Curvatures and tolerances in the Euclidean motion group (Q1781914)

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 \(SE_d\).