Everyday Nav Math

What math do I use daily?

This question came up recently, and it was actually enough of an answer to become a blog post. I realized I do have a number of things I do each day outside of Expedition/Predictwind/… or when answering a quick question about something.

What I found intriguing is we do teach a lot of paper charting in navigation classes, but I maybe have a paper chart out if I am on a boat during passage for backup, and that’s usually only for ocean crossings.

This is also not a comprehensive discussion of all navigation math as this is a pretty broad topic mostly covered in bowditch. These are just the things I find myself using daily.

Degrees to Decimals

Probably the most common things I do is convert degrees to decimals and decimals to degrees because different tools utilize different input formats. For instance, the windy app uses decimal degrees directly, but the Navionics Boating app takes Degrees Minutes Seconds as input.

I was asked recently why so many apps use degrees and minutes with the seconds represented as decimal minutes. That is because minutes of latitude are miles. So if you see 40º 29.715’ you know that a location is 29.715 miles poleward of the 40th parallel.

When I am converting from Yellowbrick data to cross reference something in an app like windy, we need to convert to purely degrees. Yellowbrick positions are similar to above, 40º 29.715’ so to convert that to degrees:

Degrees = Whole Degrees + minutes/60

• 40 + 29.715/60 = 40 + 0.4952667 = 40.4952667º

Another example is looking up data for a waypoint in degrees minutes seconds(DMS) We recently were really focused on Wake Island’s location and timing wind and waves at that location. DMS Lat is 19º 18’ 30.7872N, to convert that to purely decimal degrees:

Degrees = Whole Degrees + (minutes + seconds/60)/60

• 19 + (18 + 30.7872/60)/60
  = 19 + (18 + 0.51312)/60
  = 19 + 18.51312/60
  = 19 + 0.308552
  = 19.308552

Decimals to Degrees

So what about going the other way? Storm centers are often listed in pure decimal form. For instance, Hurricane Marty’s position on Aug 24 was listed 20.6N118.1W by NHC. Let’s say in monitoring the eye on satellite throughout the day we see the center and it is at 20.5487.

Degrees = Whole part

Minutes = 60 * Decimal part of Degrees

Seconds = 60 * Decimal part of minutes

eg:

Degrees = 20º

Minutes = 0.5487 * 60 = 32.922 = 30min

Seconds = 0.922 * 60 = 55.32

So we have both degrees min(20º32.922’), and dms(20º32’55.32”).

Latitude to Nautical Miles

Latitude to nautical miles is quite simple. Each minute of latitude is a nautical mile, so each degree is 60 nautical miles. Let’s say I want to determine the approximate distance to Portland from Seattle, since they are pretty close to the same longitude, both are near the 122.5º meridian.

Seattle’s latitude is 47º30.912

Portland’s latitude is 45º36.372

Subtracting and borrowing one degree we can write
46º90.912
- 45º36.372
———————
01º54.54

Nautical miles from longitude

NM = 60*degrees + minutes
NMseapdx = 60 + 54.54NM

NM seapdx = 114.54NM

Longitude to Nautical Miles

For this I generally use just use the Departure calculation. For great circle distance, it would be advisable to use the Haversine formula, but most of the time it’s a bit overkill. The departure calculation is simply:

NM = Minutes of Longitude * cos(latitude)

So this time let’s look at Eugene to Bend Oregon. They are both quite close to the 44.05N parallel.

Longitude Eugene: 123º05.208'
Longitude Bend: 121º18.918’

122º65.208
-121º 18.918
——————
1º46.29’ = 60+46.29 = 106.29’

NMeugbend = 106.29*cos(44.05) = 106.29*0.7187333227 = 76.47NM

Dead Reckoning Location

We all get really used to drawing our dead reckoning position on a paper chart looking at the edge of the chart for our latitude and longitude during nav class. Later in sailing, using a quick line on the plotter.

But I also though do a lot of work assisting people and I like to be able to run a number of DR lines to see what the difference may be in sea state, current, or wind before determining a best heading.

Here is the method I use:

Given a current latlon, heading, duration, and speed, find the ending coordinates for a given run:

LatLon:47.68344,-122.40630
Heading 30°T
Speed:8kts
Duration:45min(.75h)

First calculate the distance(speed*duration):
8*.75=6NM

Next, we need to calculate the latitude and longitude components of this. So we go back to middle school SOHCAHTOA. Sin = opp/hyp, cos = adj/hyp, and then there’s something about tangent🤓. Anyhow:

lat distance = 6*cos(heading)
= 6*cos(30)
= 6*0.8660254
= 5.196152422706632NM

lon distance = 6*sin(heading)
= 6*sin(30)
= 6*0.5
= 3NM

Adjusting latitude is pretty straight forward, we simply add 5.2min to 47°41’ and arrive at 47°46.2’

Longitude is a bit more nuanced because we need to convert the distance in nautical miles to longitude degrees at 47.77° latitude.

To do this we need to divide the distance in nautical miles by the departure at 47.77, then add it to the original longitude.

LonDegFromNM = 3.0/cos(47.77)
= 3.0/0.672108381606835
= 4.463565820779952’

Now we can simply add 4.464 min to -122° 24.378’ to arrive at -122° 19.914’

So at the end of our leg, we will ideally find ourselves near:
47°46.2’ -122°19.914’

There are a few more equations I use underway when I am onboard for celestial calculations, but we will touch on those in a later post. I hope these are something you can incorporate into your nav exercises onboard or at your workstation.