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A closed embedded smooth plane curve has at least four vertices, where a vertex is defined as an extremum of curvature.

A distribution of values of a discrete variate represented graphically by plotting points (x_1,f_1), (x_2,f_2), ..., (x_k,f_k), and drawing a set of straight line segments ...

The curvature and torsion functions along a space curve determine it up to an orientation-preserving isometry.

The Galilean spiral is the curve with polar equation r=btheta^2-a for a>0 which describes the trajectory of a point uniformly accelerated along a line rotating about a point.

Consider two closed oriented space curves f_1:C_1->R^3 and f_2:C_2->R^3, where C_1 and C_2 are distinct circles, f_1 and f_2 are differentiable C^1 functions, and f_1(C_1) ...

If a sequence of double points is passed as a closed curve is traversed, each double point appears once in an even place and once in an odd place.

A disk D in a solid torus V=S^1×D^2 is called meridinal if its boundary is a nontrivial curve in del V (so that it is a meridian). Then a closed subset X subset V is called ...

The four-dimensional version of the gradient, encountered frequently in general relativity and special relativity, is del _mu=[1/cpartial/(partialt); partial/(partialx); ...

A general term which refers to an increase (or decrease in the case of the oxymoron "negative growth") in a given quantity.

The contour C_epsilon illustrated above.