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 .

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Why are the trig functions versine, haversine, exsecant, etc, rarely used in modern mathematics?

I was browsing through a Wikipedia article about the trigonometric identities, when I came across something that caught my attention, namely forgotten trigonometric functions. The versine (arguably the most basic of the functions), coversine,…
sun_Jiaoliao
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Step forward, turn left, step forward, turn left ... where do you end up?

Take $1$ step forward, turn $90$ degrees to the left, take $1$ step forward, turn $90$ degrees to the left ... and keep going, alternating a step forward and a $90$-degree turn to the left. Where do you end up walking? It's very easy to see that you…
Anonymous
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Solid Angle Trigonometry?

So I am interested in finding out how solid angle trigonometry works. Specifically, in 3 dimensional space, if we have three vectors reaching out from the origin, when we link the tips of the vectors together, we should get a tetrahedron. The idea…
YiFan Tey
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Intersection of circle and geodesic segment on sphere

I am trying to find an efficient way of computing the intersection point(s) of a circle and line segment on a spherical surface. Say you have a sphere of radius R. On the surface of this sphere are a circle with center ($\theta_c$,$\phi_c$) and…
qsfzy
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Probability a random spherical triangle has area $> \pi$

From Michigan State University's Herzog contest: Problem 6, 1981 Three points are taken at random on a unit sphere. What is the probability that the area of the spherical triangle exceeds the area of a great circle? I assume we always take the…
aschepler
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Solid angle: integration

Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a Jacobian but have been unsuccessful: $$\iiint dr\…
Sam Manzer
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The area of a right spherical triangle

Is there a compact formula for the area (excess angle – assuming a unit sphere) of a right spherical triangle given its side lengths $a$ and $b$? As explained in an answer to an earlier question about the area of a generic spherical triangle, the…
S.G.
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Circle On Sphere

Sorry if this sounds too silly but my math skills are very poor and I just need this problem fixed. I made this graphic with geogebra 3D and it was quite easy there but I don't know how to write the equations for this. In the image you can…
dargelos
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Combinatorics for a 3-d rotating automaton

Let's suppose that we have some kind of special 3-dimensional rotating automaton. The automaton is capable to generate rotation about selected $X$ or $Y$ or $Z$ axis (in a current frame) in steps by only constant +$\dfrac{\pi}{6}$ angle (i.e.…
Widawensen
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Alternative proof of Girard's theorem

I am looking for an alternative proof of Girard's theorem. The standard proof, which is almost trivial, relies too much on visualizing spherical triangles on the sphere. Is there a more algebraic proof, or even a proof by straightforward integration…
user54031
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Synthetic proof that, given points $A,B,C,D$ in space, $\angle BAC \leq \angle BAD+\angle DAC$?

Consider the following statement: Given points $A,B,C,D$ in space, $\angle BAC \leq \angle BAD+\angle DAC$. This seems obvious enough -- if you're rotating a beam that moves through space at a fixed angular speed, the fastest way to go from…
Carl Schildkraut
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Finding the northernmost latitude in a great circle that passes through two points on the sphere

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on p.23 of the 6th ed.): $A$ and $B$ are two places on the earth's surface with the same latitude $\phi$; the difference of longitude between $A$…
Edgar
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Triple integral of the great-circle distance function

Numerical integration suggests that $$\mathcal U=\int_0^\pi\int_0^\pi\int_0^\pi\arccos\left(\cos x\cdot\cos y+\sin x\cdot\sin y\cdot\cos z\right) dx dy dz\stackrel{\small\color{gray}?}=\frac{\pi^4}2\tag1$$ (note that the function being integrated…
Vladimir Reshetnikov
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The distribution of areas of a random triangle on the sphere - what are the second, third, etc. moments?

Suppose that we choose three points independently and uniformly at random on the surface of a unit sphere as the vertices of a triangle, and consider the area of this triangle. Call this random variable $X$. The area of such a triangle is the sum of…
RavenclawPrefect
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Find the distance between Helsinki and Seattle

How do I find the distance between Helsinki and Seattle along the shortest route? This is actually a question from a mathematically-based Astronomy book. I am not in a science or math class. I'm just trying to learn Astronomy on my own and this is…
Science_Math
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