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Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as (1,0), and the second by ...

O_h is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated ...

Consider the distribution of distances l between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated ...

The evolute of an ellipse specified parametrically by x = acost (1) y = bsint (2) is given by the parametric equations x_e = (a^2-b^2)/acos^3t (3) y_e = (b^2-a^2)/bsin^3t. ...

For a nonzero real number r and a triangle DeltaABC, swing line segment BC about the vertex B towards vertex A through an angle rB. Call the line along the rotated segment L. ...

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a ...

Let M be a Riemannian manifold, and let the topological metric on M be defined by letting the distance between two points be the infimum of the lengths of curves joining the ...

An algorithm for partitioning (or clustering) N data points into K disjoint subsets S_j containing N_j data points so as to minimize the sum-of-squares criterion ...

A lattice L is said to be oriented if there exists a rule which assigns a specified direction to any edge connecting arbitrary lattice points x_i,x_j in L. In that way, an ...

A general set of methods for integrating ordinary differential equations. Predictor-corrector methods proceed by extrapolating a polynomial fit to the derivative from the ...