the last

race is flagged by a cárd with blanks in both the first änd second columnS.

In summary, the essential functions performed by the program are:

Read in race data from cards.

Convert elapsed time to decimal hoùrs if necessary.

Compute eight corrected times and corresponding corrected velocities

using

R5

29 feet.

Determine fleet positions of each boat under each time allowance system.

Print, if desired, a listing for the fleet of each boat's name, sail

number, rating, elapsed time, and fleet position followed by corrected

velocity for each of the eight systems.

Compute and print summary data including figures of merit, and

least-squares values of

C1

and

C2

for Speed equation (10) as well as

representative values of velocity and derivative of velocity with

respect to rating from this speed curve.

Cönstruct if desired, CALC0

plots of elapsed and corrected velocities

as well as the least-squares speed curve and figures of merit.

Punch, if desired, siary data.

25

-7.

MODIFIED

NAYRTJ SYSTS

Questions regarding the optimum value of the coefficient

0.6

in the

NAYRU TOD system are well known, and this coefficient has been varied over

the years both for general use and for particular races.

Similarly, the

optimum value of the coefficient

0.0567

in the NAYRIJ TOT sysem may be

questioned.

In order to address these questions the eight time allowance

systems listed in Table 2 were replaced temporarily, in the program described

in Sectidn 6, by sequences of eight NAYRU TOD systems where the coefficient

0.6

was varied between

0.45

and 0.8

in steps of

0.05

,

and by sequences

of eight

NAYRU TOT

systems where the coeffIcient

0.0567

was replaced by a

range of values between

O and

0.1

.

All

twenty-one races were re-run

with these two modified programs, to determine the effect on the corrected

times of these changes.

To avoid voluminous output from these computer

runs, only the figures of merit were tabulated, and the magnitudes of figures

of merit

two averaged for all twenty-one races to provide quantitative

measures of the optimum values of the coefficients in the two systems.

The results of these additional investigations are plotted in Figures

3 and 4 for the

NAYRU

TOD and TOT systems, respectively, as curves of the

figure of merit vs. change in the coefficient.

Recalling that the optimum

figure of merit is a minimum value, it follows that the coefficients of the

two time allowance systems should be chosen to coincide with the minimum

points of the curves in Figures 3 and 4.

Figure 3 shows that the optimum

value of the coefficient in the

NAYRTJ

TOD system is very close to

0.6

for

the average of all U.S. races, but is on the order of

0.8

for the British

races and

0.75

for all 21 races.

Figure 4 shows the corresponding optimums

for the coefficient of the

NAYRIJ TOT

formula, with U.S. races a minimum near

0.09

,

British races near

0.01 ,

and the average of all 21 races a very

flat insensitive minimum near

0.04

.

Also we note that if the

NAYRTJ TOD

system is used with an optimum coefficient for U.S. races, it will give

approximately the same average figure of merit

0.01

as does the

NAYRU TOD

system.

But for British races the optimum TOT system is substantially better

than the optimum

TOD

system.

These results confirm our earlier conclusions

based on the comparisons of the coefficients

C1 and C2

in Figure 2, as

05

Q6

07

08

COEFFICIENT

Figure 3 -- Summary of figure of merit for NAYRU time-on-distance

.o.o3

w

Li

o O02

w

IL

004

o..o i

-

27

-O05

o.'

COEFFICI ENT

Figure 4 -- Summary of figure of. merit for.NÄYRtJ

_{time-on-timé.}

8.

$t1NMARY AND CONcLUSIONS

.

. ,.

Based onthe aata bank of race results, and a computer program for

.

analysing these in conjunction with varoús time allowance systems, it has

been possible to perform exteiisive investigations qf the race data and

various time allo-ance systems..

Further work along the stne lines can be

carried out Tjt

future race data, and a similar program can be used to

evaluate modifications of. the rating rule in conjunction. wLth the existing

race data.

It is by no means certain that the twenty-one races studied to

date are a sufficiently large sample to be statistically relevant, and the

analysis of additional race data should be made before drawing f irth conclusions.

Neve±theleàs the sample jncluded here is more extensi.ve than any kflown to

have been studied. systematically in the past, and some tentative conclusion

can be set forth for short-range purposes and discussion.

generali differeñcês bétwee

races are moré importaflt than differences

between time-allowance systems.

