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Given a map f from a space X to a space Y and another map g from a space Z to a space Y, a lift is a map h from X to Z such that gh=f. In other words, a lift of f is a map h ...

Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions psi_i, such that the support ...

For X a topological space, the presheaf F of Abelian groups (rings, ...) on X is defined such that 1. For every open subset U subset= X, an Abelian group (ring, ...) F(U), ...

An algebraic set is the locus of zeros of a collection of polynomials. For example, the circle is the set of zeros of x^2+y^2-1 and the point at (0,0) is the set of zeros of ...

A map projection which is a conformal mapping, i.e., one for which local (infinitesimal) angles on a sphere are mapped to the same angles in the projection. On maps of an ...

Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the ...

The dihedral angle is the angle theta between two planes. The dihedral angle between the planes a_1x+b_1y+c_1z+d_1 = 0 (1) a_2x+b_2y+c_2z+d_2 = 0 (2) which have normal ...

A discriminant is a quantity (usually invariant under certain classes of transformations) which characterizes certain properties of a quantity's roots. The concept of the ...

A map projection in which areas on a sphere, and the areas of any features contained on it, are mapped to the plane in such a way that two are related by a constant scaling ...

A circumconic hyperbola, which therefore passes through the orthocenter, is a rectangular hyperbola, and has center on the nine-point circle. Its circumconic parameters are ...