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To find the minimum distance between a point in the plane (x_0,y_0) and a quadratic plane curve y=a_0+a_1x+a_2x^2, (1) note that the square of the distance is r^2 = ...

A tangent vector v_(p)=v_1x_u+v_2x_v is a principal vector iff det[v_2^2 -v_1v_2 v_1^2; E F G; e f g]=0, where e, f, and g are coefficients of the first fundamental form and ...

A pseudoanalytic function is a function defined using generalized Cauchy-Riemann equations. Pseudoanalytic functions come as close as possible to having complex derivatives ...

The whole neighborhood of any point y_i of an algebraic curve may be uniformly represented by a certain finite number of convergent developments in power series, ...

A point where a curve intersects itself along four arcs. The above plot shows the quadruple point at the origin of the quadrifolium (x^2+y^2)^3-4x^2y^2=0.

A type of cusp as illustrated above for the curve x^4+x^2y^2-2x^2y-xy^2+y^2=0.

The space of currents arising from rectifiable sets by integrating a differential form is called the space of two-dimensional rectifiable currents. For C a closed bounded ...

An area-minimizing surface (rectifiable current) bounded by a smooth curve in R^3 is a smooth submanifold with boundary.

In a complete metric space, a countable union of nowhere dense sets is said to be meager; the complement of such a set is a residual set.

A manifold possessing a metric tensor. For a complete Riemannian manifold, the metric d(x,y) is defined as the length of the shortest curve (geodesic) between x and y. Every ...