Problem research of display the orthogonal projection of four dimensional hypersurface defined by the equation in an implicit form one and two hyperplane. To communicate and bypass hypersurface Discriminants assumes that the hypersurface of the two-parameter family of two-dimensional display received surfaces in five-dimensional space. The analysis of kriminant four dimensional hypersurface orthogonal projection onto the coordinate hyperplane in two coordinate axes. The intersection of three dimensional hypersurfaces axiom (kriminant) defines a two-dimensional surface that is the envelope of the two-parameter family of two-dimensional surfaces. Are the necessary and sufficient conditions of existence of this envelope.
The results obtained are used to study the Discriminants four-dimensional hypersurface, the display of two-parameter family of spheres in a five-dimensional space.  

Keywords: a family of surfaces, hypersurface, envelope, the display features, а discriminant.

  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

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