handicap.js](https://github.com/morgan3d/sailing/blob/main/racing/handicap.js)
and the examples in [handicap.md.html](https://raw.githubusercontent.com/morgan3d/sailing/refs/heads/main/racing/handicap.md.html)
Conversion
========================================================================
Approximate conversions between [Performance Handicap Racing Fleet](https://www.ussailing.org/competition/offshore/phrf/)(PHRF),
[RYA/US Sailing Portsmouth Numbers](https://www.rya.org.uk/racing/technical/handicap-systems/portsmouth-yardstick) (PN), and
[US Sailing Dixie Portsmouth Numbers](https://www.ussailing.org/wp-content/uploads/2018/01/2017-Portsmouth-Precalculated-Classes.pdf) (DPN) are below. The DPN to PN
conversion is based on fitting a line to the handicaps for the same
boats under the different systems, and could vary slightly depending
on the year in which it is computed.
\begin{eqnarray}
PHRF &=& 6 \times DPN - 330 \\
DPN &=& PHRF \div 6 + 55 \\
PN &=& \tfrac{100}{6} \times DPN - 462 \\
PN &=& 100 \times PHRF - 455 \\
PHRF &=& (PN - 455) \div 100
\end{eqnarray}
Portsmouth
========================================================================
The Portsmouth Number $PN$ is a unitless quantity in the RYA
Portsmouth Yardstick system. It is adjusted regionally each year based
on collected performance across clubs, and then per club in some
cases. See [single-handed](../dinghies/dinghy1.md.html) and
[double-handed](../dinghies/dinghy2.md.html) dinghy handicap tables in
the respective articles.
PNs happen to be calibrated such that almost all boats have a higher
corrected time _CT_ than elapsed time _ET_. That is, PN makes most
boats virtually slower and is a disadvantage-based system.
Popular midrange performance dinghies including 420, ILCA 7 (Laser),
Wayfarer, GP14, Hadron H2, RS200, RS Aero 7, and the Europe are all
relatively close to each other around $PN = 1100$. Lower $PN$ numbers
are faster boats, higher numbers are slower boats.
Corrected Time
---------------------------------------------------------------
The corrected time _CT_ for a boat with elapsed time _ET_ and
Portsmouth number _PN_ is given by:
\begin{equation}
CT = ET \times \dfrac{1000}{PN}
\end{equation}
Lead Times
------------------------------------------------------------------
How much time do you need to beat another boat by to rank higher
than it?
A faster boat _f_ ranks higher in the race than a slower boat _s_
if _s_ finishes more than
\begin{equation}
\Delta T > ET_f \left(\dfrac{PN_s}{PN_f} - 1\right)
\end{equation}
seconds later. Conversely, boat _s_ ranks higher if it finishes
less than $\Delta T$ seconds later.
For example, in a race that a Laser 6 finishes in
$ET$ = 20 minutes, these boats need to finish by the following
elapsed times to tie it:
Pursuit Offsets
----------------------------------------------------------------
In a pursuit race, the start times for each class are offset
so that all boats should finish at the same time. This means
that the rankings are based finish order and no correction
is needed after the race.
Let $PN_Z$ be the Portsmouth number of the slowest, "scratch" boat
class and $ET_Z$ be the anticipated elapsed time for
a boat in that class on the course. The offset start time
for another class is:
\begin{equation}
\Delta T = ET_Z \times \left(1 - \dfrac{PN}{PN_Z}\right)
\end{equation}
For example, in a pursuit race that an Alpha One will start at 13:00 (1:00 PM) and
is expected to complete in 2.0 hours, the start times for the other
boats should be:
This should match your intuition. The RS 800 $PN=799$ is about 2/3 of the Alpha One $PN=1250$,
so the RS 800's expected elapsed time is about 2/3 as much as the Alpha One and it starts
about 45 minutes later for a 120 minute race.
#### Derivation
For the derivation, I shorten the standard notation to $C = CT, E = ET, P = PN$
and leave out the multiplication signs.
The corrected times for the scratch boat $Z$ and another, faster boat
are given by:
\begin{eqnarray}
\label{eqn:CZ} C_Z &=& E_Z \dfrac{1000}{P_Z} \\
C &=& E \dfrac{1000}{P} \\
\end{eqnarray}
Our goal is for all boats to finish at the same wall clock time of day
with staggered starts. That means for each boat
\begin{equation}
E + \Delta T = E_Z
\end{equation}
The elapsed time $E_Z$ of the scratch boat is fixed. We need to find
the $\Delta T$ such that on corrected time, $C_Z = C$ and all boats
would tie. Rearranging to expose the key variables gives:
\begin{eqnarray}
\Delta T &=& E_Z - E \\
E &=& C \dfrac{P}{1000} \\
E_Z &=& C_Z \dfrac{P_Z}{1000} \\
C &=& C_Z
\end{eqnarray}
Substituting and simplifying provides the solution:
\begin{eqnarray}
\Delta T &=& C_Z \dfrac{P_Z}{1000} - C_Z \dfrac{P}{1000}\\
\Delta T &=& C_Z \dfrac{P_Z - P}{1000}
\end{eqnarray}
We need this in terms of the actual elapsed time of the scratch boat, $E_Z$,
so substitute that from equation \ref{eqn:CZ}:
\begin{eqnarray}
\Delta T &=& E_Z \dfrac{1000}{P_Z} \dfrac{P_Z - P}{1000}\\
\Delta T &=& E_Z \dfrac{P_Z - P}{P_Z}\\
\Delta T &=& E_Z \left(1 - \dfrac{P}{P_Z}\right)
\end{eqnarray}
Portsmouth Yardstick Table
---------------------------------------------------
PHRF
========================================================================
The PHRF handicap has units of seconds per nautical mile [s / NM]. A
fast boat has a low or negative rating. A slow boat has a high,
positive rating.
The difference between two boats' PHRF ratings is the expected
difference in their sailing times per mile.
PHRFs happen to be calibrated such that almost all boats have a lower
corrected time _CT_ than elapsed time _ET_. That is, PHRF makes most
boats virtually faster and is an advantage-based system.
For reference, here are some PHRFs for common boats in their standard
configuration. Note that these values are adjusted for different
styles of racing, rigs, crew weight, etc. on specific boats--every
individual boat must explicitly be assigned a PHRF to race.
| Boat | PHRF ----------------|-------------: Farr 36 | 0 J/111 | 39 Dehler 30 | 45 Henderson 30 | 45 C+C 34 | 87 Jeanneau 36 Sun Fast | 93 Hunter 40 | 96 X-99 | 114 Jeanneau SO 349 | 120 Morgan 38 | 138 | Boat | PHRF ----------------|-------------: Beneteau 343 | 147 Swan 36 | 150 Catalina 34 | 150 Nonsuch 33 | 168 Island Packet 38 | 168 O'Day 31 | 171 J/22 | 177 Alberg 30 | 228 Rhodes 19 | 234 Catalina 22 | 276 |