Spherical Geometry (1 Viewer)

[Muzzaw]

This relates to the 2005 Success One HSC book.

M7: Spherical Geometry.

Question 6 (a) (iii) Find the distance between Tokyo and Teheran along the parallel of latitude.

Now, I know how to do that question, but what I'm confused about is: why can't I apply the same formula to calculate Question 7 (b) (i)?

What's the difference between what the questions are asking for?

Cheers.

 

[Muzzaw]

One more thing.

How is Question 6 (a) (ii) different from Question 6 (a) (iii)?

Arn't they asking for the same thing?

 

[danieljarvis]

i dont have the book, but just a few things it COULD be...

transfer to M ( nautical miles ) or vice versa... to k's

x/360 x pie x Diameter or just angle x 60 to get M , first is more accurate.. both give fairly similar answers but..

 

[PC]

There's two ways to work out the distance between two places. If you know the angular distance (theta), then you can do one of the following:

1. Use the arc length of a circle formula (on formula sheet)
Length = theta/360 x 2πr, where r is the radius of the circle

2. Use the fact that 1 minute is equivalent to 1 nautical mile, so 1° is 60 nautical miles.
Length = theta x 60 (nautical miles)
Then multiply by 1.852 to get kilometres.

BUT

Method 2 ONLY works for great circles. You can not use it for small circles. Teheran and Tokyo lie along the same line of latitude which is a small circle.

Method 1 works for any distance along any line of latitude or longitude, but if you're travelling along a small circle then its radius is NOT 6400 km, so you need to be given the radius of the small circle.

I think Q6 (a) (ii) might be asking you to work out the radius of a small circle. This was part of the old MIS course, but no longer in the General course.

 

[Muzzaw]

If you know the angular distance (theta).

Is the angular distance in Question 6 (a) (iii) 140 - 50 = 90 degrees?

Or is the angular distance only for great circles? I.e North - South?

Just to confirm, great circles are only longitude, except for the equator?

Excellent explanation. Thanks alot.

 

[PC]

Just to confirm, great circles are only longitude, except for the equator?

Spot on!

I don't have the book, so I can't check your first question, angular distance can be along any line of latitude or longitude.

 

[Muzzaw]

Sorry mate, but I'm not 100% on what you mean by 'angular distance'.

Explanation please?

 

[PC]

Town A is (45°N,13°W)
Town B is (45°N,43°E)
Town C is (6°N,43°E)

A and B are on the same line of latitude (45°N - a small circle**) but on different lines of longitude.
Angular distance = 13 + 43 = 56°

B and C are on the same line of longitude (43°E - a great circle) but on different lines of latitiude.
Angular distance = 45 – 6 = 39°

Distance BC = 39/360 x 2π x 6400 = 4356 km
or
Distance BC = 39 x 60 = 2340 nautical miles = 2340 x 1.852 = 4334 km




**Note: The radius of the small circle is given by r = R cos a = 6400 cos 45 = 4525 km
So now we can find distance AB
AB = 56/360 x 2π x 4525 = 4423 km

 

[Muzzaw]

Cheers, mate.