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Let R be a plane region bounded above by a continuous curve y=f(x), below by the x-axis, and on the left and right by x=a and x=b, then the volume of the solid of revolution ...

The Minkowski measure of a bounded, closed set is the same as its Lebesgue measure.

A mixed graph in which both directed and undirected edges may exist. If only directed edges exist, the graph is called a directed graph. If only undirected edges exist, it is ...

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

If two curves phi and psi of multiplicities r_i!=0 and s_i!=0 have only ordinary points or ordinary singular points and cusps in common, then every curve which has at least ...

Any irreducible curve may be carried by a factorable Cremona transformation into one with none but ordinary singular points.

A ruled surface M is a normal developable of a curve y if M can be parameterized by x(u,v)=y(u)+vN^^(u), where N is the normal vector (Gray 1993, pp. 352-354; first edition ...

Any continuous cumulative frequency curve, such as the one illustrated above in the right figure.

A curve y(x) is osculating to f(x) at x_0 if it is tangent at x_0 and has the same curvature there. Osculating curves therefore satisfy y^((k))(x_0)=f^((k))(x_0) for k=0, 1, ...

For a parabola oriented vertically and opening upwards, the vertex is the point where the curve reaches a minimum.