Thus race results are not stroly depeident

on which of the eight time alIowanòe systems is used, to. the same extent

that the results depend on race conditions

Thi

is apparent quantitatively

from Table 4 where the variation of the figures of merit is much greater

among the different races than it

is among the various time allowance systms

for-a given racé.

In this context unjustified criticism has been directed

at particular time allowance systems, as in a magazine account of the 1974

Morgan Cup Race where the victory

of "BES" is

attributed to the Performance

Factor System; from the results for this race shown in the Appendix it can

be confirmed that the same boat would win under all but one of the eight

tithe-allowance systems.

.

In order to provide a quantitative description of the varying race

conditions we have used a two-parameter family of "speed curves" derIved as

a generalization from the NAYRTJ time-on-distance formula.

The two parameters

C1. and

C2. which thereby characterize a particular race are plotted in

Figure .2, and listed, in the Appendix.

Figure. 2 displays vividly the differences

between BrLtish and Ameridan racés, and the associated advantages of

time-on-time vs. time-on-time-on-distance systems respectively.

But a unifying trend is

also apparent in Figure 2, na.ely a relátion between the two parameters

9-British and American races, and also for some unusual race conditions such

as were met in the 1974 Lipton Cup Race.

This type of analysis suggests

that we are still far short of an optimum time allowance system.

Our study of time allowance systems has led to the development of a

new time-on-time formula, based on the same general family of speed curves

as the NAYRU time-on-distance system.

This new time-on-time system, denoted

for convenience as the NAYRU TOT system, has been selected by the RORÇ for

use in Britain during the forthcoming season, including the Admiral's Cup.

Thus our comparison here of the results to be expected from this system, in

comparison with the NAYRU TOD system and other British an4 Australian time

allowance formulae, may be used to give some indications of the results to.

be anticipated from the 1975 RORC races.

Detailed computer listings for each of the twenty-one races and eight

time allowance systems are included in the Appendix, and the essential results

of these computations are plotted in a more digestable form in Figures 5-25,

which are shown on the following pages.

Yachtsmen curious to see how they

would have fared "if only the race had been sailed under the right time

allowance system" may wish to study the Appendix.

Others can evaluate the

time allowance systems and race ch.aracteristics in the more descriptive form

of Figures 5-25, where anonymity of boats may also contribute to the

objec-tivity of the conclusions.

REFERENCE S

"Handicapping Systems for Ocean Racing Yachts", by J. N. Newman, paper

to be presented to the Royal Institution of Naval Architects 20th March 1975

meeting and published in the July 1975 issue of The Naval Architect.

"A Fundamental Approach tó Ocean-Racing Handicap Rules", by J. N. Newman,

Proceedings of the Symposium on Yacht Architecture, HISWA, Amsterdam, 1973,

p. 187.

>-I..

C)

D

w

>

H-3

D

-2

i. I I I I I I f

20

30

110

50

60

70

HATING

(FEET)

Figure 5 -- Velocity plots for the 1973 Seine Bay Race.

Figure a

(above) shows the actual velocity of each boat in knots, defined

as race distance divided by elapsed time.

Also shown in this

f igre is the least-square speed curve represented by equation

10, with the coefficients C1 and C2 determined from equations

20-21. Figure

b (opposite page) shows the corrected velocities

of each boat in knots, defined as the race distance divided

by corrected time, under each of the eight time allowance systems

listed in Table 2.

Also shown in this figure are the

least-square straight lines corresponding to figure of merit two,

these straight lines being weighted for each boat by its fleet

position.

6 D D H-D

z

-6 D

-D H- LI-6 (n -Lj_ X X

1973 SEINE BAI RACE

31

- H-D - H-L) >-L)

1973 SEINE BIA RACE.

2Ó 30 Lb 50 60 70 20 O- LIb 50

RATING CEET)

RATING

(FEET)

F-D

H

X X X

1.973 HRHkICHHI OF H.O.LLRND RCE

H'

I I

Figure 6

Velocity plots for the 1973 Harwich-Hook Q

Holland

Race

Figure a (above) shows the actual velocity of each boat

in knots, defined as race distance divided by elapsed time..

Also shown in this figure is the least-square speed curve represented

by equation 10, with the coefficients C1 and C2 determined from

équations 20-21.

