Affine mappings on simplices (Q2878908)

The paper deals with affine mappings on domains in \(\mathbb{R}^3\) that are unions of simplices.NEWLINENEWLINELet \(Q\) be a domain in \(\mathbb{R}^3\) that is a union of simplices \(Q_i\) whose interiors are disjoint. Consider an affine mapping on each simplex \(Q_i\) of the form \(f_i(x)=A_ix\), where \(A_i\) is a \(3 \times 3\) transformation matrix. The authors study the conditions that these matrices must satisfy in order to obtain a continuous piecewise affine mapping \(f\) on the domain \(Q\) such that \(f\mid Q_i=f_i\).NEWLINENEWLINEBy the singular value decomposition theorem, each \(A_i\) can be written as \(U_iD_iV_i^T\), where \(U_i\) and \(V_i\) are orthogonal matrices and \(D_i\) is a diagonal matrix. Taking into account that \(A_i^TA_i=V_iD_i^2V_i^T\) is independent from \(U_i\), for obtaining necessary and sufficient conditions for \(A_i\) so that \(f\) is continuous, the authors analyze the conditions for \(D_i\) and \(V_i\) that affirm the existence of \(U_i\) such that matrix \(A_i\) satisfies the desired property.NEWLINENEWLINEFirstly, the simplest case when the domain \(Q\) is a union of two simplices sharing a common face is studied. After that, the authors analyze the case of a domain that is a union of four simplices sharing a common edge. Finally, a result for a domain that is a union of three simplices is presented.