# Questions tagged [spherical-trigonometry]

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

177 questions
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### How to find the distances between P0 to P1 in this 3 dimensional ellipsoid? [on hold]

(Kindly think your idea before looking at details file ) How to find the distances between P0 to P1 in this 3 dimensional ellipsoid? Can give example please ? How to find the distance D between two ...
1answer
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### How to find the nearest point inside the intersection of two circles to any given point on the surface of a sphere

This drawing I made when I was thinking about the problem shows that my initial idea was simply to calculate the nearest point to the circle whats center point is closest to the target, then from that ...
1answer
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### Do we have a formula for spherical quadrilateral like a triangle one?

Given 3 angle of spherical triangle we could find a solution for arclength of each side with the cosine rule So, given 4 arbitrary angles, is it possible to find 4 arclength for each side in the same ...
1answer
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### Why does y=cot(theta) have zeros at the points where y=tan(theta) has asymptotes

y=tan(theta) has an asymptote when theta= pi/2 because 1/0 is undefined, and the Taylor Series for tan approaching this point just goes on indefinitely I'm guessing (I'm in gr 11 so I'm new to all ...
4answers
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### How to you use the trigonometric functions without a calculator?

Every single time I do anything with circles/triangles I always run into the primary trig ratios. With radians, I found some hope, but it was short-lived, because yet again we needed trig functions to ...
2answers
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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 ...
2answers
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### On a sphere, what is the formula for a great circle in latitude and longitude

Let $\theta$ be latitude, $\phi$ be longitude. I need to find the formula for the great circle passing ($\theta_0$, 0) and (0, $\phi_0$). This seems a easy and common problem, but I can not find any ...
1answer
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### Condition on the existence of a spherical triangle

It is known that $a,b,c>0$ are the sides of a triangle in the Euclidean plane if and only if $$a+b>c,\hspace{0.3cm} a+c>b,\hspace{0.3cm} b+c>a.$$ I would like to give a similar condition ...
1answer
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### “Height” of an equilateral spherical triangle

consider an equilateral spherical triangle (living on a unit sphere) defined by the interior angle of each of its corners. How can I compute the arc length of one of its vertices to the mid-point of ...
1answer
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### Acute triangle on sphere

In excersise 3.7 from Geometry and Topology by Reid M. and Szendroi B. they ask me to prove that $(p,q,r)$ must have a specific form when you have an acute angled spherical triangle whose angles are ...
0answers
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### Surface area of a sphere segment. [duplicate]

I got this problem and solution in a paper but cannot find how they have solved it. Consider I have a sphere that is equally divided into two different patch(P, S). The sphere can rotate and translate(...
1answer
34 views

### Surface area of spherical section delineated by 2 perpendicular circular planes/central angles

The problem concerns visible area based on a field of view from the center of a sphere. I was never taught spherical trigonometry so even basic terminology is hard. After trying to figure out the ...
1answer
27 views

### spherical polar geometry change in elevation angle

how to calculate change in elevation angle if you know coordinates of two point on surface of sphere. let us say assume that a point move on the surface of sphere from [x1 y1 z1 ] = [0.1 0.1 0.9899] ...
1answer
59 views

### Iterative algorithm to draw an ellipse on sphere

I am trying to understand a formula in the drawEllipse function of KDE Marble. This function draws an ellipse, given a center ...
0answers
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3answers
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### Deriving the Sine Formula for spherical trigonometry without using the Cosine Formula

Is it possible to derive the sine formula for spherical triangle without the use of the cosine formula ? Every book on spherical trigonometry derives it from the cosine formula. Kindly provide any ...
0answers
32 views

### How to flatten out a curve based on the dot product to be linear?

I am interpreting a value based on the amount an object is "facing" a wall in a performance critical computer simulation. I take the dot product between the object's forward vector and the negated ...
0answers
66 views

1answer
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### Proofs for the Spherical Laws of sines and Cosines

