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Curvatures and tolerances in the Euclidean motion group. (English) Zbl 1075.53007
The authors investigate by their “tolerance zones” imprecisely defined geometric objects where toleranced input data provide toleranced output data. A determining role play the tolerance zones for the $$d(d+ 1)$$-dimensional linear space $$SA_d$$ of all affine mappings $$x\mapsto Gx+ g$$ of $$\mathbb R^d$$, given by the metric of the scalar product $$\langle(G, g), (H, h)\rangle:= \int_{\mathbb R^d}\langle Gx+ g,Hx +h\rangle d\mu(x)$$, where $$\mu$$ is a finite positive Borel measure. Thus, for example, a ball $$\Gamma(\gamma)$$ in $$SA_d$$ of radius $$r$$ as tolerance zone of its centre $$\gamma$$ provides for sufficiently small $$r$$ the interior of a hyperboloid of revolution as the tolerance zone of a line in $$\mathbb R_d$$, i.e., its orbit under $$\Gamma(\gamma)$$. – The main part of the paper deals with tolerance zones $$\Gamma(\gamma)\cap SE_d(\gamma\in SE_d)$$ of the Euclidean motion group $$SE_d\cap SA_d$$. In order to circumvent computational difficulties the authors replace here the Euclidean tolerance zones by the linearized zones $$\Gamma(\gamma)\cap T_\gamma SE_d$$ where $$T_\gamma SE_d$$ is the tangent space of $$SE_d$$ at $$\gamma$$. They compute an upper bound for the linearization error, made in this process, which requires an investigation of the curvature of curves covering $$SE_d$$.

##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 65G40 General methods in interval analysis
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