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

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Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where ...
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Area of a sphere bounded by hyperplanes

Say we have a sphere in d-dimensional space, and k hyperplanes (d-1 dimensional) all passing through the origin. Is there a way to calculate (or approximate) the area of the surface of the sphere ...
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Deriving the Surface Area of a Spherical Triangle

A triangle on a sphere is composed of points $A$, $B$ and $C$. The $\alpha$, $\beta$ and $\gamma$ denote the angles at the corresponding points of the triangle: The Girard's theorem states that the ...
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36 views

Plotting a skew ellipse

I tried to make a geometers sketchpad toolkit for spherical geometry that is more exact (and easier to understand) than the present available one. but then I soon realised my background is not good ...
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Computing a double integral over a surface S, where S is the unit sphere,

$$ \int \int_S (x^2+y^2)d\sigma$$ Where S is the unit sphere centered at (0,0,0), and $\sigma$ is surface area. I arrived at the correct answer of $\large \frac{8\pi}{3}$, but I took an (educated?) ...
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Is this algorithm for 3D spherical interpolation correct?

I am attempting to write a spherical interpolation algorithm for for the application of smooth 3D animation in a game. The scripting language that the game engine uses is Lua. It is often easier for ...
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145 views

Find third geographic coordinate in triangle using spherical earth model

I'm trying to solve triangulation problems using geographic coordinates from a GPS. all calculations must use the spherical earth model (great circle distance). Given the points and lengths: Point ...
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42 views

How can I find the center of gravity of a hollow spherical cap?

I am looking to find the center of gravity for a hollow spherical cap. Could I use that point as the point at which the entire mass of the spherical cap is for newtonian gravity problems?
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341 views

How to calculate the area covered by any spherical rectangle?

Is there any analytic or generalized formula to calculate area covered by any rectangle having length $l$ & width $b$ each as a great circle arc on a sphere with a radius $R$? Note: Spherical ...
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46 views

Why is the polar triangle useful in spherical geometry?

We can solve many problems in spherical geometry by using the polar triangle. I am looking for an intuition why (and when) this is easier than working in the original triangle.
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30 views

mean square displacement on the 3-sphere

I would like to compute the mean square displacement (MSD) for a particle moving on the surface of a 3-sphere of radius R. I see that I could eventually use the polar coordinates and get a polar ...
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74 views

Orthogonal transformation of a triangle on $S^2$

Let $v_1, v_2, v_3$ and $w_1, w_2, w_3$ denote the vertices of two spherical triangles $\bigtriangleup_1, \bigtriangleup_2$ with the property, that $\|v_i - v_j \| = \|w_i - w_j \|$, e.g. their ...
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lat/lon spherical coordinates to equidistant spherical coordinates

How to transform spherical data expressed in latitude/longitude pairs (parallels/meridians) in a new set of pair expressed just in parallels pairs? In other words, I need to transform data expressed ...
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Proof for spherical polar law of cosine

I'm reading my textbook and for some reason, it does not present the proof for the spherical polar law of cosine which is: $$ \cos(a)=\frac{\cos(A)+\cos(B)\cos(C)}{\sin(B) \sin(C)}$$ It does present ...
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The congruence of Two Spherical Triangles

I found in Wikipedea following claim : Two Spherical triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent). I know that the Area of Two triangles are ...
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1answer
28 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system (r, θ, φ), θ can range from 0 to 2pi, but φ only varies from 0 to pi. Why is that?
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37 views

Direction of rotation to transform from Point a to b on a unit sphere

If I have two points $a$, $b$, on a unit sphere, I believe I can determine the angle between them, expressed as vectors, as follows: $$\theta = \arccos\left(\frac{a\cdot b}{\|a\| \|b\|}\right)$$ ...
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Stereographic projection and lengths

We know the Stereographic Projection doesn't preserve areas except for the points on the plane such that $x^2+y^2=1$, because that's where $dA=dxdy$. I was wondering what would happen with lengths: ...
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332 views

Spherical Harmonics expansion of a Dirac Delta at the North Pole

I think all the coefficients for the spherical harmonic expansion of a delta function at the north pole should be a constant (presumably 1), but I'm having difficulties calculating them. Could someone ...
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130 views

Instruct geometer moths so you can learn about their true geometry.

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...
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1answer
28 views

constructing a spherical triangle using only the laws of sines and cosines

I have a spherical triangle with corners $A,B,C$, angles $\alpha, \beta, \gamma$ and sides $a,b,c$ (which are opposite to the corresponding corners/angles). I am given $a,b$ (with $a>b$) and ...
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507 views

Navigating though the surface of a hypersphere in a computer game

People in StackOverflow seems not so into this theme, so I thought I could have better luck in here. I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
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163 views

Volume of a sphere “corner”

I would like to find the formula of the volume of the "corner" of a sphere of radius R, more specifically the volume delimited in a sphere by the intersection of two perpendicular planes, one parallel ...
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1answer
56 views

Test to know if a vector is inside a spherical triangle

Given a spherical triangle defined by $3$ unit vectors on a sphere, how can we test if a vector is contained inside the spherical triangle?
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Maximal Triangle on Sphere [closed]

If en equilateral triangle is drawn on the surface of a sphere and expanded till its three vertexes coincide in one point, how many sections result?
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Spherical Geometry and Playfair's Axiom.

