Questions tagged [spherical-trigonometry]

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

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2
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1answer
27 views

How to find the coordinates of a point on a great circle defined by two known points that is a given distance away.

Background: I have three known points: A, B, and C—each with known spherical coordinates, $\rho$ and $\lambda$ on the surface of a sphere of unit radius that form a spherical triangle. I want to find ...
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32 views

Derivation of General form of Map Projections

In this paper, the following formulae are derived for the cylindrical and conic projections: Cylindrical: $T(\phi, \lambda)= (\lambda, h(\phi))$ for some function $h$. Examples include: Equi-...
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1answer
43 views

How do I account for the tilting of Earth's axis in right ascension and declination in a model solar system?

I am building a simplified model solar system in GeoGebra. Celestial objects are placed in a heliocentric coordinate system with the sun at the origin, the x-y-plane as the ecliptic, and the x-axis ...
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16 views

Satellite Look Angles

Can anyone point me to a good textbook resource with strong derivations for calculating look angles from a satellite sensor or antenna to a point on the ground? I'm looking for something detailed ...
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1answer
50 views

Reference request on the relationship between inscribed polytopes and shadows of their facets

Two chords of a circle of unit radius have equal lengths if their corresponding arcs have equal lengths. Suppose a polytope is the convex hull of finitely many points on the unit $(n-1)$-sphere in $\...
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32 views

Explanation of the formula for horizontal sundials

I am researching sundials and the maths behind them. For horizontal sundials (ones that stand perpendicular to the equator) there is a formula to compute the hour angles (the angle between the shadow ...
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1answer
21 views

How to calculate the solid angle spanned by 3 vectors? [duplicate]

I find an equation from wiki https://en.wikipedia.org/wiki/Solid_angle#cite_note-6, $$\tan{\frac{\Omega}{2}}=\frac{|\vec{a}\cdot(\vec{b}\times\vec{c})|}{|a||b||c|+(\vec{a}\cdot\vec{b})|c|+(\vec{a}\...
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1answer
63 views

Third vertex of equilateral SPHERICAL triangle

I’m trying to solve Fermat’s problem on sphere for the given triangle ABC using wolfram.I already made out,that in order to find a Fermat’s point i need to build three equilateral triangles on each ...
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1answer
36 views

Spherical Trigonometry: RA-Dec to Alt/AZ

While this question is related to astronomy, I believe it's a spherical trigonometry problem as outlined below. Below are two different sets of equations for computing the Altitude and Azimuth of a ...
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1answer
23 views

Solid angle subtended by polar cap

Solid angle subtended by polar cap at unit sphere center latitude $\phi$ is $$ 2 \pi (1- \sin \phi_c)$$ What is the solid angle it subtends at other unsymmetric points inside the sphere like ...
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66 views

Unusual proof for the spherical cosine rule

We have $0\lt a\leq b\leq c\lt \frac{\pi}{2}$, and $A$=($0$, $0$, $1$), $B$=($\sin c$, $0$, $\cos c$) on the unit sphere. Also $$C_\phi=(\sin b\cos \phi, \sin b\sin \phi,\cos b)$$ is given such that $\...
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1answer
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Geometry(length of an angle bisector) [closed]

The length of two sides of a triangle are $b$ and $c$. Let $s$ be the length of the angle bisector of the angle between the two given sides. The length of the third side of the triangle is:
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9 views

Sum two cartesian (from a sphere) coordinates that store orientations without Quaternions

Besides they store a spherical point they are not spherical coordinates just a Vector3 that sets a point in the sphere surface of radius 1 with the origin at 0 0 0. This is the common way to deal with ...
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13 views

Simple proof of spherical triangle inequality

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 ...
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1answer
54 views

Proof that a certain expression is the Dirac delta in spherical coordinates

My professor during the lesson said that the following expression is the 3D delta function written in spherical coordinates: $$\sum_{l=0}^{\infty}\sum_{m=-l}^l \frac{\delta(p'-p)}{p^2}Y_{lm}(\alpha',\...
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1answer
110 views

Analytic formula for integral $I_p(\theta) := \int_0^{2\pi}\cos(t)^p\cos(t-\theta)^pdt$

Let $p$ be a nonnegative integer and $\theta \in [0, \pi]$. Question. What is an analytic formula for the integral $I_p(\theta) := \int_0^{2\pi}\cos(t)^p\cos(t-\theta)^pdt$ ? Note. My ultimate goal ...
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What happens with the convex hull of $6$ random points on a sphere?

