NavList:

   

Reply

Just to elaborate a little bit on Pierre's answer:

Most offshore navigation courses teach "sailings" (or at least used to
teach them pre-GPS).  These are ways of answering the question "given a
starting L/Lo and destination L/Lo, what (rhumb line) course should I
steer to get to my destination and how far away is it?"   (There are two
assumptions here:  (1) you're not interested in a great circle course
with its constant need for changes in course direction and (2) you don't
want to plot the course on a chart and/or that the scale of available
charts are inappropriate for answering the question).

A rhumb line (constant-direction) course stem from and are easily found
by plotting them on a Mercator chart.  But on a Mercator chart the
distance between parallels of latitude increase with distance from the
equator in order to compensate for drawing meridians of longitude
parallel to one another despite the fact that they converge on a real
globe.  This change in distance between parallels makes calculating
course and distance mathematically challenging.

The very easiest way is to calculate the "latitude expansion factor"
using one's starting latitude.  It's simply 1/cos(Lat) (in other words,
if two meridians of longitude a minute apart are drawn an inch apart on
a Mercator chart, two parallels of latitude a minute apart will be
1/cos(Lat) inches apart on the chart (and note that 1/cos(x) is greater
than or equal to 1)).  But if there's any significant change in latitude
at all, the answer will be off using this method because the "expansion
factor" is changes as latitude changes.  An almost-as-easy way to
calculate a rhumb-line course that's much more accurate is using what is
called a mid-latitude sailing, which is what was described by Pierre.
Here the "latitude expansion factor" is approximated by calculating it
for a latitude half-way between one's starting and ending latitudes:
1/cos(mid-Lat).

Another sailing is a Mercator Sailing.  In this, tables of the expansion
factor used in drawing Mercator charts are used to get the most accurate
estimate of the north-south expansion.   They basically say "if a minute
of longitude is x units on your chart, multiply x by this amount to get
the distance between one minute of latitude" for one-minute increments
of latitude from the equator to the pole.   Mercator sailings are harder
to work out because of the need to utilize these tables, but give a more
accurate answer when a course involves a large change in latitude.

Descriptions of Sailings can be found in Chapter 24 of the current
edition of Bowditch (http://www.i-DEADLINK-com/bowditch/pdf/chapt24.pdf)

So far we've talked about "expanding" latitude to match the expansion of
longitude found in a Mercator chart.   At the risk of making things seem
even more complex, another approach is to ask the opposite question --
"what is the size of a degree (or minute) of longitude as compared to a
degree of latitude?"

Of what use is this?   Suppose I sail on a 45 degree course.  For every
mile I sail north, I'll also sail a mile east.   It's easy to calculate
my change in latitude, it's just one minute for every nautical mile.
But what about my change in longitude?  Change in longitude depends on
latitude.  Every mile sailed east or west corresponds to a longitude
change of 1/cos(Lat) minutes!

Now back to Guy's original question.  This is really the inverse problem
from sailings.  Sailings answer "I'm at this L/Lo and want to go to
another L/Lo, what's my course and distance?" Guy's question is "I was
at this L/Lo and sailed a certain course for a certain distance, what's
my new L/Lo?"

Any course between a starting and ending point can be drawn as a right
triangle with the actual course leg being the hypotenuse and the other
sides being the north-south and east-west differences between the
starting and ending points.  This triangle is commonly labeled with the
symbols l, p, D, and C:

          p
     ------------
     |         /
     |        /
     |       /
   l |      /
     |     /  D
     |    /
    >| C /<
     |  /
     | /
     |/


where p is the east-west side of the triangle, l is the north-south side
of the triangle, D is the actual distance and hypotenuse of the
triangle, and C is the course angle.   At this point it's important to
note that p is measured in miles (as are D and l) and not degrees of
longitude.

Calculating l and p knowing D and C is straightforward trig:
     l = D * cos (C)
     p = D * sin (C)

But a navigator usually doesn't want to know how many *miles* east or
west his vessel has traveled, he/she wants to know by how much it's
longitude has changed.

That takes us right back to the formula several paragraphs ago for
converting east-west distance into change of longitude.

Putting it all together:

1.  Calculate change in latitude and therefore endpoint latitude

2.  Calculate mid-latitude by simply averaging the starting and ending
latitudes (or, alternatively, adding half of l (expressed as minutes) to
the starting latitude).

3.  Calculate change in longitude, DLo = p * 1/cos(mid-lat)

Mid-latitude sailings are considered to be reasonably accurate for
distances of several hundred miles (more if course is nearly east-west,
less if it's mainly north-south).   At that point, I hope one is
considering getting a fix!!

Hope this helps.  Forced me to go back and review a lot of stuff on
sailings.


Lu Abel


Pierre Boucher wrote:
> 40 deg 07 min N -> 40.12 (rounded)
> 44 deg 37 min W -> 44.62 (rounded) not needed
> Course 090 -> Course N 90 E
> distance traveled 56.88 Nm
>
> To find DLat (difference of latitude in minutes of latitude)
> distance traveled x cos(course) = Dlat
> 56.88 x cos 90 = 0
>
> To find DLo (difference of longitude in minutes of longitude)
> distance travel / cos(mid latitude)
> 56.88 / cos 40.12 = 74.4 = 1 deg 14.4 min (Eastward)
>
> New DR?
> New latitude
> 40 deg 07 min N + 0 = 40 deg 07 min N
> New longitude
> 44 deg 37 min W - 1 deg 14.4 min = 43 deg 22.6 min W
>
> Pierre Boucher
>
> Guy Schwartz a ?crit :
>
>> This has probably been discussed here before, but hopefully someone
>> will help me out. I'm trying to find the formula for caculating your
>> DR after traveling from a know position.
>> For instance:
>> Last Fix
>> 40 deg 07 min N
>> 44 deg 37 min W
>> Course 090
>> distance traveled 56.88 Nm
>> New DR?
>> It would be a big help to me if someone that knows the formula could
>> incorparate the above example so I could follow along.
>> Thank you,
>> Guy
>
>
>

   

Reply