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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, ...

The plane spanned by the three points x(t), x(t+h_1), and x(t+h_2) on a curve as h_1,h_2->0. Let z be a point on the osculating plane, then [(z-x),x^',x^('')]=0, where ...

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

An apodization function similar to the Bartlett function.

Li and Yorke (1975) proved that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as ...