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

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121 views

Formula for surface measure of spherical cap on $S^n$.

Can you show me an easy to use exact formula, or good lower and upper estimates, for the measure of a spherical cap of height $h$ on the sphere $S^n$?
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2answers
210 views

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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0answers
34 views

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 ...
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1answer
257 views

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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1answer
1k views

Ratio of volume to radius of a sphere

If I have two spheres of a known volume $V$ and radius $r$ and I make one bigger sphere out of them, what will the new sphere's radius be? For example: Sphere $1$: $r_1=2$, $V_1=8$. Sphere $2$: ...
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3answers
446 views

The unsolved mathematical light beam problem

I have the following problem: Imagine that you have a sphere sitting at the interface of two media(like water and oil). And the position(the heigth) of the interface to the center of the sphere is ...
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1answer
145 views

surface area of a slice of a hemisphere

Imagine a hemisphere on it's base in the horizontal plane (center at the origin). Imagine another plane P which passes through the center (origin) and inclined at alpha to the horizontal (the base of ...
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46 views

integral over two spherical Bessel function

I am now having a problem regarding the integral over two spherical Bessel function. If anyone can give any help, it would be so nice of you. Thank you so much for any help. Specifically, I intend to ...
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2answers
120 views

How can I “move through a hypersphere?”

A man walking along a 2 dimensional circle will take a periodic path that begins and ends at the same point. Since he can travel in only a single direction, let's say how far along he is in his ...
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57 views

“Great Circle” distance [duplicate]

Given two points on a sphere, then the "great circle distance" between two points is the length of the smallest arc of a great circle containing both points. Assume that $\Sigma$ is a sphere of radius ...
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1answer
178 views

Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
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2answers
308 views

Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere ...
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60 views

Finding the coordinates of the corners of an aligned pole-centered spherical square

Given a spherical square of radius $1$, with edge midpoints at $(1, x, 0)$, $(1, x, \pi/2$), $(1, x, \pi)$ and $(1, x,3 \pi/2)$ (in the spherical coordinate system of (radial distance, polar angle, ...
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1answer
165 views

What is the maximum entropy distribution of points on a sphere that has a fixed non-zero average cosine of the polar angle?

Suppose we have a unit vector in 3D space whose orientation has some unknown distribution $p(\theta,\phi)$. All we know about this distribution is the average value of $cos(\theta)$: ...
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1answer
139 views

Laplacian on Sphere of Function Only Depending on Angle Between Points

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) ...
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1answer
204 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 ...
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2answers
181 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 ...
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3answers
233 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 ...
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1answer
279 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?
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1answer
197 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 ...
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2answers
1k 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 ...
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2answers
254 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 ...
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2answers
312 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, ...
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1answer
260 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 ...
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4answers
1k views

Intersection of two arcs on sphere

I have two arcs on a sphere that are defined as pair of points: (θ₀, φ₀), (θ₁, φ₁). I need to find a point where they intersect, or some indication if they don't. What is important is that they are ...
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1answer
886 views

What is the metric tensor on the n-sphere (hypersphere)?

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 ...
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2answers
66 views

Ray-Lens Intersection

So imagine that I have a ray parameterized as $\vec{R} = \vec{O} + t\vec{D}$, where $\vec{O}$ = origin, $t$ = parameter and $\vec{D}$ = direction vector. I also have a spherical lens with aperture ...
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973 views

Spherical Trigonometry: Spherical triangle

ABC is an equilateral spherical triangle in which small displacements are made, in the sides and angles, of such a nature that the triangle remains equilateral. Prove that $$ \frac{da}{dA} = ...
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1answer
38 views

Find position on surface of a lens

If I have a lens with coordinates UV on the lens surface where U, V are [-1, 1] and I want to find the real-world (x,y,z) coordinates of the UV point, how would I do that if I have the following ...
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1answer
53 views

Normal intersecting a sphere

Let $\textbf{x}$ and $\textbf{y}$ be two points on the sphere. Show that the normal to the plane determined by the great circle through $\textbf{x}$ and $\textbf{y}$ intersects the sphere at the ...
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1answer
410 views

A different way of calculating the surface area of the sphere

I just don't understand this. I know $A_{sphere} = 4\pi r^2$ and the circumference $C_{sphere} = 2\pi r$, so why can't I just sum up (integrate) all the circumferences to get the area? That is, why ...
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1answer
295 views

Spherical geometry - relating angles of lunes and segments of great circles

Consider the picture below. I have a sphere of radius $r$, centered at $C$. The angle $\varphi$ is the dihedral angle between the plane defined by the shaded area and a plane through the indicated ...
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1answer
123 views

Proving that the only glide reflection which maps more than one line to itself on a sphere is an antipodal map

I know that in Spherical geometry, a rotation is the same as a translation. So a glide reflection is the same as a rotation-reflection or translation-reflection. Also, geodesics in $S^2$, are great ...
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2answers
144 views

Given a latitude how many miles is the corresponding longitude?