I am looking for "classical" proofs for the spherical laws of sines and cosines. A proof that relies only on knowledge that was common to the ancient greek geometers, not containing analytical ...
2answers
138 views

### Determine orientation of spherical polygon without trig functions

Is there a way of testing the orientation of a spherical polygon given an ordered list of its vertices that doesn’t involve computing (inverse) trigonometric functions? The polygon is not necessarily ...
0answers
35 views

### How to calculate angle between line segment and line passed through the middle of the segment

There are three points on sphere with latitude/longitude coordinates: ${a}_1, {a}_2, {b}_1$. Let ${a}_m$ is a center of spherical line segment $\overline{{a}_1{a}_2}$. How to calculate an angle ...
1answer
234 views

### How to discretize a sphere?

I would like to discretize a sphere into icosahedra whose vertices are equidistant, i.e., I want to plot $n$ equidistant points on the surface of a sphere. I am familiar with R, Python, and Matlab. ...
1answer
310 views

0answers
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### Spherical collision detection with Longitude and Latitude

Given the following information: A coordinate of an object, along with its bearing A zone, defined by two coordinates Where a coordinate is a latitude, longitude pair How do I calculate whether the ...
1answer
75 views

### Spheric trigonometry - large circle on earth (sphere)

I'm trying to draw a very large circle on the earth surface and my formula fails either when the circle goes above the north pole or when some component reaches 90 degree length (I'm not sure which ...
0answers
49 views

### Make object orbit around another with Degrees only (Not radians)

I'm currently developing something that requires objects to orbit around another. Though, I am limited. I cannot use Radians to achieve this. I have access to sin and cos, and the degrees. But I ...
2answers
104 views

### Dividing spherical triangles on sphere into 4 self similar smaller spherical triangles?

Starting from an intersection of the vertices of a tetrahedron with a sphere; Is it possible to recursively divide the 4 spherical triangles into 4*4 = 16 smaller triangles according to the pattern ...
4answers
158 views

### 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 ...
1answer
984 views

### Expressing angle between two vectors in 3D in terms of spherical polar coordinates

I wanted to express angle between two 3D vectors pointing in arbitrary direction say $\vec{r}$ and $\vec{R}$. If I take the z axis along any other direction (other than the direction of $\vec{R}$ and ...
1answer
71 views

### Find Latitude x miles north of starting latitude using ellipsoid earth model

Let us say I am given a starting (latitude, longitude)=(lat,lon) coordinate in degrees. The objective is to compute the new latitude, ...
1answer
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### Spherical Trigonometry for Horizontal coordinate system

Find the latitude of an observer in the northern hemisphere if it is known, at a certain time, the time angle of the Sun $H_S = 1^h34^m24^s$ , the altitude of the Sun $h_S = 40º10 '$and the ...
1answer
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1answer
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### Integrate the normal vector over a spherical polygon

Given a polygon $P$, with geodesic edges, on the surface of the unit sphere in $\mathbb R^3$, what is the integral of the unit normal vector $\hat n$ over the polygon's area? (The normal vector is ...
3answers
854 views

### 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 ...
1answer
175 views

### Convert a vector in Lambert Conformal Conical Projection to Cartesian

I have wind vectors with 2 components $(u, v)$ that are in the Lambert Conformal Conical Projection. I want to compare the vectors with observations of the wind $(u_{obs}, v_{obs})$ that are created ...
0answers
39 views

### How to find the length of vector to the point on surface of sphere

For my work on ray tracing, I have a light source positioned at <0,0,0.5> which produce the light ray defined by two angles theta (around XY plane) and phi(around YZ plane). Here the ray makes a ...
1answer
72 views

### Spherical Triangles

I'm trying to figure out how to calculate coordinates on a globe, and I would like to ask for some help. Let's say I have POINT A on the globe with the following coordinates: POINT A Latitude 45° ...