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A basepoint is the beginning and ending point of a loop. The fundamental group of a topological space is always with respect to a particular choice of basepoint.

Two curves which, at any point, have a common principal normal vector are called Bertrand curves. The product of the torsions of Bertrand curves is a constant.

If P(x) is an irreducible cubic polynomial all of whose roots are real, then to obtain them by radicals, you must take roots of nonreal numbers at some point.

The radial curve of the catenary x = t (1) y = cosht (2) with radiant point (x_0,y_0) is given by x_r = x_0-coshtsinht (3) y_r = y_0+cosht. (4)

Given a point lattice, a cluster is a group of filled cells which are all connected to their neighbors vertically or horizontally.

The four medians of a tetrahedron concur in a point which divides each tetrahedron median in the ratio 1:3, the longer segment being on the side of the vertex of the ...

The cubic groups are the point groups T_h and O_h together with their pure rotation subgroups T_d, T, and O (Cotton 1990, pp. 433-434).

The dihedral group D_2 is a point group that is isomorphic to the vierergruppe and the finite group C_2×C_2.

Given a vector bundle pi:E->M, its dual bundle is a vector bundle pi^*:E^*->M. The fiber bundle of E^* over a point p in M is the dual vector space to the fiber of E.

A point p on a regular surface M in R^3 is said to be elliptic if the Gaussian curvature K(p)>0 or equivalently, the principal curvatures kappa_1 and kappa_2 have the same ...