# 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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### 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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### 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 ...
136 views

### 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 ...
504 views

### 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 ...
217 views

### 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 ...
287 views

### 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 ...
1 vote
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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 ...
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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### 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 ...
106 views

### 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 ...
1 vote
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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 ...
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 \$\...