# Questions tagged [spherical-trigonometry]

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

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $d(x,y)+d(y,z)>= d(x,z)... 0answers 215 views ### For spherical triangle$\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. 0answers 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 ... 0answers 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 ... 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 $$a+b+c=\pi/2$$. 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 ... 0answers 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 ... 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$\...
I am trying to calculate the signed spherical angle between two intersecting great circle arcs. In my specific case, I am considering a unit sphere ($r=1$). The first great circle arc goes from \$A(\...