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right angles (spherical triangle rectangle)

geodesist
(@geodesist)
10+ posts Member

Dear colleagues

do all intersections between latitudes and longitudes form right angles (90 °) or only the original meridian with the equator forming a right angle (90 °)?

Thanks

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Topic starter Posted : June 16, 2019 1:11 pm
Bill93
(@bill93)
5,000+ posts Member

All meridians form right angles with the tangent to each parallel of latitude.

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Posted : June 16, 2019 1:29 pm
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geodesist
(@geodesist)
10+ posts Member

Thanks, please what about (90°-Â) in the right spherical triangles (Napier Mnemoic)?

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Topic starter Posted : June 16, 2019 3:16 pm

Dave Lindell
(@dave-lindell)
1,000+ posts Supporter

The thing about spherical angles is that they are computed from the triangle sides, which are themselves measured in angular form.

A right-angled spherical triangle can appear anywhere on the sphere.  (Looking at a globe, it could be in Mali with the legs pointing in the direction of Greenland and Brazil, for example.)

The easiest one to picture is the one with all three angles being 90°: from the north pole south along the Greenwich meridian to the equator, thence west along the equator to the 90th meridian, thence north along the 90th meridian to the north pole.  (You can make one by cutting a melon through its "equator" and then slicing it into four equal parts.)

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Posted : June 16, 2019 5:54 pm
RADAR, dms330, MathTeacher and 1 people liked
geodesist
(@geodesist)
10+ posts Member

Thank you, for your interesting response.

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Topic starter Posted : June 17, 2019 1:45 am
GeeOddMike
(@geeoddmike)
1,000+ posts Member

Beyond the explicit question in the original post, not sure what you are curious about. Napier’s rules and mnemonic relates to solving quantities in spherical trigonometry. 

Here is an interesting book on spherical trigonometry you might find of interest: https://www.amazon.com/gp/product/0691148929/ref=as_li_ss_tl?ie=UTF8&tag=theende-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0691148929

“Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry” by Glen Van Brummelen. The Kindle e-book version is only $9.99.

Lost art indeed. I am surprised that the responses did not mention spherical excess...

IMO Computations using a spherical representation of the earth rather than an ellipse reduces its utility in positioning except in navigation or approximations.

Cheers,

DMM

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Posted : June 20, 2019 11:46 am

Larry Scott
(@larry-scott)
500+ posts Member

No matter you are, from due north/south to due east/west is 90 deg. 

Due E/W at a point is the great circle, the prime vertical. 

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Posted : June 22, 2019 1:01 pm
Dave Lindell
(@dave-lindell)
1,000+ posts Supporter

Wrong on all accounts.

The tangents of north-south to east-west are a right angle.  The angle on the sphere is totally different.

The only east-west great circle is the equator. To travel east or west from a point on the sphere is to maintain the same latitude, that of a small circle.

All longitude meridians are great circles.

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Posted : June 22, 2019 2:10 pm
Bill93
(@bill93)
5,000+ posts Member

The only great circle that is always E-W is the equator. At any point except a pole there is a great circle which heads E-W along the tangent to the parallel of latitude, but then its direction does not continue along the parallel.

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Posted : June 22, 2019 4:35 pm

Larry Scott
(@larry-scott)
500+ posts Member

No. Due E/W is the direction 90 from north. Take one step in that direction and it’s no longer E/W. And keep going and you’ll go below the equator 180 deg longitude later, and come back up north of the EQ  that’s a great circle. The prime vertical  

A straight line, is a line of CONSTANTLY changing bearing. 

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Posted : June 22, 2019 4:51 pm
Larry Scott
(@larry-scott)
500+ posts Member

There are infinite great circles on the globe. Meridians and the EQ are just a few examples. 

Back site due N, turn 90, set a point a off in the distance. The forward azimuth is due E,  by definition. As you know, the back azimuth is not due W, and no point along the line due E or W. 

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Posted : June 22, 2019 5:21 pm
Dave Lindell
(@dave-lindell)
1,000+ posts Supporter

And just to show you what spherical triangle can do, you can have a "triangle" with only two sides: start at 0° longitude at its intersection with the equator, travel along the equator to 180° longitude, then along the great circle through the pole to the point of beginning.

Are there two 90° angles and a 180° angle in this lune ?  Two 90° and two 180° ?

All spherical triangle sides by the way are along great circles.

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Posted : June 22, 2019 7:45 pm

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