On the Area of a Region on a Developable Surface

1. A family of tangent lines to a space curve Γ forms a developable surface (DS). The position of a point A on DS is defined by distance u = AM along the tangent to Γ at the point M, and by the arc length v locating the point of tangency M on Γ. The coefficients of the first differential Gaussian form for DS are known [1]: E=1, F=1, G=1+(u/ρ)2, where 1/ρ is the curvature of Γ. We introduce new coordinates (u, φ), where φ is the angle swept on DS by the tangent line to Γ from its initial position φ = 0. Then the area of a region Ω on DS, bounded by two curves u (φ) and Γ, u(φ) = 0, and two tangent lines with angles φ1 and φ2, is given by the expression S = EG − F