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

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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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1answer
171 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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123 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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82 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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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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83 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
142 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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221 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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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
56 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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1answer
42 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
165 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
52 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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108 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
165 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
87 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
50 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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110 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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122 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
166 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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2answers
115 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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1answer
56 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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64 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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2answers
353 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
92 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
81 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 ...
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2answers
62 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 ...
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2answers
2k 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° ...
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1answer
120 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 ...
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1answer
341 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: ...
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2answers
295 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 ...
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1answer
232 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.... ...
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61 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 ...
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44 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 ...
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1answer
74 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 ...
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162 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 ...
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1answer
51 views

What are the umbilic hypersurfaces in a sphere?

It is a well-known result that all umbilic hypersurfaces (complete and connected, say) of $\mathbb{R}^n$ are spheres or planes. But what can we say about umbilic hypersurfaces of a constant curvature ...
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73 views

Intersection of Two Parabolas on a Sphere

I'm trying to implement the algorithm in this paper which describes an implementation of Fortune's algorithm on a sphere, and I'm getting hung up on the math explaining how to calculate the ...
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2answers
39 views

Implicit Representation of Surfaces - Basic Quesion

I am reading on implicit representations of surfaces and cant quite come around the following example. Take $F : \mathbb{R^3} \rightarrow \mathbb{R}$, where $F(x,y,z)=x^2+y^2+z^2$. Now we want to ...
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1answer
41 views

Curving points to a sphere

I think math.stackexchange is the right place to post this, but if not, feel free to tell me. I have a series of points to be plotted on a sphere (Each one has a latitude and longitude value). These ...
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1answer
45 views

Closed Operations in Spherical Space

Euclidean vector arithmetic (addition/subtraction) is not closed under a spherical space. For example: $\mathbb{S}^2=\{v \in{\mathbb{R}^3}|\|v\|=1\}$ We have $(1, 0, 0)\in\mathbb{S}^2$ and $(0, ...
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121 views

Computing distance from line to point in geodetic environment

Supposing to be in a cartesian plan and that I have the following point: $$A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}), D(x_{4},y_{4})$$ $$P(x_{0},y_{0})$$ Now immagine two lines, the fist one ...
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570 views

How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
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2answers
2k views

Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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147 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
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1answer
36 views

Verify whether the centre of a sphere is outside a pyramid whose vertices are given

I have a maths question where the equation of a sphere $S$ is said to be $x^2+y^2+z^2-12x-6y-4z = 0$, and I'm asked to show that the centre is outside the pyramid whose vertices are $A(12,0,0)$, $B ...
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1answer
23 views

Equation of a sphere passing 3 points and tangential to a line

I got a maths question that gives you 3 points, $A (6,0,0)$ and $B (6,6,0)$ and $C (0,6,0)$ and a line DG, D being $(0,0,6)$ and $G (0,6,6)$ so the equation of DG is $\vec r$ = 6$\hat i$ -$6t\hat j$ . ...
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1answer
44 views

Suppose that a triangle T on a unit sphere has area equal to π/2. Which of the following is necessarily true?

Suppose that a triangle T on a unit sphere has area equal to π/2. Which of the following is necessarily true? ...
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92 views

Volume of a spherical tetrahedron

In the paper Jun Murakami, The volume formulas for a spherical tetrahedron a formula for the volume of a spherical tetrahedron is given. I am trying to work through the details for the specific ...
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1answer
101 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 ...