May 072010

As I was flying to Japan today, I crossed 11 time zones. When I was in the Navy, I had to really understand time zones because I was regularly working with other commands across many different timezones. Today’s posting is intended to teach you a little more about time zones.
There are 25 world time zones. Each time zone is centered upon every 15 degrees of longitude (7-1/2 degrees on each side of the 15 degree increment) except for the two time zones on either side of the international date line. Those are each only 7-1/2 degrees wide.
Each time zone is represented numerically by a number and a plus or minus sign, such as -5. That number indicates what you need to add to GMT in order to calculate the time in that time zone. In the example of -5 (which happens to be the time zone for EST, or the east coast of the United States in the summer time), we need to add -5 hours to the local time in GMT to determine the local time in the -5 time zone. GMT is centered on the prime meridian, or 0 degrees longitude. It is twelve hours from the international dateline.
The time zones are also represented in the US military by a letter. This works out great since there are 25 time zones and there are 26 letters (want to guess which letter is not used?). GMT (also called UTC for Universal Time Code) is represented by the letter Z, or the word ZULU. The first zone to the west of GMT is labeled N. But it is 1 hour before GMT, it is also labeled -1. To put it all together, we usually call that time -1N (and we usually use the phonetic spelling of N, which is NOVEMBER). So if you were talking about that time zone, you would refer to it as “minus one November”. The next time zone to the west is mike, and it is -2. The first time zone to the east of GMT is A (or alpha) and is labeled +1. And so on. See the table below for the list of all time zones and it’s letter designation.

Abbreviation Full name Time zone Start Long. Center Long. End Long.
Z Zulu Time Zone UTC -7.5 0 7.5
Y Yankee Time Zone UTC – 12 hours -180 -176.25 -172.5
X X-ray Time Zone UTC – 11 hours -172.5 -165 -157.5
W Whiskey Time Zone UTC – 10 hours -157.5 -150 -142.5
V Victor Time Zone UTC – 9 hours -142.5 -135 -127.5
U Uniform Time Zone UTC – 8 hours -127.5 -120 -112.5
T Tango Time Zone UTC – 7 hours -112.5 -105 -097.5
S Sierra Time Zone UTC – 6 hours -097.5 -090 -082.5
R Romeo Time Zone UTC – 5 hours -082.5 -075 -067.5
Q Quebec Time Zone UTC – 4 hours -067.5 -060 -052.5
P Papa Time Zone UTC – 3 hours -052.5 -045 -037.5
O Oscar Time Zone UTC – 2 hours -037.5 -030 -022.5
N November Time Zone UTC – 1 hour -022.5 -015 -007.5
A Alpha Time Zone UTC + 1 hour 007.5 015 022.5
B Bravo Time Zone UTC + 2 hours 022.5 030 037.5
C Charlie Time Zone UTC + 3 hours 037.5 045 052.5
D Delta Time Zone UTC + 4 hours 052.5 060 067.5
E Echo Time Zone UTC + 5 hours 067.5 075 082.5
F Foxtrot Time Zone UTC + 6 hours 082.5 090 097.5
G Golf Time Zone UTC + 7 hours 097.5 105 112.5
H Hotel Time Zone UTC + 8 hours 112.5 120 127.5
I India Time Zone UTC + 9 hours 127.5 135 142.5
K Kilo Time Zone UTC + 10 hours 142.5 150 157.5
L Lima Time Zone UTC + 11 hours 157.5 165 172.5
M Mike Time Zone UTC + 12 hours 172.5 176.25 180

So, what happens when daylight savings kicks in? Daylight savings (also known as “Summer Time”) affects the zone descriptions. Assume one day I am on standard time and I have a particular zone description (say, -5R). When daylight savings kicks in, I will then have a different description (in this case, -4Q).
The time zones do not follow the 15 degree +/- 7-1/2 longitude lines exactly. Sometimes the lines are adjusted for political reasons or convenience.
What’s more, there are some areas that are 1/2 hour ahead/behind the nearest time zone instead of the usual hour and there are even some areas that follow quarter-hour differences.

