Questions tagged [spherical-trigonometry]
For geometric questions about solving spherical triangles and spherical polygons on spheres.
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Finding common tangent lines of two circle in spherical surface [closed]
I am able to find common tangent lines of two circles in 2D space ( point is x,y ).
but the problem is can I use the equation to be used in spherical surfaces ( ...
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What is the parametric path of an equalateral spherical triangle in the first octant on the unit sphere with the edges equadistance from each plane?
I want to clarify a bit. The vertices need to be the same distance a path that follows from (1,0,0), (0,1,0), and (0,0,1) and converges at ($\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}...
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Diameter of Voronoi cells of regular simplex in sphere
Let $\mathbb{S}^n\subseteq\mathbb{R}^{n+1}$ be the sphere with geodesic distance, and let $p_1,\dots,p_{n+2}\in\mathbb{S}^n$ be distinct points forming a regular simplex in $\mathbb{R}^{n+2}$. For $i=...
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Spherical Triangle with right-angle calculating an angle with one side and one angle [closed]
Given the following spherical triangle, is it possible to calculate B
given side b, angle A and angle C = 90 degrees?
If so, which formula is it.
I've tried the sine Rule and cosine rule but I need to ...
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Triple integral of the great-circle distance function
Numerical integration suggests that
$$\mathcal U=\int_0^\pi\int_0^\pi\int_0^\pi\arccos\left(\cos x\cdot\cos y+\sin x\cdot\sin y\cdot\cos z\right) dx dy dz\stackrel{\small\color{gray}?}=\frac{\pi^4}2\...
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A cow is tied to the outside of a raised square platform of side 10m, with a rope of 25m. what is the area the cow can graze? [closed]
This is similar to a number of such quesiton, but the overlapping areas are a bit tricky. Ive a attached a diagram which i think is correct
(may not be)
overlapping bits
The file is also available ...
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Solar declination as a function of solar longitude [duplicate]
I'm trying to find the relationship between solar declination and solar longitude. Solar declination is the angle between the line Earth-Sun and the equatorial plane of the Earth. Solar longitude is ...
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Longest distance between spherical segments
In the following, I use the spherical distance in $\mathbb{S}^n$.
Let $a,b,c,d\in\mathbb{S}^n$ and suppose we know the pairwise distances between them. Are there well known formulas for the maximal ...
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with only the 200 meters height and 5º depression angle, how to discover the curve D (distance from the lighthouse to the boat, on earths surface) [closed]
with only the 200 meters lighthouse and 5º depression angle of view, how to discover the curve D, distance from the lighthouse to the boat, on earths surface, having in consideration the earths ...
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I need to find the optimal coordinates for a pyramid apex
The problem I want to solve is to find the coordinates of the apex of a generic square pyramid from the coordinates of the four points that form the base. And the condition that has to be fulfilled is ...
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How do you construct a great circle arc that best fits (in the least-squared distance sense) three or more points on the surface of a sphere?
I am interested in solving a celestial navigation for myself. This reference https://aa.usno.navy.mil/downloads/reports/Kaplan1996b.pdf by George Kaplan describes how the US Navy’s STELLA software ...
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Find coordinate for the third point of a triangle on the unit sphere, given the coordinates of the two points and their angles
Suppose the Cartesian coordinates of the three points are $A,B,C,\|A\|=\|B\|=\|C\|=1$.
Given $A$,$B$ and $b=dist(B,C)$, $c=dist(A,C)$, how to find $C$?
Here, $dist(\cdot,\cdot)$ is the distance on the ...
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Calculating the intersection area for circles on spheres
How would one calculate the intersection area of two circles on the surface of a unit sphere, defined by its direction and angle.
In the pictures there are three possible problems.
One where the ...
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Highest Latitude a Plane Reaches on a Great Circle Path. [closed]
This is a practice question for a math competition that I am lost on how to solve:
Los Angeles is located at (34°N, 118°W), and Osaka, Japan is at (34°N, 136°E). If a plane flies the great circle ...
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How can I locate the poles of a great circle given two non-opposing points? [closed]
Given two non-antipodal points on the surface of a sphere (in -lat,lon or any similar coordinate system) how do I calculate the positions of the poles (one will do, of course) of the great circle that ...
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Computing the angle in interval $[0,2\pi)$ between points on great circle
I'm trying to find a way to compute the angle that subtends the cartesian unit vector in the x direction $(1,0,0)$ and some arbitrary point on a unit sphere with spherical coordinates $(\phi,\theta)$ (...
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Given the Latitude, Longitude and Altitude of Origin and Destination points, Calculate Distance & Bearing (from origin to destination)
Realize that this is similar to many of the questions here. However, I have not been able to find the answer that incorporates the altitude differences.
Given the Latitude, Longitude and Altitude of ...
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Radius of Gyration Calculation of a Sphere
When viewing the radius of gyration of a sphere, and trying to verify my own calculations $R_g^2=\frac25R^2$ against the online material , while the later shows $R_g^2=\frac35R^2$ and does not use the ...
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A great circle on a sphere
The following proposition is from 'Spherical Geometry and Its Applications' by Marshall A. Whittlesey:
Proposition 5.6 If two distinct points on a sphere are not antipodal then there exists a unique ...
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Help with understanding a proof about spherical trigonometry
I was reading the book "Foundations of Hyperbolic manifolds" by J. Ratcliffe, where I found the cosine formula for a geodesic triangle on the sphere. The result is as follows:
If $x, y, z \...
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Spherical transformation from world coordinates
I have a camera pose (position and orientation) in a 'world coordinate' frame described by a (4, 4) transformation matrix.
I want to transform this in a spherical coordinate frame, assuming the ...
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How to calculate error propagation in spherical astronomy formula?
Distance (r) as the angle between two stars are calculated by following formula:
$$
\cos(r) = \sin(\delta_1) \sin(\delta_2) - \cos(\delta_1)\cos(\delta_2)\cos(\alpha_1-\alpha_2)
$$
$\delta$ and $\...
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Does $c^2 \leq a^2 + b^2$ hold for right triangles on the sphere?
I was hoping for a proof of something which appears to be intuitive to me, but which I can't prove.
Let $a, b$ & $c$ be lengths of the sides of a triangle. We know that on a plane, $c^2 = a^2 + b^...
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A spherical version of the generalized half-angle formulas
The following is a generalization of the half-angle formulas presented at Nabla - Applications of Trigonometry.
Generalization. Let $a$, $b$, $c$, $d$ be the sides of a general convex quadrilateral, $...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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