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

3
votes
1answer
108 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
58 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
64 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
379 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 ...
2
votes
0answers
77 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
95 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
171 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
223 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
151 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
294 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
48 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: $$ \mathbb{S}^d\times\stackrel{n}{\cdots}\times\...
0
votes
0answers
27 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
169 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
409 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
69 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
349 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
153 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
108 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
97 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
114 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
256 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
286 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 24,...
1
vote
0answers
23 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 - \cos(\beta)}...
1
vote
1answer
60 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
70 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
279 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
79 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
179 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
258 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
138 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
56 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 ...
1
vote
0answers
147 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 ...
4
votes
0answers
236 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? $$\frac{V^{...
1
vote
1answer
367 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
votes
2answers
145 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
votes
1answer
86 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 $40....
2
votes
0answers
76 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
votes
2answers
644 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 ...
1
vote
1answer
126 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 ...
1
vote
1answer
90 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
votes
2answers
71 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 ...
2
votes
2answers
4k views

Equation for calculating azimuth between two points

Does anybody know an equation or approximation for calculating the azimuth as a function of latitudes and longitudes of both the points. For example I have Princeton, NJ is at 40.3571° N, 74.6702° W,...
0
votes
1answer
194 views

Sphere Calculations: Determining the remaining surface area of the sphere when the sphere is cut by a vertical and horizontal plane

Sphere Calculations: Determining the remaining surface area of the sphere when the sphere is cut by a vertical and horizontal plane and the centre of the sphere is not on the vertical and horizontal ...
2
votes
1answer
519 views

how to find distance in longitude and latitute when center and radius is given? [closed]

How to find distance in longitude and latitute when center and radius is given? for example: ...
3
votes
2answers
563 views

Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as ...
0
votes
1answer
303 views

c.s.a of a hemisphere

Here is a question which I would like to understand. I want to know How to prove that c.s.a of a hemisphere is $2\pi r^2$ ? I'm a 10th CLASS average student,so please keep it simple. Thank you....
2
votes
3answers
65 views

Spherical coordinate transformation

We have the following picture ($r=1$. $AB$ is the prime meridian): We can find the coordinates of $C$ using: $$ x = \sin(b)\cos(a)$$ $$ y=\sin(b)\sin(a)$$ $$ z = \cos(b) $$ I understand this ...
1
vote
0answers
58 views

creating a mono-monostatic body from a basketball and acrylic tube

I'm looking for a formula to calculate if it is possible to create a mono-monostatic body out of a miniature rubber basketball and an acrylic tube of a variable length. I have PUR casting resin that I ...
0
votes
1answer
101 views

Trouble with understanding a spherical coordinate system.

We have a sphere with $r=1$, and we want the coordinates of $C$. $A$ is the north pole, and $AB$ is our prime meridian. See picture: I'm familiar with an $(x,y,z)$ coordinate system, but not so ...
5
votes
0answers
208 views

How I cut my orange - spherical volume integral

I cut my orange in six eatable pieces, following some rules. My orange is a perfect sphere, and there is a cylindrical volume down through my orange, that is not eatable. In the diagram, the orange ...