## Abstract

Geometry of part surfaces is discussed in this chapter of the book. The discussion begins with analytical description of perfect (ideal or geometrically accurate) surfaces. Here, in this text ideal surface is interpreted as a zero-thickness film. Then difference between Classical Differential Geometry and Engineering Geometry of Surfaces is analyzed. This analysis is followed by analytical description of real part surfaces, which based in much on analytical description of the corresponding ideal surface. It is shown that while remained unknown, real part surface is located in between two boundary surfaces. The said boundary surfaces are represented by two perfect surfaces of the upper tolerance, and of the lower tolerance. Specification of surfaces ends with discussion of natural representation of a desired part surface. This consideration involves the first and the second fundamental forms of a smooth regular part surface. For an analytically specified surface, elements of its local geometry are outlined. This consideration includes but is not limited to analytical representation of unit tangent vectors, tangent plane, unit normal vector, unit vectors of principal directions on a part surface and so forth. Ultimately, design parameters of part surface curvature are discussed. Mostly, equations for principal surface curvatures along with normal curvatures at a surface point are considered. In addition to that mean curvature, Gaussian curvature, absolute curvature, shape operator and curvedness of a surface at a point are considered. Classification of local part surface patches is proposed in this section of the book. The classification is followed by a circular chart comprised of all possible kinds of local part surface patches.

## References

- 1.Radzevich, S. P. (1991).
*Differential-geometrical method of surface generation*(300 p.) (Doctoral Thesis) Tula Polytechnic Institute, Tula.Google Scholar - 2.Radzevich, S. P. (2017)
*Generation of surfaces: Kinematic geometry of surface machining*(738 p.). Boca Raton, Florida: CRC Press.Google Scholar - 3.Radzevich, S. P. (2010).
*Gear cutting tools: Fundamentals of design and computation*(p. 786). Boca Raton, Florida: CRC Press.CrossRefGoogle Scholar - 4.Mortenson, M. E.:
*Geometric modeling*(763 p.). New York: Wiley, Inc.Google Scholar - 5.Bonnet, P. O. (1867)
*Journ. Ec. Polytch*., xiii, 31.Google Scholar - 6.Radzevich, S. P. (2002). Conditions of proper sculptured surface machining.
*Computer-Aided Design,**34*(10), 727–740.CrossRefGoogle Scholar - 7.Radzevich, S. P. (2001).
*Fundamentals of surface generation*(p. 592). Monograph: Kiev, Rastan.Google Scholar - 8.Radzevich, S. P. (2008).
*Kinematic geometry of surface machining*(p. 536). Boca Raton, Florida: CRC Press.Google Scholar - 9.doCarmo, M. P. (1976)
*Differential geometry of curves and surfaces*(503 p.). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - 10.Struik, D. J. (1961).
*Lectures on classical differential geometry*(2nd ed., p. 232). Massachusetts: Addison-Wesley Publishing Company Inc.zbMATHGoogle Scholar