Figure b (opposite page) shows the corrected

velocities of each boat in knôts,. definéd as the race dïstañce

divided by corrected time, under each of the eight time allowance

systems listed iñ Table 2.

Alo showr in this figure are the

least-square straight lines corresponding to figure of merit

two, these straight lines being wéighted for each boat by its

fleet position.

20

30

LID

.

50

60

70

-

33

-1973 HPRNICHHOK OF HOLLAND ARCE

1973 HPR1JICHHOO

OF HOLLANO ARCE

ARTING (FEET)

ARTING [FEET]

173 LE HAVRE-ROYAL SOVEREIGN HACE

I I

I--

I i X

F-5

-XXX

xX

X

z

X XX X

2-J.

.1:

L

i

20

30

140

50,

RATING (FEET)

Figure 7

VelocIty plots for thè 1973 Le Havre-Royal Sovereign

Race

Figure a (above) shows the actual velocity of each boat

in knots, defined as race distance divided by elapsed time

Also shown in this figure is the least-square

speed curve

repre-sented by equation 10, with the coefficients C1 and C2 determined

from equations 20-21

Figure b (opposite page) shows the corrected

velocities of each boat in knots, defined as the race

distance

divided by corrected time, under each of the eight time

allowance

systems listed in Table 2

Also shown in this figure are the

least-square straight lines corresponding to figure

of merit

two, these straight lines being weighted for each boat by its

f leét position.

WI:

70

7

1973 LE HRVE-RTRLSDVEREIGN flRCE

-

35

-I I I

21J 30 40 50. 60 70

RFITJNG (FEET)

1973 LE -RVRE-R0YRL SOVEE.TGN ARCE

20 30 14Q 50

ARTINC (FEETÌ

H-1973 MORGAN CUP RACE

X

X

X

'igure 8 -- Velocity plots for the 1973 Morgan Cup Race.

Figure a

(above) shows the actu1 velocity of each boat in knots, defined

as race distance divided by elapsed tizne.

A]so shown in this

figure is the least-square speed curve represented by equation

10, with the coefficients Cl and C2 determined from equations

20-21.

Figure b (opposite page) shows the corected velocities

of each boat in knots, defined as the race distance divided

by corrected time, under each of the eight time allowance systems

listed in Table 2.

Also shown in this figure are the

least-square straight lines corresponding to figure of rneit two,

these straight lines being weighted for each boat by its fleet

position.

I I I I I I

20

30

Lb

50

60

70

20 30. LIO 50 60

- 37

j5.T9

6-RATING

[FEET)

RATING [FEET]

70 20 . 30 LIO

.50

60 . 70

-J

3

Cr).

F

)QX

z

xx

S-,-,

*

XX

X X X

>-

x.

F-r-'

D

X

Ç.)

X X

. ,- X

LU

X X

F-

_X X

X

D

1973 CONES-DINAR.D HRC

X X X X X X X I -

-

I : I - I

20

-

30

110

50

60

70

HATING

(EEET)

Figure 9 -- Velocity plots for the 1973 Cowes-Dinard Race.

Figüre a (above) shows the actual velocity of each boat in knots,

defined as race distance divided by elapsed timé.

Also shown

in this figure is the least-square speed curve represented by

equation 10, with the coefficientS C1 and C2 determined from

equations 20-21.

Figure b (opposite page) shows the corrected

velocities of each boat in knots, defined as the race distance

dividèd by corrected time, under each of the eight time allowance

systems listed in Table 2

Also shown in this figure are the

least-square straight lines corresponding t9 figure of merit

two, theSe straight lines being weighted for eadh boat by its

fleet poSition

X

U)

D

O)

D.

:

- *

C1P2 TOT

oi -J BD!RC PFS. NAYHU TOD cil ai

:

+'

y'

''

t V * :+ + V t V, * *+*_

V.

t

. .

1

-t

'V

HORC. TOD I I I

20.

30

1973 CHANNEL RACE

Q

X X X

H

I: i::.

f I I

50

HATING

(FEET)

X

60

Figure 10 -- Velocity plots for the 1973 Chanñel Racé.

Figure a

(above) shows the actual velocity of each boat in knots, defined

as race distance divided by elapsed time

Also shown in this

figure is the least-square speed curve represented by equation

10, with the coefficients C1 and C2 determined f romequations

20-21.

