A Small Quiz

[+fizzymagic]

I was working on the documentation for GeoCalc last night, making some pictures of great circle and rhumb lines, and I thought of a few questions that would be a good test of people's intuition about travel on the Earth.

 

For these questions, assume the Earth is a sphere.

1. The direction from London Heathrow (N 51 30.0, W 0 27.0) to LAX (N 33 54.0, W 118 24.0) is:
   1. 
      
   2. SW
      
   3. NW
      
   4. Due W
      

[*]The direction from LAX to LHR is:

1. SE
   
2. NE
   
3. Due E
   

[*]If you could drill a hole straight through the Earth from LHR to LAX, it would point:

1. SW, but not the same azimuth as the above-Earth path.
   
2. NW, but not the same azimuth as the above-Earth path.
   
3. At exactly the same azimuth as the above-Earth path.
   

[*]If you started at LAX and flew with your GPS always indicating a heading of 78.4 degrees true, you would end up in London. If you kept on that path, always keeping the bearing the same, you would eventually end up:

1. At the North Pole
   
2. Back at LAX
   
3. Other
   

[*]If you started at LAX, flew the shortest path to London, and kept on that path you would eventually end up:

1. At the North Pole
   
2. Back at LAX
   
3. Other
   

[*]If you started at LHR and flew with your GPS always indicating a heading of 258.4 degrees true, you would end up at LAX. If you kept on that path, always keeping the bearing the same, you would eventually end up:

1. At the North Pole
   
2. Back at LHR
   
3. Other
   

I'll post the correct answers tonight.

[+BlueDeuce]

Man, it's the weekend.

 

If I was at work I'd be happy to rack my brain with this thing. Brain damage would be better than being a data monkey.

 

If it's still here on Tuesday I'll give you an answer.

[+fizzymagic]

If I was at work I'd be happy to rack my brain with this thing.

No math is required for this; just general knowledge about navigation. The numbers are only there for illustration purposes.

[+BlueDeuce]

No, trust me. Racked brain = Smashed pea.

 

[Hugh Jazz]

I know where you're going with #5 but are we neglecting the fuel required for the trip? Currently there's only one plane that can do it nonstop and I think it's in the smithsonian.

[+wildearth2001]

hmm, i think I have the answers, a few are devilishly tricky with the wording on fizzy's part

[+sept1c_tank]

Not sure if you wanted us to post our answers, but I'm doing it anyway.

 

1. A

2. B

3. A

4. A

5. C*

6. A

 

*It depends on where you stop; you could stop back at LAX.

 

(If I remember correctly, you refused to take my test!)

[+DocDiTTo]

Answers 1-6: I dunno.

 

BUT, since we're here and you mentioned geocalc, how about a Pocket PC version?

 

Then I could tap the stylus and probably answer all these questions when I'm out caching, when I really *need* to have the answers.

[+ParrotRobAndCeCe]

I know where you're going with #5 but are we neglecting the fuel required for the trip?  Currently there's only one plane that can do it nonstop and I think it's in the smithsonian.

 

Edit: nm, you're talking about the final destination

Edited January 15, 2006 by ParrotRob

[+fizzymagic]

(If I remember correctly, you refused to take my test!)

This is true, of course.

 

Thanks for being the first one brave enough to post your answers. I'm surprised, given the willingness of people to post on everything else, that there haven't been more.

 

You only got 2 out of 6 right, though...

 

(3 if you count your smart-you-know-what answer to number 5)

 

Edited January 15, 2006 by fizzymagic

[+ParrotRobAndCeCe]

I was working on the documentation for GeoCalc last night, making some pictures of great circle and rhumb lines, and I thought of a few questions that would be a good test of people's intuition about travel on the Earth.

 

For these questions, assume the Earth is a sphere.

