Dual motions and real spatial motions. (English) Zbl 1255.53005

Consider a smooth curve \(\alpha\) on a surface \(M\) in Euclidean three-space. The adapted frame \((T,Y,N)\) on \(M\) along \(\alpha\) consists of the tangent vector \(T\) of \(\alpha\), the surface’s normal vector \(Y\) and the vector \(N = T \times Y\). It defines a spherical motion \(A(t)\) and it is well known how properties of certain trajectories are related to properties of such curves \(\alpha\). The article under review makes a similar construction but with dual numbers instead of real numbers. Instead of spherical motions, one obtains Euclidean motions, the trajectory of a dual unit vector \(\hat\delta\) is, by means of the Study mapping, a ruled surface \(\Phi_{\hat{\delta}}\).
The special trajectories considered in this article are obtained for \(\hat{\delta} = \hat{T}\) (tangent vector), \(\hat{\delta} = \hat{Y}\) (the surface’s normal vector) and \(\hat{\delta} = \hat{N} = \hat{T} \times \hat{Y}\). For these, the following statements are equivalent: \(\Phi_{\hat{Y}}\) is developable; \(\Phi_{\hat{N}}\) is developable; \(\alpha\) is a curvature line on \(M\).
The pitch of the instantaneous screw motion is a simple function of the curvatures of \(\alpha\) and simplifies even further for geodesics and asymptotic lines. Similar formulas are given for the case where \(\alpha\) is replaced by a different curve \(\beta\) but the adapted frame remains unchanged.


53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
53A17 Differential geometric aspects in kinematics
53A25 Differential line geometry
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