September 24, 2020, 11:04:51 am

### AuthorTopic: spherical geometry help?  (Read 147 times) Tweet Share

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#### parista

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##### spherical geometry help?
« on: August 11, 2020, 08:36:02 pm »
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Hi everyone, I'm really having trouble trying to tackle this question and don't know where to start. I tried drawing diagrams but could not figure it out

Two locations X and Y have the same latitude 30◦S. The longitude of X is 145◦E and the
longitude of Y is 130◦E.

d) Find the difference in distance between the distance around the great circle and the
distance around the parallel of latitude

(whole question attached below)
if this helps, i found the answers for a,b,c and got 5542.56 km, 1451km and 6400km respectively.

any help would be appreciated, thank you so much

#### mathsTeacher82

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##### Re: spherical geometry help?
« Reply #1 on: August 12, 2020, 08:03:39 am »
+2
Hi Parista,

You need to find the circumference of the small circle with latitude 30 degrees.
So you need to find the radius of this small circle first.

I have attached a diagram showing the triangle which you can use to find this radius.

Let me know if it helps, or if you still have any questions...
« Last Edit: August 12, 2020, 08:30:59 am by mathsTeacher82 »

#### parista

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##### Re: spherical geometry help?
« Reply #2 on: August 12, 2020, 09:40:22 am »
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Hi Parista,

You need to find the circumference of the small circle with latitude 30 degrees.
So you need to find the radius of this small circle first.

I have attached a diagram showing the triangle which you can use to find this radius.

Let me know if it helps, or if you still have any questions...

hello!
yes, I've found out the radius of the small circle in part a) however, i'm not quite sure what i am supposed to do after that?

#### mathsTeacher82

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##### Re: spherical geometry help?
« Reply #3 on: August 12, 2020, 10:06:20 am »
+1
OK I see. It's asking for the distance between X and Y, either around the great circle or the small circle with 30 degree parallel of latitude...

Below is an updated diagram to show the triangle and sector you would need to use.

You would use the arc length formula with the radius OX = 6400, but the tricky part is finding the angle <XOY.
« Last Edit: August 12, 2020, 04:14:37 pm by mathsTeacher82 »

#### parista

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##### Re: spherical geometry help?
« Reply #4 on: August 12, 2020, 10:13:14 am »
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OK I see.

The distance around the great circle is $C_1=2\times \pi \times 6400$

The distance around the 30 degree parallel of latitude is $C_2=2\times \pi \times r$ , where r is the radius of the small circle.

Then subtract to find the difference...

ahh..okay i see what you mean!
however, this was the sample solution (photo attached) which is very convoluted:-\

#### mathsTeacher82

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##### Re: spherical geometry help?
« Reply #5 on: August 12, 2020, 04:07:34 pm »
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Yes the answer you attached is correct, although there are a few typos in the working which I have corrected (in red).

But to be honest this question should not appear on your exam. Here is the relevant dot point from the VCAA study design:

"use of a great circle to determine the shortest distance between two points on the surface of the earth that have the same longitude"

If the points are on the same longitude, the required angle in the arc length formula is just the difference in longitudes, and you would not have to go through the "convoluted" process in this problem.
« Last Edit: August 12, 2020, 04:09:14 pm by mathsTeacher82 »

#### parista

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##### Re: spherical geometry help?
« Reply #6 on: August 12, 2020, 04:12:37 pm »
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Yes the answer you attached is correct, although there are a few typos in the working which I have corrected (in red).

But to be honest this question should not appear on your exam. Here is the relevant dot point from the VCAA study design:

"use of a great circle to determine the shortest distance between two points on the surface of the earth that have the same longitude"