1. The direction from London Heathrow (N 51 30.0, W 0 27.0) to LAX (N 33 54.0, W 118 24.0) is:
   1. 
      
   2. SW
      
   3. NW
      
   4. Due W
      

[*]The direction from LAX to LHR is:

1. SE
   
2. NE
   
3. Due E
   

[*]If you could drill a hole straight through the Earth from LHR to LAX, it would point:

1. SW, but not the same azimuth as the above-Earth path.
   
2. NW, but not the same azimuth as the above-Earth path.
   
3. At exactly the same azimuth as the above-Earth path.
   

[*]If you started at LAX and flew with your GPS always indicating a heading of 78.4 degrees true, you would end up in London. If you kept on that path, always keeping the bearing the same, you would eventually end up:

1. At the North Pole
   
2. Back at LAX
   
3. Other
   

[*]If you started at LAX, flew the shortest path to London, and kept on that path you would eventually end up:

1. At the North Pole
   
2. Back at LAX
   
3. Other
   

[*]If you started at LHR and flew with your GPS always indicating a heading of 258.4 degrees true, you would end up at LAX. If you kept on that path, always keeping the bearing the same, you would eventually end up:

1. At the North Pole
   
2. Back at LHR
   
3. Other
   

I'll post the correct answers tonight.

OK, I'll take a stab at it.

 

1) Generally, SW. More W than S though, so more like WSW

2) NE, More like ENE though

3) exactly the same azimuth

4) At the north pole, via a spiral path

5) Back at LAX, via a great circle

6) Other. I believe you would follow a spiral path to the south pole.

[+fizzymagic]

1) Generally, SW. More W than S though, so more like WSW

2) NE, More like ENE though

3) exactly the same azimuth

4) At the north pole, via a spiral path

5) Back at LAX, via a great circle

6) Other. I believe you would follow a spiral path to the south pole.

Not bad, 5 out of 6, though I ought to ding you for your answer to #2. Yeah, it's sufficiently incorrect we'll leave it as 4 out of 6.

[darwinmay]

I'm probably wrong, but...

 

1. A

2. B

3. A

4. B

5. B

6. B

Edited January 15, 2006 by BlueNinja

[+matfam]

i'll guess

 

1. a

2. b

3. a

4. b

5. c

6. b

[+Mudfrog]

Im sure im way off but here goes:

 

1. a

2. b

3. c

4. a

5. b

6. c

[+El Diablo]

What they said.

 

El Diablo

[+fizzymagic]

Im sure im way off but here goes:

Congrats, Mudfrog! You only got one wrong!

 

The answers are:

1. Direction from LHR to LAX:
   Answer: B. To go straight from London to LA, you head almost exactly NW.
   
2. Direction from LAX to LHR:
   Answer: B. The azimuth is actually 34 degrees, which is almost NNE.
   
3. Drill a hole. What's the bearing?
   Answer: C. Exactly the same azimuth as for question 1. This answer is the only one that depends on approximating the Earth as a sphere; for the ellipsoid, the two are slightly different.
   
4. Constant heading LAX-LHR, continued:
   Answer: A. You'd end up at the North Pole.
   
5. Shortest route LAX-LHR, continued:
   Answer: B. Great-circle routes always go exactly around the Earth.
   
6. Constant heading, LHR-LAX, continued:
   Answer: C. You'd end up at the South Pole.
   

Discuss.

 

I'll post some pictures to illustrate as I finish them tonight.

Edited January 15, 2006 by fizzymagic

[+Moose Mob]

And I was hoping for more ways to look at our "all finds" PQ files (e.g. our finds vs the 500 caches closest to home.)

[+fizzymagic]

And I was hoping for more ways to look at our "all finds" PQ files (e.g. our finds vs the 500 caches closest to home.)

Heehee.

 

What else do you need?

 

Here are the promised illustrations:

A view of the Earth with the paths shown on it:

 

Another view of the same thing, from above the great circle:

 

The same paths shown on an equirectangular projection:

 

Notice that on the map, the great circle looks a lot longer, while on the globe, it's obviously shorter. In fact, the great circle distance is 5441 miles, while the rhumb line distance is 6044 miles - more than 600 miles longer!