May 042010

This week I am flying to Japan for a business trip. I have only been there once before for another business trip and I really enjoyed it, so I am really looking forward to going again. Except for the whole 15-hour flight part. That part isn’t so great. But it is good to know that Great Circle Routes can shorten the travel distance (and time) considerably.  Compare the great circle route shown in this picture with the straight heading route from the image below.  Note that it almost passes north of Canada and Alaska, and skirts along the eastern coast of Siberia.  In fact, when we take off, we will be heading close to a northerly direction (approximately 332 degrees).  Along the way, the plane will have to alter its heading to stay on this route.  In fact, as we approach Japan, we will be heading almost due south.  The entire route will be a big sweeping left turn.  Every 30 minutes or so the plane will alter course by about 8 degrees to the left.  This seems a little confusing because the equator is a great circle, and if I flew a path between two cities on the equator, that would indeed be the great circle route and the shortest route.  But I wouldn’t have to change course from either due east or west.  That’s only because the equator just happens to be a great circle that lies on a cardinal heading (east-west).  Also, if you were flying a great circle route that had you going either due north or due south, you would not have to alter your heading during the trip.  In this case you would be flying along a line of longitude.  Other than these two special cases, all other great circle trips require periodic heading alterations to stay on course.  So let’s analyze my trip.

The Newark airport is at 40° 41′ N 074° 10′ W and the Narita airport in Tokyo is at 34° 45′ N 140° 22′ E.

Using regular trigonometry we can calculate the distance between these two airports using the fact that on average a degree of latitude is about 69.1 miles long and a degree of longitude is about 53.0 miles long. So all we do is calculate how many degrees of change in latitude and how many degrees of change in longitude there are between the two points, and multiply it by the mileage factors. Now, this formula makes one big (incorrect) assumption. That is, that the distance between two points long a line of longitude is the same at all latitudes.  In just a minute, we will correct for that. So, using Pythagorean, and taking the north-south change as the “X” and the east-west change as the “Y”, we can calculate the length of the hypotenuse.

We are basically trying to calculate the length of this line:

That is the route you would take if you pointed your plane right at Tokyo and flew straight at it the whole way.  As you know, when you fly on a great circle route, you have to periodically alter your heading  to stay on the route.

x= 69.1 * (lat2 - lat1)
y = 53.0 * (lon2 - lon1)

For the math to work out, north latitudes are positive and south are negative; east longitudes are positive and west are negative, and we need to convert the degree-minutes-seconds into decimal degrees

dist = \sqrt{x^2 + y^2}

Newark, NJ
lat1 = 40+41/60 = 40.6833
lon1 = -74+10/60 = -74.1667

Tokyo, Japan
lat2 = 34+45/60 = 34.75
lon2 = 140+22/60 = 140.3667

x = 69.1 * (34.75 - 40.6833) = -409.9910
y = 53.0 * (140.3667 - -74.1667) = 11370.2702

dist = \sqrt{11370.2702^2 + -409.9910^2} = 11377.6596 miles

But now let’s try and adjust for the difference for different distances at different degrees of latitude.

The improved formulas are:

AvgLat = (lat2 + lat1)/2
x = 69.1 * (lat2 - lat1)
y = 69.1 * (lon2 - lon1) * cos(AvgLat)

When taking the cos calculation, make sure you are calculating in degrees mode, not radians.

dist = sqrt(x^2 + y^2)

Substituting our values in

AvgLat = (40.6833 - 34.75) / 2 = 37.7167
x = 69.1 * (34.75 - 40.6833) = -409.9910
y = 69.1 * (140.3667 - (-74.1667)) * \cos(37.7167) = 11726.6589

dist = \sqrt{11726.6589^2 + (-409.9910^2)} = 11733.8239

Which, interestingly, is higher than the first approximation, yet is considered more accurate. However, it probably is pretty close to the actual distance that would be traversed if that route were indeed chosen. The number does make sense though.  The circumference of the earth is about 24,000 miles.  I am not going quite half way around, so a number a little less than half seems to be about right.  I would not recommend choosing that route though because the great circle route is shorter by several THOUSAND miles. Let’s calculate it.