Figure b (opposite page) shows the corrected velocities

of each boàt i.i

knots, defihed as the race distance divided

by corrected time, under each of the eight time allowance systems

listed in Table 2.

Also showi in this figure are the

least-square straight lines corresponding to figure of merit two,

these straight lines being weighted for each boat by its fleet

position.

(D -J w D D

z z

nl w D D pl (D -J w D D

z z

nl f-

J

D D nl NAYFU TOT ACRC2 TOT

ROFIC Toi

CIClI TOT

)

O) a O) O)

U)

D

z

OE

D

D3

1.973 FASINET RPE

X

2Ò

30

140

50

60

70

HATING

(FEET)

Figure 11 -- Velocity plots for the 1973 Fastnet

Race. Figure a

(above) shows the actual velocity of each

boat in kñots, defined

as race distance divided by

elapsed time.

Also shown in this

figure is the least-square speed curve represented by equation

10, with the coefficients Cl and C2

determined from equations

20-21.

Figure b (opposite page) shows the corrected

velocities

of

ach boat in knots, defiñed as the race distance

divided

by corrected time, under each of the eight

time allowance systems

lïsted in Table 2.

Also shown in this figure are the

least-square straight lines corresponding to figure of merit two,

these straight lines being weighted for

eách boat by. its fleet

OE L)

*. I.»

b

LI

.' D D

F

L) D +

r

* t

1973 FRSTNET RRCE

3,0 LI0 50 aO RAT 1N 'IFEET]

= 43

- >-L)

6-q

1973 FRSTNET R.PE

I 'I .1 F. F--D

-*

=5--- * 30 ' LIO 50 ' 'SO ' 70 -, RT'1'NÒ .(FET)

1973 HONOLULU RACE

-

MOCÓ RATINGS

.50.

60

RAT ING

(FEET)

Figure 12 -- Velocity plts for the 1973 Honolulu Raqe.

Figure a

(aboyé) shows the actual velocity of each bOat in knots, defined

as race distance divided by elapsed time

Also shown in this

figure is the least-square speed curve represented by equation

10, with the coefficients C1 and C2 determined from equations

20-21.

Figure b (opposite page) shows the corrected velocities

of each boat in knots, defined as the race distance divided

by correctéd time, üñder éach of the eight time allowance Systems

listéd in Table -2.

Also shown in this figure are the

least-square straight lines corresponding to figure of merit two,

tiese straight lines being weighted for each boat by its fleet

position..

D D Li

7-Li Li 5 20 30 L0 50

RATING

[FEET)

1973 HONOLULU RACE - MOO. RATINGS

t. I I t I I

:.

:

45

-

7-6-

'+

4+,.

Li D D

-5 7 7

1973 HONOLULU RACE - MOO. RATINGS

,

I I -t -4 I

60 70 20 30 Lb .. 50 60 70

x

X

XX

X

XX

XXX

X

1973

SYDNEI-HOBRHT RPÌCE

X* X X X X X X X X X X X X X X

Figúre 13 -

Velocity plots for the 1973 Sydney-Hobart Ràce.

Figure a (above) shows the actual velocity of each boat in knots,

defined as race distance divided by elapsed time.

Also shown

in this figure is the least-square speed curve represented by

equation 10, with the coefficients C1 and C2 determined f röm

'equations 20-21. Figure b (opposite page)

hows the corrected

velocities of each boat in knots, defined as the raöe distance

divided by corrected time, under each of the eight time allowance

systems liSted in Table 2.

Also shown in this figure are the

least-square straight lines corresponding to figure of merit

two, these straight lines being weighted for each boat by its

fleet position.

20

30

LIO

50

60

7q

RRTING (FEED

U-)

F

z

XX X X X, X X

R)

e D

U)

D

:ij

- D

D

NDYRU TOT -J 0) FORC2 TOT 0) -J CD HOFC TOT 0) -J co coi CYCA TOT -J

:

:

:

:

, Q. e e X X

.::

:

e co (J) D

z

Pl 7< D 03 D 33 :u 33

D

ru U) -C

D

Pl

z

D

cci D Ji Ji 33,.

n

_nl CYCR2 TOT -J CD ROAC PFS o) -J

NRIAU TOD

0) -J CD RORC TOD co cii

j