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

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1answer
200 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}= ...
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2answers
62 views

Non congruent (spherical) quadrilateral with same angles

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 ...
2
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1answer
105 views

Computing distance on a sphere

Let's say I want to compute the distance between two far points on Earth, say Toronto and Brazil. I can do this by getting in my car, setting my odometer to zero and then driving to Brazil. For me, ...
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1answer
53 views

Does there exist a spherical quadrilateral with all angles pi/2?

Does there exist a spherical quadrilateral with all angles pi/2? I do not think so but I am not sure. I am unable to really visualize this. Please offer suggestions. Thank you.
3
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1answer
70 views

Formula for Determinant of Vectors given in spherical coordinates

In 2D, one has an easy formula for the determinant of two vectors given in spherical coordinates, i.e. $\begin{vmatrix} \cos(\phi_1) &\cos(\phi_2)\\ \sin(\phi_1) &\sin(\phi_2)\end{vmatrix} ...
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1answer
27 views

Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter [duplicate]

Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter. I know that diameter is equal to the twice of radius. How can you possibly solve ...
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0answers
39 views

Find the length of any great circle on $\mathbb{S}^2$

I am speculating that the length is $2\pi$ because the circumference of a unit circle in $\mathbb{R}^2$ is $2\pi$. From my understanding, a great circle in $\mathbb{S}^2$ would be a circle centered ...
1
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2answers
153 views

Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
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0answers
55 views

Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
0
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1answer
49 views

Do the $2^n$ hyper-octants of a $n$-sphere always have a $n$-dimensional right angle? Is $\pi/2$ only fundamental in $2$ and $3$ dimensions?

In $2$ dimensions, a $2$-sphere can be divided into $2^2 = 4$ congruent pieces, the $4$ quadrants, each of angle $\pi/2$ radians. In $3$ dimensions, a $3$-sphere can be divided into $2^3 = 8$ ...
1
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1answer
35 views

Line-of-Sight Angle on Sphere

I'm trying to calculate the angle (in degrees) between two latitude/longitude pairs, but with a twist. Most calculations I see use the Great Circle / bearing method, but this does not seem correct ...
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0answers
38 views

Loxodrome : found an error on wolfram MathWorld web site?

Could it be?... We find this claim on Wolfram MathWorld site http://mathworld.wolfram.com/SphericalSpiral.html The claim is that this curve (given in oblate spheroidal coordinates in the limit where ...
0
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1answer
91 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
0
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0answers
7 views

Question on setting up Gosset for a spherical caps search

I just obtained Neil Sloane's Gosset program which does spherical designs, but I am having trouble setting it up to look for n optimal spherical caps on the unit sphere S2. Has anyone here used Gosset ...
2
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1answer
97 views

Great circles on a unit sphere

Any hints as to how I can find the equation of all great circles passing through a given point (polar angle $\theta$, azimuthal angle $\phi$) on the surface of a unit sphere? Thanks.
3
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2answers
108 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 ...
0
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1answer
35 views

Is there any special name for a $n$-torus made by products of hyperspheres?

I was wondering if there exist an accepted name for an $n$-torus made by the product of hyperspheres $\mathbb{S}^d$, that is for the following set: $$ ...
0
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0answers
24 views

Rotation invariant method to compare points on spherical surface

Is there any rotation invariant method that I can use to compare the similarity between the three groups of "A" points as shown below?
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0answers
54 views

Looking for algorithm for spherical point in polygon that works across meridian and anti-meridian

I need to process millions of latitude/longitude points every day to see if they are located within a defined lat/lon bounded polygon. The polygon may be rectangular, or it may be some irregular 3.. ...
2
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0answers
127 views

Finding the leftmost, rightmost, top, and bottom, points, on a surface, of a sphere.

So I'm making a 3D game, and the player is inside a glass sphere. I'm projecting a bunch of points onto the sphere, and I need to find the leftmost, rightmost, topmost, and bottommost points, so I can ...
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0answers
28 views

Spherical geometry/trigonometry: lat/lon of intersection between line of sight from a given lat/lon and altitude above ground

Originally posted in GIS, but not sure if it belongs there. Given a starting latitude, longitude and altitude, and a line of sight defined by azimuth and elevation, I want to find the latitude and ...
0
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1answer
113 views

Determine if one point lies between two other points on a sphere

My question is rather simple. Can I use the dot product to determine if a coordinate lies between two others? With coordinates I mean a Point P(latitude, longitude) on the surface of the sphere. I ...
6
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2answers
101 views

Spherical geometry as an example of non euclidean geometry

I have recently been learning some hyperbolic geometry and the professor briefly mentioned spherical geometry. From a modern, naive point of view, it seems quite easy to show that spherical geometry ...
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0answers
73 views

A point minimizing total great circle distance to a given set of points on a hemisphere

If you have a set of points on a hemisphere, how do you find a point on that hemisphere that has the minimum total great circle distance to the points in the set.
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0answers
74 views

intersection of a line and plane on a 3-sphere

Suppose I have two 4D points, $\mathbf{a}=(a_1,a_2,a_3,a_4)$ and $\mathbf{b}=(b_1,b_2,b_3,b_4)$, that both lie on a unit 3-sphere (i.e. unit distance from origin). In addition, I have a 2-D plane that ...
0
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1answer
63 views

Granted I have NE and SW coordinates for a rectangle, how do I get the center point?

I've got the NE and SW coordinates/points for a minimum bounding rectangle. How do I calculate the center point of this rectangle? At first thought, I could calculate this using simple division. ...
0
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1answer
104 views

Converting quaternions to spherical angles

Consider a situation where a beam is shot at a cube C from an arbitrary position P. The cube detects the angle of incidence relative to its $ x $ axis. The cube can be rotated and moved, and the ...
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5answers
186 views

How to work out miles between Longitude values based on a Latitude value.