First, convert all the lats and longs to radians.

Newark, NJ
lat1 = 40.6833 / 57.29577951 = 0.7101
lon1 = -74.1667 / 57.29577951 = -1.2945

Tokyo, Japan
lat2 = 34.75 / 57.29577951 = 0.6065
lon2 = 140.3667 / 57.29577951 = 2.4499

The formula for great circle distance is

3963.0 * arccos[\sin(lat1) * \sin(lat2) + \cos(lat1) * \cos(lat2) * \cos(lon2 - lon1)] 3963.0 * arccos[\sin(.7101) * \sin(.6065) + \cos(.7101) * \cos(2.4499-(-1.2945))] = 6788.5935

For these calculations, you will need to do the trig functions in radians mode.

Google earth calculated 6724.76 miles. It looks like they are off by about 60 miles :)

Actually, I did some rounding along the way here, and these calculations are very prone to precision errors.

But what a difference!  As you can clearly see, the great circle route is the way to go here.

Finally, even though this is the shortest route, it may not be the fastest route. The airlines are pretty smart about taking other things into consideration, namely the high altitude weather. They like to fly with the wind on their tails to give them a little boost. Sometimes it can be quite significant.

Now, I have not gone into why great circle routes are the shortest routes.  Hopefully you already understand that, but if you are a little rusty on that concept, there is plenty of reading material available on the web.  But as you can see from our simple calculations here, the distance saved is significant by choosing a great circle route.

The formulas for calculating great circle distances are well documented and are known as the spherical laws of cosines.  I used this web page for most of my verification.

Apr 292010

Have you ever planned out a single photo?  I mean, really planned out single photo?  If you are like most people, you take snapshots of your friends and family and interesting things you see as you travel, but you never really planned a single shot. The most satisfying pictures that I have ever taken are some that I literally took hours, if not days, to research, to get just the perfect picture.  They aren’t my BEST photos that I have ever taken, but they were very satisfying because in the end, I knew my research paid off and I got the shot that I envisioned.
Every day I drive over some railroad tracks near my house.  One night when I was returning home from work, I noticed that the sun was about to set a few degrees to the left of the track.  I wondered if the sun would ever set “right on” the track.  I wondered if I had just missed it, perhaps the week before.  I couldn’t figure out if the sun was moving progressively to the right or to the left each night.  How would I calculate all these things?
WARNING–This paragraph is a little technical with respect to astronomical terms, but I will try to keep it simple.  If you want to skip it, I won’t blame you.  When I got home, the first thing I did was open google earth and find the railroad in my neighborhood. Using the “Measure” tool, I calculated the angle of the railroad track.  It happened to be 260 degrees west, or 80 degrees east.  I was looking to shoot my picture at sunset, so I was interested in the westerly direction.  I then went to, entered in my location (Chesapeake, Virginia) and found out that the sun would set at the azimuth of 260 degrees on Feb 25th and 26th.  Furthermore, the numbers were increasing each day, so I knew that I did not miss my shot.  Of course, I could always wait until next year (or more accurately, on October 14th when the sun comes back from the other direction, but I was happy that I did not have to wait).
You still with me?  Good.  So, now that I calculated the dates I could take my shot (Feb 25th & 26th), I was ready to go.  If you remember, February 25th was a rainy day.  I was not able to go out and shoot that day.  I did however, get out the next day.  You can see here the shot I got.  And if you look carefully, you will see that the sun is a little to the right of the tracks.  I think February 25th would have been a better day.  Also, another option is to shoot the sun rising on the track.  The reciprocal of 260 degrees is 80 degrees.  It turns out that the sun does rise at that azimuth on April 7th and 8th, and September 2, 3 and 4.
The punchline here is, if you want to be a master of photography, you will have to also be a master of your subject.  If you want to get great horse racing, or food, or portrait pictures that don’t look like the same pictures that every other person with a camera gets, you have to know every in and out, up and down of the subject, and how it correlates to lighting and photography.  Few people in this hobby just have the knack and can just point their camera at something and have it spit out money.  It doesn’t work that way.  It takes time and effort to really get to know your subject and how to best photograph that subject.