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Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy ...

Sphere line picking is the selection of pairs of points corresponding to vertices of a line segment with endpoints on the surface of a sphere. n random line segments can be ...

Define the packing density eta of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for ...

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates theta and phi from uniform distributions theta in [0,2pi) and phi in ...

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. n random ...

Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of ...

A spheric section is the curve formed by the intersection of a plane with a sphere. Excluding the degenerate cases of the plane tangent to the sphere or the plane not ...

Take the Helmholtz differential equation del ^2F+k^2F=0 (1) in spherical coordinates. This is just Laplace's equation in spherical coordinates with an additional term, (2) ...

A solution to the spherical Bessel differential equation. The two types of solutions are denoted j_n(x) (spherical Bessel function of the first kind) or n_n(x) (spherical ...

The spherical Bessel function of the first kind, denoted j_nu(z), is defined by j_nu(z)=sqrt(pi/(2z))J_(nu+1/2)(z), (1) where J_nu(z) is a Bessel function of the first kind ...