Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [spherical-trigonometry]

For geometric questions about solving spherical triangles and spherical polygons on spheres.

Spherical trigonometry is the area within that studies the of spherical polygons—most notably, spherical triangles—which are bounded by great arcs on . Its importance to spherical geometry is akin to that of to .

279 questions
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Precise statement of Legendre's theorem on spherical triangles

This Google Books search finds a page that says this: 74. Legendre's theorem ${}\qquad\qquad{}$ If each of the angles of a spherical triangle whose sides are small when compared with the radius of the sphere be diminished by one third of the…
Michael Hardy
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How can I locate the poles of a great circle given two non-opposing points?

Given two non-antipodal points on the surface of a sphere (in -lat,lon or any similar coordinate system) how do I calculate the positions of the poles (one will do, of course) of the great circle that runs through them? I can work out trigonometric…
Roger Neyman
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A great circle on a sphere

The following proposition is from 'Spherical Geometry and Its Applications' by Marshall A. Whittlesey: Proposition 5.6 If two distinct points on a sphere are not antipodal then there exists a unique great circle passing through them [1] Let there be…
Ali Kıral
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distance from athens to san francisco

I have a question, how to calculate distance from athens to san francisco? here is what I did, but it is different than the answer I searched on google San Francisco & Athena both have 38 degree North Latitude; San Francisco has 122 degree West…
Rex Rau
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Help with direct tunnel distance between two lat /long coordinates.

My brother wants to take a sign to the Sign Post Forest in Canada's Yukon. He wants it to show the distance to London, but directly through a tunnel that only exists in his head. There's not a lot in his head. So it's the mathematically shortest…
Paul Uszak
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What is your idea on "3-d trig ratios" for surface area?

Let's say we take a triple integral and use it to find "3d angles", which use the idea of of taking a portion of a 3-d sphere, compared to the entire sphere. If 2-d trig ratios can be used to find side-lengths of 2-d triangles, is it possible we can…
Arbuja
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Determining the solid angle of 3 overlapping spherical caps of same angular radius

Please consider figure 1 which displays 3 spherical caps slightly overlapping on the unit sphere $S2$ with a spherical triangle intersection area hightlighted in green. Let $\vec{U} = [u_x,u_y,u_z]$; $\vec{V}=[v_x,v_y,v_z]$ and…
Randall
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Relationship between the length of the tangent line through a point on sphere and great-circle distance

As an aviator I'm familiar with the concept of great-circle navigation because when we fly a route between 2 points on the globe we know the shortest distance between these two points is the great circle distance. I'm developing a navigation app in…
rgraulus
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Stewart theorem validity on a sphere

EDIT1: On a spherical surface radius $R$ a geodesic triangle is drawn: Let a,b, and c be the lengths of the sides of the geodesic triangle. Let d be the geodesic arc length of a cevian to the side of length a. The cevian divides the side of length a…
Narasimham
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Three great circles meet at a point

Find a relation between three great circle segments of length $(a,b,c)$ meeting at a point when the sum of integral curvatures of the three triangles equals $ 2\pi $ steridians.
Narasimham
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Is there a proof of the spherical law of sines?

I know that the spherical law of sines is $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$ but how does one prove this rule?
Henry Lock
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Express sine of 1/2 angle of a spherical triangle as a function of the sides derivation

In reading through Spherical Trigonometry (Todhunter) he makes a jump from $1 - \dfrac{\cos a - \cos b \cos c}{\sin b \sin c}$ being equal to $\dfrac{\cos(b - c) - \cos a}{\sin b \sin c}$. I do not understand how he equates the numerators? Any…
Craig Matthews
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Find the cosines of the (arc) lengths of the sides. How many sides have an arc length less than $90^{\circ}$?

Question: A Spherical triangle has angles of $120^{\circ}$, $60^{\circ}$ and $45^{\circ}$. Find the cosines of the (arc) lengths of the sides. How many sides have an arc length greater than $90^{\circ}$? I applied the law of cosine which is $$\cos…
Justin
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Find the centroid of a spherical triangle

Calculate the latitude and longitude of the centroid O of a spherical triangle ABC with the following vertices: A) Miami International Airport 25.793333, -80.290556, B) John F. Kennedy International Airport 40.639722, -73.778889, and C) Los Angeles…
Thomas E Jones
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Elusive common geometric parameter / property in spherical triangles set

A constant length $(=a)$ segment of a variable great circle has its extremities $(P,Q)$ moving along fixed Longitudes $(L1,L2)$ passing through North pole $N.$ Please help to identify the geometric parameter with invariant property $\dfrac{\sin…
Narasimham
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