## 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

### Keywords:

curvature; geodesic torsion; ruled surface; spherical dual curve
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