Questions tagged [spherical-trigonometry]

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

Filter by
Sorted by
Tagged with
-4
votes
0answers
15 views

“Diameter” of a sphere [closed]

A sphere is on a table. At a height of 5cm from the table's surface is the sphere center, i.e. the sphere diameter is 10cm. What is it "diameter" at a height of, say, 4cm from the table's ...
2
votes
1answer
42 views

Determining the lune angles of a spherical quadrilateral

Suppose that we have a convex spherical quadrilateral, and we know its internal angles $\alpha,\beta,\gamma$ and $\delta$. Pairs of opposite sides of the quadrilateral are pieces of distinct great ...
0
votes
1answer
21 views

Dihedral in a regular spherical polygon

Planes rotate around a central symmetry axis pass through a sphere centre to intersect on the sphere forming a regular spherical polygon of $n$ sides. A small circle forms as base of cone semi-...
0
votes
1answer
36 views

Manipulating the product of the dot product of multiple vectors is producing a paradox

Unless otherwise indicated all defined vectors lie on the surface of the unit sphere (i.e their norm is 1). Let's take the following expression: $(V\cdot Z_1)(V\cdot Z_2) = V^TZ_1V^TZ_2$ Given that ...
0
votes
0answers
20 views

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 ...
0
votes
1answer
24 views

Calculus I - Ballon Variation when perforated by a needle

A spherical balance is pierced by a needle so that its volume comes to decrease a rate of $50\, \mathrm{cm}^3 / \mathrm{min}$. Determine an index of variation of your surface area on the moment when ...
1
vote
1answer
28 views

Fun thing to think about: Parametrize a sphere from the outside

Suppose I have a sphere of radius R centered at $(x_0,0,0)$ where $x_0 > R$. I would like to parametrize the "face" of the sphere closest to the origin, using the angles $\theta, \phi$, where $\...
0
votes
0answers
18 views

Reconstruct a sphere from 6 patches

Let's say that I need to reconstruct a surface that could be cloned 6 times to create a perfect sphere. It is sampled in some finite N elements per side so the patch will have NxN elements (vertex) ...
0
votes
0answers
17 views

Spherical geometry (optimizing functioin)

I'm currently working on the following problem: Given points $H$ and ${X}_{1},\dots,{X}_{n}$ on a sphere and let ${d}_{i}$ denote the spherical distance between $H$ and ${X}_{i}$. Further let $f$ be ...
0
votes
0answers
16 views

Find the maximum spherical convex polyhedron area given great circles

Consider a unit sphere centered at 0 and $n$ hyper-planes containing the point 0. The intersection between the sphere and the hyper-planes are great circles. These great circles partition the surface ...
1
vote
1answer
14 views

Limiting Case of Distance Between Points on a Sphere

So, the formula for the angular separation between two points on a sphere is $$\cos d=\sin\delta_1\sin\delta_2+\cos\delta_1\cos\delta_2\cos(\alpha_2-\alpha_1)$$ where $\delta$ denotes the latitude of ...
0
votes
1answer
30 views

Average angle between a point on a sphere and the line of sight

I want to know what the average angle is for a random point on a sphere with respect to our line of sight. I simulated it by doing the following (Image link): Randomly sample points on a surface of a ...
0
votes
1answer
105 views

Formula for the Volume of a Spherical Triangle given the Solid Angle

I'm working on a problem for which I have to calculate the volume of a spherical triangle given the Solid Angle Ω. Specifically, if the Solid Angle Ω (in steradians) span by 3 vectors $\...
1
vote
1answer
60 views

Haversine Formula prove inequality triangle

I've been trying that distance using haversine formula is metric space. I can prove first and second conditon, but i have problem with prove that for haversine is true that $ d(x,y)+d(y,z)>= d(x,z)...
1
vote
0answers
100 views

For spherical triangle $\triangle ABC$, show that $\pi<A+B+C<3 \pi$.

For spherical triangle $\triangle ABC$, show that $\pi<A+B+C<3 \pi$. I know that sum of the spherical triangle's sides are less than $2\pi$ but cannot get a start to above problem.
0
votes
0answers
19 views

Best books/lectures for learning Origami polyhedral folding trigonometry.

I would like to understand the trigonometry behind how these equations were derived from the presented folding model of Muira ori origami pattern. My 3D trigonometry is not as good as it can be.
1
vote
0answers
139 views

Determine the compound angle, vertical angle, horizontal angle after rotating a bend pipe.

