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Questions tagged [spherical-trigonometry]

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

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27 views

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 ...
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1answer
31 views

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 ...
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1answer
37 views

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 ...
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1answer
25 views

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 ...
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4answers
70 views

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 ...
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2answers
5k views

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 ...
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2answers
71 views

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 ...
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1answer
30 views

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 ...
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1answer
28 views

“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 ...
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1answer
32 views

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 ...
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0answers
46 views

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(...
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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 ...
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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] ...
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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 ...
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0answers
38 views

Intersection points between a circle and a straight line on a sphere

I have a circle on the surface of a sphere. I need to check whether the circle intersects with a given straight line or not. The center of the circle $c$ is given in terms of latitude and longitude $(\...
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1answer
33 views

Spherical Triangles: Area and mapping to Euclidean space

If we take a sphere of radius 1 and travel a quarter-circumference south from the north pole, turn 90 degrees, travel another quarter circumference, then return North, we form a triangle with an angle ...
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1answer
42 views

Spherical cap problem - trigonometry / circle theorems problem / surface area

Graph here I am trying to derive the following equation from a paper I am studying, which the author has derived from the graph above. The two slightly curved lines here are modelled as the surfaces ...
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0answers
12 views

What is the way to go from point 1 to 2 while remaining in the region defined by half torus?

Basically this is a trajectory planning problem. As shown in figure below, how can I ensure that in moving from a point (x,y,z) to another point within torus, I don't get out of that torus region? go ...
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4answers
154 views

Angles in a spherical triangle.

Just seeking advice here! I have 3 coordinates; $A(-0.52992,0.84805,0),\\ B(0.84805,0,0.52992),\\C(0.15461,0.47553,0.86603)$. I want to find the angles at $A$, $B$ and $C$. Hence, I find the normal ...
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1answer
60 views

Closest point on line segment of a great circle

If I have a sphere of radius R, and two points $A$ and $B$ on its surface, at $(R, \theta_A,\phi_A)$ and $(R, \theta_B,\phi_B)$ respectively in spherical coordinates. Call $AB$ the geodesic from $A$ ...
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1answer
32 views

Projection of points on the spherical triangle

I'm looking for a formula to project (x,y,z) point of a spherical triangle (v1, v2, v3) onto the point (a,b) of planar triangle defined by same vertices (v1, v2, v3) and vice versa: sTriangle(x,y,z) &...
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1answer
36 views

Solving stereographic projection for central latitude $\phi_1$ and central longitude $\lambda_0$

For a given longitude and latitude and their projected point I want to know the central longitude and latitude of the stereographic projection. I used the formulars from wolfram alpha: http://...
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3answers
134 views

relationship between a great circle arc and a latitude circle arc at a given latitude

My spherical geometry is a very rusty but looking at the figure below: ... my intuition tells me that angles $\phi$ and $\theta$ (measured in radians) are connected with the following equation: $\...
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3answers
131 views

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 ...
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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 ...
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0answers
66 views

Spherical symmetry vs. integration

$\textbf{Notation:}$ $\bullet$ For $N$ vectors $\textbf{u}_1, \ldots, \textbf{u}_N \in S^{n-1} \subset \mathbb{R}^n$, let $\alpha_{ij}$ ($i < j$) denote the angle between $\textbf{u}_i$ and $\...
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1answer
57 views

Angles and area of a triangle defined on a sphere?

Consider the spherical triangle $\mathcal{P}$ with vertices $P_1 = (1,0,0)$, $P_2 = (0,1,0)$ and $P_3 = (1/\sqrt{3}, 1/\sqrt{3},1/\sqrt{3})$. Find the angles $\phi_1, \phi_2, \phi_3$ of $\mathcal{P}$ ...
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1answer
102 views

Pull Constant from Atan2 Function

I am trying to calculate intersection Latitude value based on given coordinates and intersect Longitude. $\begin {align} b &= \text {is bearing in radians} \\ Lat_1 &= \text {is Latitude 1} \\...
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1answer
212 views

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 ...
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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 ...
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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 ...
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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. ...
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1answer
310 views

How to calculate angle between two directions on sphere

There are four points on sphere with latitude/longitude coordinates: ${a}_1, {a}_2, {b}_1, {b}_2$. How to calculate an angle between two vectors on sphere: $a = \overline{{a}_1{a}_2}, b =\overline{{b}...
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1answer
233 views

Great-circle distance using haversine formula [closed]

I've been trying to calculate the distance between two locations following the haversine formula. I believe the formula is: $\Delta_\mathrm{lon}=\mathrm{lon}_1-\mathrm{lon}_2$ $\Delta_\mathrm{lat}=\...
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62 views

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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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1answer
34 views

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 ...
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1answer
450 views

Angle between two points on a sphere

Suppose I am standing at point $A$ on Earth (a 3D sphere of a known radius) and suppose $B$ is another point a few miles away. The longitude and latitude of points $A$ and $B$ are $(\lambda_1,\...
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2answers
79 views

Spherical trigometry for Sailing Problems

A man travel from Pekin (ΦP = 39◦54′N, ΛP = 116◦23′E) to Singapur (ΦS = 1◦17′N, ΛS = 103◦51′E). Suppose the trajectory is a great circle: a) find the distance in nmi betwen Pekin and Singapur b) ...
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4answers
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Why is $e^{i\pi}= -1$ [closed]

Why does $ e^{i\pi} = -1 $ ? I know that this form can be used to for instance act on a bloch's sphere (quantum mechanics) using it as $ e^{i\pi/4} $ will do a $ \frac{\pi}{4} $ rotation on the $x-...
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1answer
133 views

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 ...
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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 ...
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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 ...
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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 ...
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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° ...