right angles (spher...
 
Share:
Notifications
Clear all

right angles (spherical triangle rectangle)

Page 1 / 2

geodesist
Posts: 42
Member
Topic starter
(@geodesist)
10+ posts
Joined: 3 years ago

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

11 Replies
Bill93
Posts: 8096
Member
(@bill93)
5,000+ posts
Joined: 11 years ago

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

Reply
1 Reply
geodesist
Member
(@geodesist)
Joined: 3 years ago

10+ posts
Posts: 42

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

Reply
Dave Lindell
Posts: 1360
Supporter
(@dave-lindell)
1,000+ posts
Joined: 11 years ago

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.)

Reply
1 Reply
geodesist
Member
(@geodesist)
Joined: 3 years ago

10+ posts
Posts: 42

Thank you, for your interesting response.

Reply

GeeOddMike
Posts: 1252
Member
(@geeoddmike)
1,000+ posts
Joined: 11 years ago

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

Reply
Larry Scott
Posts: 881
Member
(@larry-scott)
500+ posts
Joined: 7 years ago

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. 

Reply
3 Replies
Dave Lindell
Supporter
(@dave-lindell)
Joined: 11 years ago

1,000+ posts
Posts: 1360

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.

Reply
Larry Scott
Member
(@larry-scott)
Joined: 7 years ago

500+ posts
Posts: 881

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. 

Reply
Larry Scott
Member
(@larry-scott)
Joined: 7 years ago

500+ posts
Posts: 881

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. 

Reply
Page 1 / 2
Share: