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

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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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2answers
27 views

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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308 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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1answer
121 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
23 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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492 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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2answers
139 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
46 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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1answer
21 views

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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20 views

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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3answers
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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
48 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
11 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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18 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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16 views

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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2answers
46 views

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
92 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
35 views

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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20 views

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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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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72 views

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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1answer
65 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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1answer
80 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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22 views

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
34 views

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
50 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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2answers
68 views

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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2answers
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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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1answer
54 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
21 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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3answers
601 views

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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0answers
34 views

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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1answer
32 views

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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3answers
270 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 ...
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1answer
66 views

Area of Spherical Polygon

It appears to me that after repeated applications of Girard's theorem on the area of spherical triangles that we can obtain the surface area of a spherical polygon with interior angles ...
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4answers
216 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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On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
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Parametrization of sphere including constant inclination $(\theta, i)$ geodesics

Find parametrization of sphere with respect to $\theta$ = constant meridians and i = constant inclination geodesic circles passing through N-S axis and E-W axis respectively. The Earth does not rotate ...
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1answer
36 views

Stereographic Projection from an Arbitrary Point

Let $p \in \mathbb{S}^{n}$, then the stereogaphic projection is a diffeomorpshim $h:\mathbb{S}^{n} \setminus \{p\} \to \mathbb{R}^{n-1}$. Suppose that $p$ is the 'north pole' ($p = (0,0,..,1)$), then ...
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find the equation of a sphere with endpoints A and B where B is the point of tangency of the sphere and the plane

Find the equation of a sphere with a diameter that has endpoints $A(1, 8, −2)$ and $B$, where $B$ is the point of tangency of the sphere with the plane $−9x +6y + 2z = 2$. Now i know that i can get ...
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2answers
66 views

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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1answer
29 views

The locus of points forming a right angle, in nonzero curvature

Given a line segment $AB$ in the Euclidean plane, the locus of points which form a right angle with $A$ and $B$ is known to be a circle, with $AB$ as a diameter. Is this also true for a geodesic ...
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1answer
44 views

Given angles and area, how to find sides of a spherical triangle?

So, given angles and area, how to find the sides of a spherical triangle? I only know that the angles uniquely determine the sides, but what is the relation?
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Nearest point on Spherical Cap

Let $A \subset \mathbb{S}^n$ be a spherical cap. More specifically, there exists a point $v \in \mathbb{S}^n$ and $\epsilon > 0$ such that $A = \{u \in \mathbb{S}^{n}\mid v\cdot u \geq \epsilon\}$. ...
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Solve trigonometric equations with sin and cos

I have two equations for two unknowns $u_k$ and $v_k$: $\tan{u_1} \tan{u_k} + \cos{(v_1 - v_k)} = 0$ $\tan{u_2} \tan{u_k} + \cos{(v_2 - v_k)} = 0$ where $u_1$, $v_1$, $u_2$ and $v_2$ are ...
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0answers
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Analytical solution for volume when a plane cuts a hemi-sphere

I need to find the analytical solution when the plane $ P: z = grad\cdot y + z_{cut} $ cuts the hemi-sphere $ S: x^2 + y^2 + z^2 = r^2;\:y \leq 0 $. I constructed two 3D images in MatLab of the ...
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distance in n-dimensional space

According to answer of this question : Distance between 2 points in 3D space (in spherical polar coordinates) The distance between 2 points in 3 dimensional space is : $$ ...