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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.

For a unit speed curve on a surface, the length of the surface-tangential component of acceleration is the geodesic curvature kappa_g. Curves with kappa_g=0 are called ...

Let Gamma be an algebraic curve in a projective space of dimension n, and let p be the prime ideal defining Gamma, and let chi(p,m) be the number of linearly independent ...

Let a space curve have line elements ds_N, ds_T, and ds_B along the normal, tangent, and binormal vectors respectively, then ds_N^2=ds_T^2+ds_B^2, (1) where ds_N^2 = ...

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).

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 ...

The Pappus spiral is the name given to the conical spiral with parametric equations x(t) = asin(alphat)cost (1) y(t) = asin(alphat)sint (2) x(t) = acos(alphat) (3) by ...

The class m, curve order n, number of ordinary double points delta, number of cusps kappa, number of inflection points (inflection points) iota, number of bitangents tau, and ...