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A necessary and sufficient condition for a curve to be a helix is that the ratio of curvature to torsion be constant.

A function giving the distribution of the interpoint distances of a curve. It is defined by p(r)=1/Nsum_(ij)delta_(r_(ij)=r).

A level set in three dimensions.

The plane spanned by the normal vector N and the binormal vector B.

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 ...

If del xF=0 (i.e., F(x) is an irrotational field) in a simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the gradient of a scalar field phi(x), ...

sigma=1/tau, where tau is the torsion. The symbol phi is also sometimes used instead of sigma.

The plane spanned by the tangent vector T and binormal vector B.

dN^^+kappa_idr=0, where N^^ is the unit normal vector and kappa_i is one of the two principal curvatures.

One of the straight lines sweeping out a ruled surface. The rulings on a ruled surface are asymptotic curves.