Edited January 15, 2006 by fizzymagic

[+Mudfrog]

The answers are:

Direction from LHR to LAX:

Answer: B. To go straight from London to LA, you head almost exactly NW.

 

Well heck, that was the one i thought i may have gotten right. You had me scratching my head for a minute there but it was easy to figure out where i thought wrong when i saw your diagram.

 

Thanks for the lessons!

[+sept1c_tank]

I knew that, but I didn't want to give away the correct answers for everyone else!

[+RockyRaab]

For those not understanding why or how an apparently straight path can become a spiral, it's simple geometry.

 

If you cross every line of longitude at the same geometric angle (except due east or west), you MUST constantly turn. The reason is because lines of longitude aren't parallel. They all cross at the poles. Therefore, the effective ground path will spiral toward the pole, getting tighter as the latitude increases.

 

Make sense?

 

Clever quiz, Mr. cartographer-man!

[+sept1c_tank]

For those not understanding why or how an apparently straight path can become a spiral, it's simple geometry.

 

If you cross every line of longitude at the same geometric angle (except due east or west), you MUST constantly turn...

Your explanation that longitudinal lines are not parallel is correct, but I believe that, following a constant heading, it's not you that is turning. You are actually following a straight line that spirals (apparently) because it is on a sphere.

[+ParrotRobAndCeCe]

For those not understanding why or how an apparently straight path can become a spiral, it's simple geometry.

 

If you cross every line of longitude at the same geometric angle (except due east or west), you MUST constantly turn. The reason is because lines of longitude aren't parallel. They all cross at the poles. Therefore, the effective ground path will spiral toward the pole, getting tighter as the latitude increases.

 

Make sense?

Or to put it in practical terms, wrap a string around a cylinder, like a paper towel roll, and keep going. Unless you're exactly perpendicular to the roll, you'll eventually run off of one end (pole) or the other.

[+fizzymagic]

Your explanation that longitudinal lines are not parallel is correct, but I believe that, following a constant heading, it's not you that is turning. You are actually following a straight line that spirals (apparently) because it is on a sphere.

Actually, as you can see from the picture of the sphere, it's the great circle path that is straight. That's why I asked the question about the hole through the Earth! If you shine a laser parallel to the ground, the beam will get higher and higher as the Earth curves down away from it, but the path you would follow if you stayed exactly beneath the beam, not turning either left or right, is a great circle.

 

The path of constant heading (known as a rhumb line) requires that you turn constantly. That's because the position of the North Pole relative to you is changing as you move.

 

Here's a good analogy. Suppose instead of the North Pole, you used a nearby lightpole as your reference. As you walk past the lightpole in a straight line, the angle between you and the lightpole changes dramatically! If you took a path with a constant angle between you and the lightpole, it would be a spiral.

[+tozainamboku]

Here's a good analogy.  Suppose instead of the North Pole, you used a nearby lightpole as your reference.  As you walk past the lightpole in a straight line, the angle between you and the lightpole changes dramatically!  If you took a path with a constant angle between you and the lightpole, it would be a spiral.

Or a circle or a line. If you are traveling directly toward or away from the lightpole your path will be a straight line. If you are traveling at 90 degrees from the lightpole your path is a circle. Anything else is a spiral, either toward the light pole or away from it. But any geocacher would know to travel straight toward the lightpole because that's where the cache is most likely to be hidden.

 

In fact when I looked at fizzymagic's quiz, I wondered what difference it made to the average geocacher. Now, if you were an airline pilot, you would have to know this stuff.

 

Edit to add:

The above is also true of rhumb line routes. If your heading is 0 or 180 degrees, a rhumb line route will follow the line of constant longitude and is the same as the great circle route. If your heading is 90 or 270 degrees, the rhumb line follows a line of constant latitude (and if you are on the equator will be the same as the great circle route). Other headings spiral to either the north or south pole.

Edited January 16, 2006 by tozainamboku