In many applications of profiling cutting tool, define the envelope surface collection. Along with the classic approach to the definition of the envelope of the recently used and new. So, if your family schedule projected two-dimensional surfaces in space R4, then get some three dimensional hypersurface Σ. Kriminanta this is the envelope of family of surfaces. The authors study the surface Σ when setting its parametric equations and equation in implicit form held in the works.  Found some new properties of such surfaces. Due to the fact that when profiling cutting tool set family of surfaces, coordinate transformation equations, an important task is the study received thus hypersurface. On the basis of article studies obtained in a general manner the necessary payment according to calculate kriminanty and diskriminanty hypersurface.  They show examples of how to define envelopes two families of spheres. These are family formulas transform coordinates defining the translational and helical movement. The results are illustrated by computer polygonal models.  

Keywords: a family of surfaces, hypersurface, profiling, feature display, cutting tool