# Tagged Questions

geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect

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### Curving points to a sphere

I think math.stackexchange is the right place to post this, but if not, feel free to tell me. I have a series of points to be plotted on a sphere (Each one has a latitude and longitude value). These ...
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### How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
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### Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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### Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
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### can a great circle route be predicted from initial condition?

Assuming the world is a sphere with no wind, can the great circle route of a vessel be predicted from the current position $\{\phi_i,\lambda_i\}$ and the current true course $\theta_i$? Presently, I'...
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### spherical geometry

A mobile, on the surface of the earth, is at a point A. Travels 200 km south arriving at a point B. Later moves 200 km west arriving at a point C. Finally moves over 200 kilometers to the north, back ...
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### General Formula for Volume of Spherical Triangle

I'm working on a problem for which I'm trying to divide a sphere into layers defined by integer radius values ($r\in{1, 2, ...}$) such that the segments in each layer all have the same volume. Doing ...
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### Map Earth surface so straight line distance is great circle distance?

Is there a function $f:$ (latitude, longitude) $\longrightarrow \mathbb{R}^n$ (for any finite $n$) such that the linear distance between $f(x)$ and $f(y)$ is the great circle distance between $x$ ...
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### Make a balloon with least total length of seams

I'd like to build a small hot-air balloon out of flame-retardant plastic sheeting to suspend a camera. The plastic sheeting (plastic film) is commonly sold in a long roll in a width of 20ft (6 meters)....
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### Ratio of Circumference to Diameter on a sphere

I was listening to an audiobook of Einstein when they started discussing spherical geometry and how Pi was no longer the ratio of a circle's circumference to its diameter, so I set out to find the ...
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Consider a function $f:S^2 \to \mathbb{R}$ , with $S^2$ the unit $2$-sphere in $\mathbb{R}^3$. Let's say that $f$ depends only on the polar angle $\theta$ from the north pole (e.g., $f(r,\theta,\phi) =... 1answer 252 views ### Convert spherical coordinates to Cartesian coordinates for a vector So let's say I have a normalized vector$N$given in cartesian coordinates and I have another normalized vector$V$, defined in spherical coordinates relative to the vector$N$. So$\theta_V$is the ... 2answers 284 views ### Gauss Theorem example Verify the Gauss theorem for the vector field$F(x)=\frac{x }{\|x\|},$where$x \in W \subset \mathbb{R}^3$and$$W=\left\{(x,y,z) \in \mathbb{R}^3 \left/ a^2\right.\leqslant x^2 + y^2 + z^2 \leqslant ... 3answers 449 views ### How to understand and create quaternions? I have to multiply two quaternions to calculate a so called spherical linear interpolation between two$R^3$coordinate systems within the interval$t = [0, 1]$. I understand how to do the ... 1answer 405 views ### Spherical coordinates of a unit vector around a normal$N$So if I have a unit normal for a surface$N(x,y,z)$and an incident unit vector$V(x,y,z)$to that surface, how would I represent the vector V in spherical coordinates relative to the normal? 1answer 365 views ### Directions in spherical coordinates Say I have a system with standard spherical coordinates. There's a man on that sphere and he's standing on the equator facing east. He chooses a random angle$0°-360°$and turns that much in the clock ... 2answers 3k views ### Determine depth of a partially filled hemisphere Recently came across a question in a Year 9 math book of which there was no "working out" supplied and offers now description on how they obtained the answer. The question goes like this: A bowl ... 2answers 445 views ### Vector Picking on the Unit Sphere Imagine a vector from the center of a unit sphere to its surface: Now imagine a second vector generated in indentical fashion. Given the first vector, how can I generate vectors to uniformally ... 2answers 757 views ### “Center” of a spherical triangle I have a very deficient background in geometry, so I come across questions like these and I'm not sure how to verify my intuition. Consider three points in$\mathbb{R}^3$, given by position vectors, ... 1answer 393 views ### Spherical coordinate system I can easily write$z$axis value is$r\cos\theta$but what will be for$x$and$y$axis, explain a bit please. From the above how can I write the area element as$d\vec{a} = r^2\sin\theta d\theta d\...
I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and \$...