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1n fact angle arbitrary curve arc QF arcsin arctan binormal boundary conditions conditions 1-14 conjugate construction coordinate axis corresponding surface course curve A(v curve A0 curve ft curves C(u defined determined developable surface differentiated with respect Doctor of Philosophy end points envelopes of families explosive families of cones Finally 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 parameter parametric equation partial differential equation plane sheet point F point Q(0 principal normal problem Q and Q rectangle R(0,v rectifying developable rectilinear geodesic circle ruled surfaces segment VQ sin2 sin2i sin2t sinfy sinV Substituting surface 5(A0 surface of revolution theorem Theorema Egregium tion triangle FAB vanishes identically vector versor vertex yields