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angle arbitrary curve arc QF arctan axis binormal boundary conditions cone F cone F(0 conjugate construction contained coordinate axis course curve A0 curves A(v curves C(u curves cr defined determined developable surface Differential Geometry differentiated with respect Doctor of Philosophy envelopes of families explosive fact families F(i families of cones Frenet equations function g(u,v fundamental form Gaussian curvature geodesic curvature geometrical given gives hence intersection interval introduce isometric joining the origin length limit conditions meridian obtained as envelopes one-parameter family orthogonal osculating plane P(cr parameter parametric equation partial differential equation plane sheet point F principal normal problem Q and Q rectangle fl(0,v rectifying developable rectilinear geodesic circle ruled surfaces segment VQ sin2 sin2i sin2t sinV solutions Substituting surface of revolution tangent theorem Theorema Egregium tion triangle FAB vector versor yields