# Tagged Questions

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

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### 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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### 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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### Analytical solution for volume when a plane cuts a hemi-sphere

I need to find the analytical solution when the plane $P: z = grad\cdot y + z_{cut}$ cuts the hemi-sphere $S: x^2 + y^2 + z^2 = r^2;\:y \leq 0$. I constructed two 3D images in MatLab of the ...
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### 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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### Internal angle of a vertex of degree $d$ in $\mathbb{E}^2$ and $\mathbb{S}^2$

I am currently working on determining the maximum number of times the minimum spherical distance can occur among $n$ points in $\mathbb{S}^2$, and I have the following question. In $\mathbb{E}^2$, ...
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### How to calculate rotated global location coordinates (Long, Lat)?

Given the current global location coordinate system: -180 <-> 180 Longitude -90 <-> 90 Latitude A rotation of the globe 90 degrees counter-clockwise around the Y-axis would bring the north ...
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### Showing there is no local isometry between spheres of different radii

I wish to show there is no is no local isometry between 2-dim spheres of different radii, without the use of curvature, as it is not in my knowledge yet. Could you provide directions? If such ...
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### congruency of triangles in hyperbolic and spherical geometry

In Euclidean geometry, we have the following congruencies of triangles: side-side-side, side-angle-side, angle-angle-side = angle-side-angle (because of the angle sum) and side-side-angle (only if the ...
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### Sample from distribution taking spherical statespace

I have a probability distribution over a 2-sphere, with density function $f(\phi)$, a function of polar angle only. Is there an efficient way to sample from this distribution?
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### Area of a sphere bounded by hyperplanes

Say we have a sphere in d-dimensional space, and k hyperplanes (d-1 dimensional) all passing through the origin. Is there a way to calculate (or approximate) the area of the surface of the sphere ...
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### Continuity of maximum distance between geodesics on a smooth manifold

I am working on my own version of a proof of the Jordan Separation Theorem (just for fun - I know it's been proved countless times) and in the course of so doing I use the apparently fairly obvious ...
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### Find the sum of the sides in a spherical right triangle

In a spherical triangle the angles at α, β and γ are π/5, π/3, π/2. Find the sum of the sides, we shall call the sides a,b,c So I'm looking at the formulas and I see one of Napier's rule which ...
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### 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 ...
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### 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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### 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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### 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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### 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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### Extremal constant width curves on sphere

Definition Given some length $w\in\mathbb R$, I'm interested in closed convex sets $S$ of points with the following properties: For all pairs of points from $S$, the distance between them will be ...
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### integral over two spherical Bessel function

I am now having a problem regarding the integral over two spherical Bessel function. If anyone can give any help, it would be so nice of you. Thank you so much for any help. Specifically, I intend to ...
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### Finding the coordinates of the corners of an aligned pole-centered spherical square

Given a spherical square of radius $1$, with edge midpoints at $(1, x, 0)$, $(1, x, \pi/2$), $(1, x, \pi)$ and $(1, x,3 \pi/2)$ (in the spherical coordinate system of (radial distance, polar angle, ...
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### Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
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### Average distance from point to spherical square?

I need to calculate the average distance from a point to a $4$ sided spherical polygon. Can someone point me to the right direction? I guess either the average point of a spherical square or centroid ...
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### calculating the area on the surface os a sphere created by intersection of two spherical caps!

Consider a spherical object composed of two compartments (A and B, not necessarily hemispheres) sitting at the interface which is characterized by a plane separating 1 and 2. For this case, ...
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### Find third geographic coordinate in triangle using spherical earth model

I'm trying to solve triangulation problems using geographic coordinates from a GPS. all calculations must use the spherical earth model (great circle distance). Given the points and lengths: Point $A$...
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### projection of a sphere onto a plane

Consider you have a sphere centered at the origin.The sphere has a diameter of $\frac{1}{2} \sqrt{\frac{3}{2}}$. This means that the inscribed cube has an edge of 1. Take any point from the plane (1,...
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### Geographic coordinates to pitch+yaw+pitch

I'm creating a very basic simulation which involves air travel across the world and am trying to correctly position and orient my aircraft in a rendered 3D representation. I am representing the Earth ...
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### Can one define 'geodesic' solely in terms of the betweenness relations among the points on that geodesic?

In the Euclidean plane (though I assume the following result can be generalized to any Euclidean n-space), Tarski showed that one can define what it is to be a straight line solely in terms of the ...
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### Hypersurfaces of a hemisphere

Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$...
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### Looking for a particular parameterization of $S^n$

Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$. Is there any ...
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### rotation of spherical surface in spherical coordinates

I need to plot a spherical surface in computer (like the surface of a lens). I know the normal vector (as an example, say $\ n=(1,2,3)$) of this surface and it originates from the centre of the ...
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### Non commutative flows on the 2-sphere.

The title says everything, really. I'm looking for some flows on $S^2$ such that They do not commute. They are of some interest, or they are peculiar in some ways. Thanks
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### Locate a point on sphere with equal distance

Given 3 points A (lat1, lon1), B(lat2, lon2), O(lat3,lon3) on earth with geometric location longitude and latitude and a distance d, where O is middle point of A and B. Let GCD denote the great circle ...
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### Placement of protons and neutrons in the nucleus

So, I'm creating a program that would represent a given atom (also different isotopes) in 3d view. I'd need some kind of formula to calculate the position of protons and neutrons to form a nucleus. ...
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### Orthogonal transformation of a triangle on $S^2$

Let $v_1, v_2, v_3$ and $w_1, w_2, w_3$ denote the vertices of two spherical triangles $\bigtriangleup_1, \bigtriangleup_2$ with the property, that $\|v_i - v_j \| = \|w_i - w_j \|$, e.g. their sides ...
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### How can I find the center of gravity of a hollow spherical cap?

I am looking to find the center of gravity for a hollow spherical cap. Could I use that point as the point at which the entire mass of the spherical cap is for newtonian gravity problems?
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### 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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### 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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### Make a balloon with least total length of seams

I'd like to build a small hot-air balloon out of flame-retardant plastic sheeting to suspend a camera. The plastic sheeting (plastic film) is commonly sold in a long roll in a width of 20ft (6 meters)....