OK so lines of longitude (the distance/circumference around the earth horizontally) differ based on what latitude you are at (0 at north and south poles up to ~25k at the equator.) So given a ...
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116 views

plot $(x,y)$ coordinates to a sphere

I have a planar mesh of hexagons (or any other shape) that I want to bend into the shape of a half-sphere. For this purpose I want to loop through each vertice in my mesh and find the proper z ...
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1answer
44 views

Help to find spherical Line

What is the spherical line through the points $(0,-1,0)$ and $\left(0,\frac{1}{2},\frac{\sqrt{3}}{2}\right)$? I solved: $G = \{(x,y,z)\in S^2 \mid \exists\ a,b,c \in \mathbb{R}, ax+by+cz = 0\}$ ...
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2answers
119 views

Find an atlas of charts on $S^2$ with certain properties

Find an atlas of charts on $S^2$ for which each chart preserves area, and the transition functions relating charts have derivatives with determinant 1. I have been thinking that I should consider the ...
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1answer
155 views

Geometry Question - Trihedral angles, planar geometry, spherical geometry

Two rays, $OX$ and $OY$, are drawn in the horizontal plane $\pi$, and the third ray, $OZ$, is drawn in space so that the rays $OX$, $OY$, and $OZ$ form a trihedral angle $OXYZ$. The planar angles ...
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1answer
854 views

Area of a geopolygon or polygon defined by longitude/latitude points

How would you go about finding the real area of a polygon which is defined by latitude/longitude points? Remember that the real area is different as a map is distorted towards the poles.
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377 views

Finding a third coordinate on a sphere that is equidistant from two known coordinates

Here is my problem that I'm having some trouble with: I have the coordinates (latitude and longitude) of two points on Earth. I have no problem finding the great circle distance between the two ...
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1answer
39 views

approximation of law sines from spherical case to planar case

we know for plane triangle cosine rule is $\cos C=\frac{a^+b^2-c^2}{2ab}$ and on spherical triangle is $ \cos C=\frac{\cos c - \cos a \cos b} {\sin a\sin b}$ suppose $a,b,c<\epsilon$ which are ...
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1answer
77 views

The law of cosines for a sphere

$\cos(c) = \cos(a)\cos(b) + \sin(a)\sin(b)\cos(C)$ Prove that if $a$, $b$, and $c$ is approximately $0$, then $c^2 = a^2 + b^2 - 2ab~\cos(C)$. I wasn't sure how to prove this. One thought I had was ...
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1answer
113 views

Law sines in Spherical Triangle $\rightarrow$ Law sines in plane triangle

Could any one tell me how to estimate or get law of sines in Spherical Triangle to The Law of Sines in Plane Triangle? i.e $\frac{\sin a}{\sin A}=\frac{sin b}{\sin B}=\frac{\sin c}{\sin C}$ to ...
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65 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
508 views

compute minimum distance between point and great arc on sphere

Suppose I have a point $P$ on a unit sphere whose spherical coordinates are $(\theta, \varphi)$, and a great arc from point $Q$ to point $R$, also specified in spherical coordinates. I want to find ...
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1answer
245 views

Relation between area of a triangle on a sphere and plane

We know area of a plane triangle $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\frac{a+b+c}{2}$. I was just thinking: let we have a triangle with arc length $a,b,c$ on a sphere of radius $r$, do we have ...
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110 views

to find volume of spherical tetrahedron and triangle area

I have two question though they are different in some way Could any one tell me how to find area of spherical triangle in a easiest way? How to find the volume of Spherical tetrahedron! which is in ...
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1answer
2k views

Spherical Geometry (SG) Vs Euclidean Geometry(EG)

Could any one tell me what are the fundamental contrasts with postulates of Euclidean Geometry and Spherical Geometry? I myself see these things, please tell me if there are more: Lines in EG are ...
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1answer
117 views

problem or doubt regarding visualizing angles of spherical triangle

I must confess that I am not able to visualize or understand what is the angle of a spherical triangle say $ABC$ where $A,B,C$ are vertices of the triangle which is formed by intersection of three ...
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1answer
79 views

How to minimize the length of a curve on $S^2$

The length of a curve $\gamma$ starting from a point $p$ and ending at another point $q$ on $S^2$ is given by the formula $$l_{\gamma}(S^2)=\int_{0}^{1}\sqrt{(d\phi/dt)^2+ \sin^2\phi ...