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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.

MSC:

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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