Given a collection of points on the sphere, we can consider their spherical convex hull: add all points on the shortest path between two points in the set, repeat until the resulting set does not ...
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1answer
17 views

Formulas to find the end point of an arc given the start point and midpoint on the unit sphere?

So the radius and the center of the arc are $1$ and $(0, 0, 0)$ respectively because the points and the rac is on the unit sphere $S^2$. Given the start and the midpoint of the arc, is there any ...
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2answers
123 views

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 ...
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1answer
24 views

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 $\vec{U} = [u_x,u_y,u_z]$; $...
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2answers
93 views

How do I find the x and y position given a latitude and longitude coordinate?

I am building an app that allows me to track the user's position inside a building. For this I use GPS and an image of the floor plan. I have the latitude and longitude of each of the four corners and ...
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47 views

Calculate the rotation angle for a point on a small circle (tilted from XY Plane) to a known point on XY Plane

There are two great circles which are unit circles: Great Circle 1 (GC1): On the X-Y Plane Great Circle 2 (GC2): Tilted 45 degrees about the Y axis Here are some references and constraints: 3D ...
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20 views

How do you map a state change in cartesian coordinate to polar/spherical coordinates? (Research)

Describing a line segment using polar coordinates I am stuck on equation (19). Basically in the question, a fibre or a rod $(x,y,z)$ or $(r,\theta,\phi)$ changes state by delta (strain) to $(x',y',z')$...
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1answer
172 views

Volume of a pyramid with a spherical base

Given the attached figure, we are interested in finding the volume of this red pyramid ($ABCV$) with the spherical base. We assume that the Cartesian coordinates of the points $A, B, C, V$ are known ...
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1answer
117 views

Calculating Point on a Sphere

Given a sphere, I am attempting to calculate the coordinates of a point on that sphere given some initial point, the radius of the sphere, and some arbitrary distance d the point will travel (along ...
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2answers
66 views

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 ...
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51 views

Pursuit curve on a sphere

One day I thought about chasing curve on a sphere. I wanted to find parametric equations such as θ(t) and ϕ(t) of running dog always towards a rushing cat on a big spherical world. Orange point is ...
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9 views

Finding the minimum time for one object to intercept another object on an unit sphere

I am trying to find the minimum time, $t$, for one object A, at point $A$, to intercept another object C, at point $C$ on a unit sphere. The point of interception is $B$. A has a constant velocity of ...
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69 views

direct conversion from az/el to ecliptic coordinates

Background: I'm trying to build a lightweight antenna tracker with two servos. For mechanical reasons, I'm first mounting servo 1 on a base so that it tilts forward/backwards, then mount servo 2 on it ...
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1answer
239 views

How to calculate the arc lengths of a great circle inclined with the equator at $\phi°$ broken into $12$ arcs by longitudes $30°$ apart?

A great circle lies at $\phi°$ inclination to the equator. Longitudes $30°$ apart are drawn which divides the equator in $12$ equal arcs of size (radius of earth$*30$). The corresponding arcs on the ...
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30 views

Is there a faster way to find distances on a sphere than Haversine?

My task is to find the distance between the arbitrary point and many others on the sphere. Unfortunately, Haversine, even in vectorized form, does not satisfy in terms of speed, and I would like to ...
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1answer
111 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 ...
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1answer
60 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-...
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1answer
41 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 ...
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1answer
68 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 ...
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1answer
25 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 ...
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1answer
32 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 $\...
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24 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) ...
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1answer
22 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 ...
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1answer
57 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 ...
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1answer
209 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 $\...
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1answer
187 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)...
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215 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.
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459 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 ...
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43 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 ...
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1answer
52 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 $$a+b+c=\pi/2$$.
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1answer
57 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 ...
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19 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 ...
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1answer
43 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 $\...
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1answer
110 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(\...

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