We know that when Latitude is 0, the distance between Longitude values is roughly 69 miles. When the Latitude is +/-90, Longitude values are 0 miles. At 0 Latitude, the earths circumference is ...
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0answers
93 views

list of points of a circle on a sphere, given a coordinate and radii

Here's the setup: a sphere with radius R. Now choose a single point or coordinate, C, on the sphere (in latitude and longitude). Now choose a smaller radius, S, and draw a circle around the point C. ...
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0answers
20 views

Trigonometric rule on a spherical square

Consider a square on a sphere (in three dimensions), with edges of length $a$ and angles $\beta$. I want to prove the following formula: $$ \cos(a) = \cot^2(a) = \frac{1 + \cos(\beta)}{1 - ...
1
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1answer
54 views

Non-Euclidean Geometry: Objects on which every line is a closed curve, e.g. a sphere

For any point $P$ on a sphere $S$, every line (geodesic?) containing $P$ is closed, i.e. wraps around $S$ and passes through $P$ "again." 1) Are there other objects besides spheres for which this ...
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0answers
78 views

Is it possible to calculate azimuth based on known elevation and central angle?

based on the wiki page - http://en.wikipedia.org/wiki/Great-circle_distance#Formulas it's possible to calculate central angle from known azimuth and elevation of two points (on unit sphere). But is it ...
0
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1answer
34 views

Area of triangle on a sphere (not spherical triangle)

How do I find the area of a triangle on a sphere, and the triangle is not a spherical triangle, for example, the triangle is formed with two geodesics and a line of latitude. Is there a specific ...
1
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1answer
124 views

Geodesic on n-dimensional sphere

I have a flow on n-dimensional sphere which has a stabilizing action. The tangential velocity will not be a constant, it will indeed decrease to zero as the desired point is reached. First the ...
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2answers
48 views

n-spherical geometry

I'm interested in spherical geometry on the n-sphere. Surely this has been done, but I can't find anything online. Where? No luck with n-spherical geometry, hyperspherical geometry, or higher ...
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0answers
94 views

Area covered by multiple (possibly intersecting) circles on surface of sphere

I have a number of circles of same radius on surface of sphere (Google Maps API). I'm trying to calculate the total area covered by these possibly intersecting circles. My current solution is ...
0
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1answer
125 views

Intersection of perpendicular bisectors of a spherical triangle

I have 3 points on a unit sphere identified by their XYZ coordinates. They form a spherical triangle. If I'm not mistaken, perpendicular bisectors of a spherical triangle intersect in a single point, ...
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1answer
73 views

Implicit partial derivative of a spherical cap

Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld) $A(a,h) = \pi(a^2 +h^2)$, ...
2
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1answer
49 views

How to reason about two points on the unit sphere.

I've recently been thinking about various problems involving two points on the surface of a unit sphere. Let's specify them with a pair of unit 3-vectors ${\bf \hat a}$ an ${\bf \hat b}$. Is there ...
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0answers
85 views

How to find Latitudes and Longitudes of projections of the vertices of a rectangular plane below earth's surface?

I want to find out the latitudes and longitudes of projections of the vertices of a rectangular plane inside the earth's surface. I know dimensions of rectangle, angles of orientation and latitude and ...
3
votes
0answers
84 views

Ratio of the Volume of n-spherical cap to the volume of n-sphere

Assume an n-dimensional sphere with radius $R$ and volume $V^{(n)}_s$. Also assume a corresponding n-spherical cap with height $h$ and volume $V^{(n)}_c$. what is the ratio of two volumes? ...
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1answer
115 views

Integrating a discrete 3D surface, in spherical coordinates

I have an matrix which contains height information for a sheet suspended in air. Like a checkerboard, each value in the matrix represents a sampled height. Here's the hard parts: the data in the ...
0
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2answers
97 views

Is it true that the formulas of the volume and surface area of an n-dimensional sphere are best expressed in terms of $\eta = \frac{\pi}{2}$?

Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$. . In terms of ...
0
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1answer
34 views

Convert Degrees of Latitude to Feet

I need to check this formula I have to compute the distance of a point with latitude $lat$ from the equator: $$ \mathrm{feet} = \mathrm{lat} * 10000 \times 3280 / 90 $$ Example: A point at ...
2
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0answers
58 views

Zero-distortion map projection

Is it possible to take a limit of map projections (from a sphere to a plane) with ever-smaller distortion factors to get some kind of dendritic limit projection that has zero distortion everywhere? My ...
2
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2answers
256 views

Curve on a basketball

The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of ...
0
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0answers
122 views

A basic question on surface area of spherical cap of sphere

Consider a sphere of radius 1. Now chop a spherical cap with latitude line $\phi$ at the bottom of the cap is removed from top (say $0<\phi<\frac{\pi}{2}$). I want to know the surface area of ...
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1answer
71 views

Do spherical triangles with the same base and altitude have the same height?

If two spherical triangles have the same base $\theta$ and the same altitude $\phi$, do they have the same area. Initially I believed they would have by the same logic flat triangles do. However I'm ...
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1answer
75 views

How many unit vectors can tile an n-sphere with a given angle?

Given a unit radius $n$-sphere, and a constant $c = cos(\theta)$, $0 \le \theta \le \pi$, what is the size of the largest possible set of unit vectors $U = \{u_1, u_2, ..., u_n\}$ such that $u_i \cdot ...
0
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2answers
57 views

Intersecting arcs on a sphere

I'm working through this paper and I'm hung up on Proposition $3.1$. To strip away the context of the problem and present it in another light: suppose there are two intersecting arcs $ab$ and $cd$ on ...