Questions tagged [spherical-trigonometry]

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

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What is the formula for the orbital velocity of the Earth from x, y, z coordinates of ephemerides at set time intervals? [closed]

For example for step size 10080 minutes x y z v 1721057.5 B.C. 0001-Jan-01 -5.83E-01 7.93E-01 3.65E-03 1721064.5 B.C. 0001-Jan-08 -6.78E-01 7.16E-...
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Conversion from NED to 'flat earth' coordinates.

I have a flat earth problem of a missile that needs to return to launch pad. The solution to this problem (using convex optimization in case you are interested) is then meant to be fed to a simulator ...
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1 vote
1 answer
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Formula to calculate number of sunsets per year based on lattitude

How can I create a formula to turn latitude into number of sunsets per year? Let's keep it relatively simple. Assuming the Earth is a smooth ball, assuming the sun is a single point, not getting ...
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1 vote
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Calculating Diameter of Metric Space Built from Spherical Polygons

$\DeclareMathOperator{diam}{diam}$ Suppose we have a convex spherical polygon $P$ and suppose that we've figured out the two vertices which are the greatest distance apart, say $u$ and $v$. Consider ...
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Spherical right triangles identities

We have a spherical triangles with angles $\alpha,\beta,\gamma$ that have opposite sides of length $a,b,c$. So our triangle is right angled as we have that $\gamma = \frac{\pi}{2}$. I have to prove ...
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2 votes
1 answer
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Worded spherical triangle problem

We consider a sphere with a radius of 4000 metres. We start at point A and travel on a spherical line segment to point B, turn 60$^{\circ}$ to our left then travel on a spherical line segment to point ...
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Spherical triangle problem: find the normal vector of the third edge

The following could be a problem that has already been answered to, I've looked at similar questions but I can't figure out the answer. Consider the sphere $S^2$ embedded in $\mathbb{R}^3$ whose ...
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Shape of largest convex spherical polygon

Are all convex spherical polygons of diameter equal to the diameter of the sphere identical in shape to a circle of the same diameter, or am I thinking about this wrong?
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Rotate a sphere about an arbitrary axis using 3 angles relative to coordinate axes?

I have a globe in 3D Euclidean space with the center of the globe at the origin, only the globe is tilted off axis by $\phi$ degrees so it doesn't rotate around the z-axis anymore, but an arbitrary ...
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Minimum Sight Distance of Incident Headlight on Vertical Curvature

I am working on the problem of calculating the distance at which the vehicle's headlight beam hits the ground, if the vehicle is traveling on a Vertical curvature. However, I am bit confused if using ...
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How can I derive $\cos s = \cos x \cdot \cos y$ from the spherical triangle?

I had read that for a "spherical" triangle i.e. all sides are equal and all angles are 90 degrees if e.g. (I am sorry for the crude diagram, didn't know how to make it better) it is: $\cos ...
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Vector Reflection in Spherical Coordinates Proof

Say I have a vector a = (1, $\theta_1, \phi_1$), and another vector b = (1, $\theta_2, \phi_2$). I want to come up with a formula for reflecting a through b in spherical coordinates which in Cartesian ...
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A point is translated along the surface of a unit sphere. How do I determine the new spherical coordinates of the point?

A point on a unit sphere at ($\theta$ = 0, $\psi$ = $\pi$/2) moves over the surface on great circle by $\alpha$ radians at an angle $\gamma$ from "north" (the north pole being at $\psi$ = 0)....
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11 votes
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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 ...
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How to solve this spherical trigonometry situation with missing information?

Suppose $b$ and $c$ were given constants. If $C = \theta - y$ (where $\theta$ is given) and $a = 90 \deg - y $, is it possible to solve for $y$? It seems like there must be a solution, since $y$ can't ...
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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 ...
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Spherical Pythagorean Theorem

In class, we came across the following relation for a right triangle on the surface of the sphere. $cos(\frac{c}{R})=cos(\frac{a}{R})cos(\frac{b}{R})$ where R is the radius of the sphere. Here a, b, c ...
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Express $\sin(\theta)$ in terms of spherical harmonics

So we can express trigonometric quantities in terms of Spherical harmonics, for example $\cos(\theta)\propto Y^{0}_{1}(\theta,\phi)$. Is there a closed expression for $\sin(\theta)$? If not, is there ...
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2 answers
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Asymptotic distribution of ratio of two coordinates of random unit-vector

Let $n$ be a large positive integer. Let $X=(X_1,X_2,\ldots,X_n)$ be uniform on the unit-sphere in $\mathbb R^n$ and define $R:=X_1/X_2$. Question. What is the distribution of $R$ ? Note. I'm really ...
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How to calculate the effective area of a solar panel

I am working on a project where I need to model the output of solar panels during the day. Because the sun moves across the horizon it won't always shine straight at the panels. For this reason, I ...
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1 vote
1 answer
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Pyramid - Cartesian Space xyz

I have a pyramid (in general with a rectangular base) like the following: with: Angle: $\widehat{AVB} = 30°$ Angle: $\widehat{BVC} = 40°$ Edge $\overline{VO} = 100$. It is in the space $xyz$, with ...
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1 vote
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Is there any spherical rectangle that is closed under intersection/complement?

In Euclidean space, the rectangles are closed under intersection but not closed under negation/complement. I was wondering whether we can define a rectangle in an unit sphere such that the spherical ...
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2 votes
2 answers
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Orthographic Projection of a Plane, Rotated on Two Axes, and Viewed from an Oblique Angle

Context: When installing a solar panel array in a landscape (assume flat terrain, and no obstructions with shade), sometimes the plot of land does not allow an orientation to due south (the optimal in ...
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4 votes
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Spherical trigonometry and statistics in higher dimensions

Until recently I never properly fully assimilated the spherical laws of sines and cosines into my understanding, and thinking about those, I see some parallels with some things in statistics. Segments ...
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2 votes
1 answer
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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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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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3 votes
1 answer
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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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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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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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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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1 answer
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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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  • 151
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1 answer
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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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1 answer
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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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2 answers
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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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3 votes
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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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2 votes
0 answers
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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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2 votes
1 answer
126 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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3 votes
1 answer
120 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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4 votes
0 answers
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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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1 answer
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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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5 votes
2 answers
173 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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1 vote
1 answer
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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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2 answers
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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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1 vote
1 answer
265 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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0 votes
1 answer
613 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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1 vote
2 answers
110 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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2 votes
0 answers
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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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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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2 votes
1 answer
383 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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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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