Recently I baked a spherical cake (3cm radius) and invited over a few friends, 6 of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took my egg slicer with equally spaced wedges(and designed to cut 6 […]

I’m trying to parameterize a sphere so it has 6 faces of equal area, like this: But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each “slice”). I can’t seem to get the $\theta$ elevation parameter correct. Help!

A sphere is painted in black and white. We are looking in the direction of the center of the sphere and see, in the direction of our vision, a point with a given color. When the sphere is rotated, at the end of the rotation we might see the same or a different color. The […]

I know about the antipodal mapping. What I want to know is what the most significant differences between the sphere and projective space are, and how to think of each of them and their relationship to one another. I come at this from a coding theory/vector quantization perspective; I’m trying to understand the difference between […]

I have two arcs on a sphere that are defined as pair of points: (θ₀, φ₀), (θ₁, φ₁). I need to find a point where they intersect, or some indication if they don’t. What is important is that they are arcs, not circles, so it is important to find intersection that is on the arcs […]

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere knowing the dimension $n$, the center of the hypersphere $\vec{x}$ and its radius $r$. How to do that ?

Given three vectors $u,v,w \in S^2$ and the triangle $[u,v,w]$ I want to find its circumscribed circle. However, I don’t know how to approach this problem. Would some one please explain? In my understanding, one needs to somehow find the altitudes of the spherical triangle and then the point of its intersection (circumcentre)? Then what […]

I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points as if it were on the equator (i.e. as if its elevation were zero). Specifically, each point is specified by […]

Foreword This question was inspired by initial mistakes in this question. I wanted to explore the strange circle with $A>\pi r^2$ and got lost into geometrical jungle. A spherical cap is usually described by it’s height $h$, radius of the base $a$ and radius of the sphere $r$. Besides we have a relation $a^2+(r-h)^2=r^2$ so […]

I would like to draw a discretized circle on the surface of a sphere (the Earth in that case). The input of the algorithm would be the center of the circle (expressed as longitude and latitude), the radius (expressed in meters), and the number of points to calculate. The output would be the coordinates of […]

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