# 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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### Four circles touching one another on a spherical surface

The diagram above shows four identical circles, each having a flat radius $r$ (i.e. flat area $\pi r^2$), touching one another at six different points (i.e. each of four identical circles touches rest ...
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### How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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### Point in a spherical triangle test

Given three latitude/longitude coordinates on a sphere forming a triangle, how do I test if a point p is inside that triangle? I know latitude and longitude implies Earth and Earth is not perfectly ...
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### What is the mathematics behind the two animations?

I found two animated GIFs from a designer's website, which looks very impressive: My questions are: what is the mathematics behind them? How to obtain the mathematical formulas and equations of ...
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### Spherical and Hyperbolic lines in the Extended Complex Plane.

We work in the Extended Complex Plane: $\mathbb{C} \cup (\infty)$. Basically, say we have two points, $z_1$ and $z_2$. It can be shown that, on stereographic projection of the Riemann Sphere onto ...
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### Spherical triangle vertices to spherical coordinates

Problem: Assume that the earth is a sphere of radius 5280 miles, find the length of the sides, the measure of the angles and the area of the spherical triangle with vertices A(70°N,10°E),B(10°S,100°E) ...
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### Finding circle of a sphere through two points

We have two points $P_1, P_2$ on a sphere $S$ of radius $R$. Suppose for $r \ll R$, the distance between $P_1$ and $P_2$ is less than $2r$. Then, $P_1$ and $P_2$ both lie on exactly two radius-$r$ ...
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### Drawing ellipse as google.maps.Polygon with 8 points

In a web page using Google Maps JavaScript API v3 (including Geometry library) I currently draw an ellipse as a "diamond" with 4 corner points by the following JavaScript code: ...
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### Finding the length of the sides, the measure of the angles and area of spherical triangles?

I'm trying to understand this problem in the textbook but I got lost in one part: Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the ...
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### Stereographic projection of a sphere

What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong. This image accompanies the problem: ...
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### Lift of isometries of spherical space forms

If we have an isometry between two spherical space forms, then it is said that it lifts to an isometry of the sphere. Why is that?
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The real spherical harmonics can be written in terms of the complex spherical harmonics: $$Y_{\ell m} = \begin{cases} \displaystyle \sqrt{2} \, (-1)^m \, \operatorname{Im}[{Y_\ell^{|m|}}] & \... 1answer 318 views ### How to cut a sphere in 3 parts of equal volume? I ran across this problem when working on an architecture design project. I know this probably involves integral math but I'm not very familiar with it. Any help would be appreciated. 1answer 150 views ### Reflections On Sphere Surface / Getting Great Circle from two 3D points I'm trying to calculate the reflection of a point across another point, both of which are on the surface of a sphere. I believe I could do this by getting the formula for the great circle of the ... 3answers 501 views ### Equal-area sparse spherical shell partitioning I'm trying to solve a particular problem that arose in a computer graphics context, but can be generalised to a bigger problem as well. I'm not entirely sure if this question belongs to MathExchange ... 0answers 113 views ### How many spherical caps of height h and base circle radius a can cover a sphere of radius R? Question How many spherical caps of height h and base circle radius a can cover a sphere \mathbb S  of radius R \quad (R \gg a)? What I have thought so far Since the area of the ... 1answer 89 views ### Surface area of a 2-sphere in Abstract Index Notation I believe the following completely specify a 2-sphere of radius 1 in AIN:$$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$It is easy enough to determine the area by ... 1answer 331 views ### Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic):$$ \frac{l(a)}{\sin\alpha}= \frac{l(b)}{\sin\beta}= \frac{l(c)}{\...
I want to construct two quadrilaterals on the unit sphere with same interior angles $\alpha_1,…,\alpha_4$ and the same perimeter, but which are not congruent to each other. Is that possible? How can ...