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

learn more… | top users | synonyms

2
votes
1answer
114 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
votes
1answer
50 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◦?
2
votes
1answer
104 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
votes
1answer
89 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) ...
1
vote
3answers
67 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
votes
1answer
636 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
votes
1answer
82 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
votes
1answer
66 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
votes
1answer
35 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?
0
votes
2answers
100 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|}}] & ...
0
votes
1answer
254 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.
1
vote
1answer
103 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
votes
3answers
416 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
votes
0answers
89 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 ...
1
vote
1answer
82 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 ...
11
votes
1answer
285 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}= ...
1
vote
2answers
166 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
votes
1answer
146 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
votes
1answer
83 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
votes
1answer
98 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} ...
0
votes
1answer
53 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 ...
1
vote
0answers
58 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
vote
2answers
332 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 ...
1
vote
0answers
66 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
votes
1answer
85 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
vote
1answer
90 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 ...
0
votes
1answer
182 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
votes
1answer
128 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
votes
2answers
227 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
votes
1answer
44 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
votes
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?
1
vote
0answers
126 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
votes
0answers
325 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 ...
1
vote
0answers
63 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
votes
1answer
272 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
votes
2answers
141 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 ...
1
vote
0answers
97 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.
1
vote
0answers
95 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
votes
1answer
104 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
votes
1answer
197 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 ...
4
votes
5answers
261 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 ...
1
vote
0answers
22 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
vote
1answer
58 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 ...
0
votes
1answer
52 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
vote
1answer
204 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 ...
2
votes
2answers
58 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 ...
0
votes
0answers
142 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
votes
1answer
220 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, ...
1
vote
1answer
123 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
votes
1answer
55 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 ...