A method of the determination of a developable ruled surface
Let \(\widehat{\mathbf x}(t)= {\mathbf x}(t)+ \varepsilon{\mathbf x}^* (t)\) be a curve on the unit dual sphere which corresponds to the ruled surface \({\mathbf m}(t,u)= {\mathbf p}(t)+ u{\mathbf x}(t)\) (Study's Übertragungsprinzip), where \(\varepsilon^2= \varepsilon^3= \cdots=0\), \(|{\mathbf x} |\equiv 1\), and \({\mathbf x}^*: ={\mathbf p}\times{\mathbf x}\). The author considers the dual angles \(\widehat\varphi= \varphi+ \varepsilon\varphi^*\) and \(\widehat\theta= \theta+ \varepsilon \theta^*\) in order to obtain a parametric representation of the curve \(\widehat{\mathbf x}(t)\) and shows that the base curve is given by \({\mathbf p}(t)= ((P_3-\theta^*) \cos\theta \text{cotan} \varphi+ \varphi^* \sin\theta\), \((P_3- \theta^*) \sin\theta \text{ cotan }\varphi-\varphi^* \cos\theta, P_3)\). Taking \(P_3=\theta^*\) one has \({\mathbf p}(t)= (\varphi^* \sin\theta, -\varphi^*\cos \theta,\theta^*)\). For a developable ruled surface the linear differential equation \({dy\over dt}+ Ay+B=0\) is satisfied for \(y(t):= \tan\varphi\) and some coefficients \(A=A(t)\) and \(B=B(t)\). Using the above, the author shows that developable ruled surfaces can be found so that the base curve is a given curve \({\mathbf p}(t)\). He finally considers the example in the case that the given curve is the cylindrical helix.
scientific article; zbMATH DE number 1331551