Recently I came across a variant of the Parallel Postulate known as Playfair's Axiom: In Euclidean (planar) geometry there is at most one line that can be drawn parallel to another given one ...
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comparing models of spherical geometry

There more than two models of spherical geometry? 1) one for the half sphere (taking the north pole as center, the boundary is the equator) 2) one for the whole sphere (taking the north pole as ...
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Area of a spherical triangle

Consider a spherical triangle with vertices $A, B$ and $C$, respectively. How to determine its area? I know the formula: $A = E R^2$, where $R$ is radius of sphere, and $E$ is the excess ...
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1answer
65 views

Tangent points on circle that placed on Earth surface

I need help about spherical geometry problem that I need to use it for my project. I try to calculate $T_1 $ and $T_2$ coordinates on $B$ centered small circle on the sphere and $AT_1$ and $AT_2$ ...
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1answer
18 views

How to find point on sphere from pitch and heading

I have a sphere of radius R and I would like to draw some vector positions on it given pitch and heading. I have a heading between 0 and 360 (0 being +x direction), and a pitch between -90 and 90 (90 ...
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19 views

how many Poincare dodecahedrons fill Poincare dodecahedral space?

I was reading Jeffrey's Weeks "shape of space" and that made me wonder: Every spherical 3d manifold (3d Sphere) has a finite volume, The Poincare dodecahedral space is a 3d Sphere. this manifold ...
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Finding the northernmost latitude in a great circle that passes through two points on the sphere [duplicate]

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on page 23 of the 6th ed.): $A$ and $B$ are two places in the earth's surface with the same ...
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Finding the northernmost latitude in a great circle that passes through two points on the sphere

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on p.23 of the 6th ed.): $A$ and $B$ are two places on the earth's surface with the same ...
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1answer
256 views

How to calculate a solid angle (in Steradians) given only Horizontal Beam angle and Vertical Beam angle data.

I would like to convert a rectangular beam shape given in Horizontal and Vertical beam angle, into solid angle representing the surface area in steradians of projected light. For example a light ...
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2answers
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Finding the shortest distance to the north of the sphere

I found these problems in Alan F Beardon's Algebra and Geometry: Verify that any point with latitude α is a spherical distance R(π/2−α) from the north pole. Suppose that an aircraft flies on the ...
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Question about trig identities in a celestial sphere problem

So the problem goes: The heavenly body X sets when passing through point F on the horizon. If is the hour angle of X at the time and D the declination, show that : cosH=-tanbtanD cosA=sinDsecb ...
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Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
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73 views

Can the intersection of two balls be described?

Suppose, two spheres intersect. Subtracting the equations of the speheres, a linear equation appears which indicates the plane conataining all points belonging to the intersection of the spheres. But ...
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106 views

Circle from three points on surface of sphere

I need to compute the circle on the surface of a sphere given three points on that very surface. It is very easy to do that in Euclidean geometry, but the sphere has no x and y, but just two angles ...
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Using azimuthal and polar angles in ECEF coordinate system

I have a physical cone, which its vertex located in some point (x,y,z) in ECEF coordinates, and I want to check if another point is inside this cone. In order to do it, I have to take into ...
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Construct the great circle (geodesic) in spherical or Riemanian geometry

Given: a circle $C$ with centre $M$ two points $P_1$ and $P_2$ inside circle $C$, so that $M$ is not on the line $P_1P_2$. Cunstruct an other circle $O$ so that: $P_1$ and $P_2$ are on ...
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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 ...
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1answer
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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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1answer
64 views

Find the intersection point of a great circle arc and latitude line

In spherical geometry, I need to know at what longitude λ a great circle arc φ1,λ1-φ2,λ2 has intersected a line of latitude φ. I have found the equivalent equation for solving latitude φ for an ...
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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 ...
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Why does every direction at the north pole point south?

Why does every direction at the north pole point south? Why doesn't this happen at any other point on (face of the) earth? Is this due to convention used by humans or is there a geometrical ...
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82 views

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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1answer
25 views

Point within a spherical triangle given areas

Consider a spherical triangle like this: where $A_1, A_2, A_3,$ and $P$ are points on the sphere and $t_1, t_2, t_3$ are the proportion of the area of the large triangle contained within the small ...
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Geometry of spherical triangle

Using the formula for the area of a spherical triangle, find and prove a formula relating the angle sum of a spherical polygon to its area Thought: Area (spherical triangle) ...
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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 ...