This investigation reveals the geometrical properties of the high-latitude boundaries surrounding the high-altitude cusp. These have been shown by recent studies to include the outer boundary between the magnetosheath and the cusp; the outer boundary between the magnetosheath and the high-latitude, closed field-line region of the dayside magnetosphere (which we call the high-latitude trapping region (HLTR)); and the inner boundaries on the equatorward and tailward edges of the high-altitude cusp region. We present a survey of the normal directions of the outer boundaries: between the magnetosheath and cusp in comparison to between the magnetosheath and the HLTR. This study is based on the magnetic field data set obtained by the Cluster quartet for a period during which the orbits passed through either the dayside cusp or the magnetopause on consecutive orbits. These initial results show some evidence of the control of cusp geometry by the IMF orientation and clear trends which are consistent with an indentation or at least a nonplanar geometry in the cusp-magnetosheath boundary for those cases which are clearly identified. The detailed geometry of all surrounding cusp boundaries, deduced from selected crossings in the data set, case by case, is also shown to support these findings and to address the issue of the identification of relative spacecraft location with respect to this highly dynamic region.