Moulding surface

A surface generated by the orthogonal trajectories of a one-parameter family of planes. Moulding surfaces have one family of planar lines of curvature that are simultaneously geodesics for the moulding surface. If the family of planes is degenerated into a bundle, then the moulding surface will be a surface of revolution. The sections of a moulding surface by planes of the family are called meridians, and the orthogonal trajectories are called parallels of the moulding surface. All meridians are congruent, so that a moulding surface can be formed by the motion of a planar line $L$ (the meridian), the plane of which moves without sliding along a certain developable surface. This surface is called the directing surface of the moulding surface and is one of the sheets of its evolute. If $\rho(u)$ is the position vector of one parallel position, then the position vector of the moulding surface will be

$$r=\rho(u)+\eta(v)p(u)+\zeta(v)q(u),$$

where $p=\nu\cos\theta+\beta\sin\theta$, $q=-\nu\sin\theta+\beta\cos\theta$, $v$ is the principal normal, $\beta$ is the binormal, $x$ is the torsion of the curve $\Gamma$, and $\theta=-\int xdu$. Its line element is given by:

$$ds^2=[1+k(\zeta\sin\theta-\eta\cos\theta)]^2du^2+(\eta'^2+\zeta'^2)dv^2,$$

where $\eta(v),\zeta(v)$ are the equations of $L$ and $k$ is the curvature of $\Gamma$.

References

[a1]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Chelsea, reprint (1972) pp. Sects. 85–87