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

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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 ...
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1answer
775 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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1answer
256 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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522 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
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73 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.
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1answer
52 views

Law of cosine in spherical trigonometry

I found from a book of mine the formula $\cos a=\cos b\cos c+\sin b\sin c\cos\alpha.$ Can this be true? If for example $a=1m,b=1m,c=1m,\alpha=1$, $m$ denotes by meter, then $\cos m=\cos^2 ...
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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 ...
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1answer
26 views

Calculate flight distance from one city to another on earth (sphere) [closed]

I've been sitting with this problem, I really cannot get solved. How do you calculate the flight distance from one city to another, given the longitude and latitude of the cities. For instance, can ...
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48 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
29 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: $$ ...
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1answer
215 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces? In ...
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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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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.. ...
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0answers
46 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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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 ...
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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
88 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 ...
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66 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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801 views

Lat/Long grid points covered by projecting rectangle onto sphere

Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view. Suppose we have a ...
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5answers
164 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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58 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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62 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 ...
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42 views

A continuous centerpoint of a convex spherical polygon

In discrete geometry, a centerpoint $c$ of a discrete set $S$ of $n$ points in the plane is such that any half plane containing $c$ contains (roughly) $n/3$ points of $S$. (Such a centerpoint always ...
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1answer
47 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. ...
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73 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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63 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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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 - ...
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1answer
47 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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52 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 ...
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1answer
24 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 ...
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1answer
55 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
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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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1answer
44 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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49 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 ...
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1answer
90 views

Any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a coclosed $1$-form?

What is meant by saying that any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a "co-closed" $1$-form? [...Since $H^1$ of $S^3$ is trivial it follows that the ...
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1answer
41 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) $A(a,h) = \pi(a^2 +h^2)$, ...
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1answer
45 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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51 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 ...
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1answer
174 views

Sine rule and equal angles

Is it true that if a triangle on a unit sphere has 2 sides with equal length then their opposit angles must be equal? I think it is true. I think we can use the spherical sine law. Call the sides with ...
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1answer
61 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 ...
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46 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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37 views

Find the angles defining an hyperspherical cap

For the hyperspherical cap of dimension $n+1$ find all the angle $\phi_1, \phi_2, \ldots, \phi_n$ which defines the cap? I mean, I know a cap is usually define by its height $h$ and its base $a$. ...
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2answers
63 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 ...
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2answers
109 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 ...
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1answer
31 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 ...
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How to show difference amount from scattering.

In our discussion, we need to show that the amount of photons at one wavelength is less than photons at a nearby wavelength. Photons (with a nearby wavelength) are created on the sun's photosphere ...
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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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0answers
43 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 ...
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1answer
51 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
255 views

Converting between spherical coordinate systems

Say I have the spherical coordinates of some locations, specifically their longitude ($0$ to 360) and latitude (latitude = $0$ at equator, $90$ at north pole, $-90$ at south pole) on a sphere with a ...