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
110 views

n-by-n degree grid on a sphere?

I've been trying to generate an evenly spaced grid centred at a given point on a sphere, such that the angular separation between any neighbouring pair of points is the same (e.g., 1 degree). The grid ...
2
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1answer
64 views

Spherical Triangle properties

In a spherical triangle ABC do the following properties hold? (a) If AB = AC are the base angles at B and C equal? Yes (b) If the angles at B and C are equal is it true that AB = AC? Yes (c) Do the ...
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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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2answers
20 views

Spherical distance

The spherical distance between two points (P1=(0,0,1) P2=($\frac{1}{2\sqrt{2}}$,$\frac{1}{2\sqrt{2}}$,$-\frac{\sqrt{3}}{2}$) ) is $\frac{5\pi }{6}$ I am at a loss as to how the spherical distance was ...
2
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1answer
58 views

Find the missing sides and angles in following case for a spherical triangle ABC:

$\bf{QUESTION}$: Find the missing sides and angles in following case for a spherical triangle ABC: $$a)a=60°,\beta=90°, \gamma=75° $$ So, if I am right my book says sides are denoted by lowercase ...
2
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1answer
31 views

Arc length in spherical triangle [closed]

A spherical triangle has angles of 120◦, 60◦ and 45◦. Find the cosines of the (arc) lengths of the sides. How many sides have an arc length larger than 90◦?
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1answer
65 views

Finding angles in spherical triangle using law of cosines

Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the angles and the area of the spherical triangle with vertices ...
3
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1answer
55 views

Spherical triangle vertices to spherical coordinates

Problem: Assume that the earth is a sphere of radius 5280 miles, find the length of the sides, the measure of the angles and the area of the spherical triangle with vertices A(70°N,10°E),B(10°S,100°E) ...
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3answers
34 views

Finding circle of a sphere through two points

We have two points $P_1, P_2$ on a sphere $S$ of radius $R$. Suppose for $r \ll R$, the distance between $P_1$ and $P_2$ is less than $2r$. Then, $P_1$ and $P_2$ both lie on exactly two radius-$r$ ...
0
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1answer
347 views

Drawing ellipse as google.maps.Polygon with 8 points

In a web page using Google Maps JavaScript API v3 (including Geometry library) I currently draw an ellipse as a "diamond" with 4 corner points by the following JavaScript code: ...
2
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1answer
51 views

Finding the length of the sides, the measure of the angles and area of spherical triangles?

I'm trying to understand this problem in the textbook but I got lost in one part: Problem: Assume that the earth is a sphere of radius $5280$ miles, find the length of the sides, the measure of the ...
3
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1answer
43 views

Stereographic projection of a sphere

What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong. This image accompanies the problem: ...
2
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1answer
31 views

Lift of isometries of spherical space forms

If we have an isometry between two spherical space forms, then it is said that it lifts to an isometry of the sphere. Why is that?
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0answers
67 views

Is the spherical harmonic representation of a 2D field independent of grid?

What I am currently unable to understand is whether the spherical harmonic representation of a 2D field is in any way tied to the nature of the grid on which decomposition/composition is performed. I ...
0
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2answers
55 views

In the real spherical harmonics, where does the sqrt(2) factor come from?

The real spherical harmonics can be written in terms of the complex spherical harmonics: $$ Y_{\ell m} = \begin{cases} \displaystyle \sqrt{2} \, (-1)^m \, \operatorname{Im}[{Y_\ell^{|m|}}] & ...
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0answers
70 views

Spherical coordinates and the Law of Cosines

I have one question on my project. I am assuming earth is a perfect sphere. How can I get from the Law of Cosines $$\cos(c)=\cos(\operatorname{lat} A)\cos(\operatorname{lat} ...
0
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1answer
126 views

How to cut a sphere in 3 parts of equal volume?

I ran across this problem when working on an architecture design project. I know this probably involves integral math but I'm not very familiar with it. Any help would be appreciated.
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1answer
65 views

Reflections On Sphere Surface / Getting Great Circle from two 3D points

I'm trying to calculate the reflection of a point across another point, both of which are on the surface of a sphere. I believe I could do this by getting the formula for the great circle of the ...
4
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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 ...
3
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0answers
54 views

How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere of radius $R$?

Question How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere $\mathbb S $ of radius $R \quad (R \gg a)$? What I have thought so far Since the area of the ...
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1answer
68 views

Surface area of a 2-sphere in Abstract Index Notation

I believe the following completely specify a 2-sphere of radius 1 in AIN: $$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$ It is easy enough to determine the area by ...
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1answer
239 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
80 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
113 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, ...
0
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1answer
66 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.
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1answer
79 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
31 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
41 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 ...
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2answers
215 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
57 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
60 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 $2^3 = 8$ ...
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1answer
49 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
47 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
120 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 ...
2
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1answer
100 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
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 ...
0
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1answer
37 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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0answers
25 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
71 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
171 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
35 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
142 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
112 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
80 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
79 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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1answer
75 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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1answer
122 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
199 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
102 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 - ...