I am a pipeliner and press and use a lot of carbon steel bends to accommodate the pipe line construction route. I face always a challenge when designing a single compound bend to compensate separate ...
0
votes
0answers
30 views

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 ...
0
votes
1answer
44 views

Question on Spherical Trigonometry [closed]

In a spherical triangle if $$\angle A=\pi/5, \angle B=\pi/3 ,and \angle C =\pi/2$$ , then prove that the sum of length of its sides is equal to $\pi/2$, i.e $$a+b+c=\pi/2$$.
0
votes
1answer
40 views

Locus of all points

What is the locus of all points equidistant from a fixed point and a fixed circle on a sphere? (By examining an "extreme" case, i.e. the fixed point being the North Pole and the fixed circle being a ...
0
votes
0answers
18 views

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 ...
0
votes
1answer
39 views

Solvable equation for angle between points on a sphere?

I'm looking for an equation that would give the coordinates (latitude $\delta$,longitude $\phi$) (on a sphere) for all the points on the surface of the sphere that have a certain angular distance $\...
1
vote
1answer
41 views

Signed spherical angle between two great circle arcs

I am trying to calculate the signed spherical angle between two intersecting great circle arcs. In my specific case, I am considering a unit sphere ($r=1$). The first great circle arc goes from $A(\...
-1
votes
1answer
22 views

Verifying Formulas of angle of a point form the origin (For All Quadrants)

i want to verify that is my formulas are correct or not for finding angle of a point form the origin. 1st => (Atan(y/x) * (180/PI)) 2nd => 180 - (Atan(y/-x) * (180/PI)) 3rd => 180 + (Atan(-y/-x)...
0
votes
0answers
50 views

Maximum number of degrees in Spherical triangles.

The maximum degrees in a sphere is said to be 540°. What assumptions are being made to arrive at that number? Looking at $L^p$ spaces, $\pi$ is defined as circumference over diameter. As the number of ...
0
votes
0answers
41 views

Great circle lenght: Appications to navigation

I am working on a problem about navigation assuming a perfect sphere and hence ignore any spheroidal effects. I would like to be able to describe the shortest trajectory between two points and be able ...
1
vote
1answer
59 views

Curved path on a sphere

I'm writing a code that constructs a road network a sphere. Building the network is done by: Roads are added using spherical coordinates. A straight road is defined by start and end coordinates, and ...
0
votes
2answers
122 views

Prove that, if 2 angles of a spherical triangle are equal, then the triangle is an isosceles spherical triangle

So the question goes: "An Isosceles Spherical Triangle is a triangle that has 2 sides of equal length. Prove that, if 2 angles of a spherical triangle are equal, then the triangle is an isosceles ...
4
votes
1answer
79 views

Need help to complete/correct a proof of the spherical law of sines

Assume we are working on a $2$-sphere of radius $1$. Suppose we have a triangle with vertices $A, B, C$ and sides $a, b, c$ opposite to the respective angles. My starting point is the spherical law ...
3
votes
0answers
42 views

Surface Area Contained Between Four Points on $S^3$

For two vectors $v_1, v_2\in S^1$, the (1 dimensional) surface area contained between these two vectors is simply $cos^{-1}(v_1\cdot v_2)$. Similarly, three vectors $v_1, v_2, v_3\in S^2$ will define ...
0
votes
0answers
57 views

Slope of the spherical cap

I have a full spherical cap (lower part), with a known R and h. It is located on three needles at points A, B and C, which form an equilateral triangle and have the same coordinates Z. Point A and B ...
1
vote
0answers
48 views

Finding rotational axis of sphere from 2 points and measurements

I feel relatively confident this type of question has been answered, but I'm a bit out of touch - I've taken a university-level introductory linear alegebra course, so I understand a little bit about ...
-1
votes
2answers
55 views

How to find the arriving angles $\alpha_b , \beta_b$ ? If we know the values of two sides b,c and angle between them $\alpha_a$ , $\beta_a$.

How to find the (arriving angles) $\alpha_b , \beta_b$ ? If we know the values of two sides of triangle $b$, $c$ and angles between them $\alpha_a , \beta_a$ . The angles $\alpha$ and $\beta$ are ...
0
votes
1answer
39 views

How do we find the distance of a point on oblate (spheroid) from its two foci ? Can anyone give example?

How do we find the distance of a point on oblate (spheroid) from its two foci ? Does it also equals to 2a like in ellipse or not? Can anyone give the example? (Given a>b=c). In 3D ellipsoid 'a' ...
0
votes
1answer
92 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 ...
2
votes
1answer
101 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 ...
0
votes
1answer
45 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 ...
0
votes
4answers
209 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 ...
44
votes
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 ...
3
votes
2answers
127 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 ...
0
votes
1answer
44 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 ...
0
votes
1answer
89 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 ...
0
votes
1answer
60 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 ...
1
vote
0answers
66 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(...
1
vote
1answer
41 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 ...
-1
votes
1answer
77 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] ...
0
votes
1answer
159 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 ...
0
votes
0answers
43 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 $(\...
0
votes
1answer
54 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 ...
0
votes
1answer
64 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 ...

1
2 3 4 5