MATHEMATICAL FORMULATION OF BODIES OF REVOLUTION 



by 



L. Landweber and M. Gertler 



ABSTPACT 



Various methods of defining bodies of revolution are considered with the con- 

 clusion that the most satisfactory method is one which defines the sectional- area curve 

 by means of a polynomial. The polynomial form possesses certain advantages in ease 

 of computation and ready application to hydrodynamical problems, such as the computa- 

 tion of theoretically derived pressure distributions. 



The degree of the polynomial fixes the number of parameters that may be pre- 

 scribed to determine a form. In order to generate the sixth- degree polynomial forms 

 the dimensionless parameters chosen are the nose and tail radii, r and r , the pris- 

 matic coefficient, C , and the position of the maximum section at x = m. It is shown 

 p. 



that the polynomial expression for the sectional-area curve is a linear combination of 



r , r , and C , with polynomials of the sixth degree as coefficients. Formulas and 



tables for these polynomial coefficients are provided, so that when r , r , C , and m 



o' i' p' 



are given, the offsets of a form may be rapidly computed. 



Not all combinations of these parameters give practical or desirable forms. 

 The range of useable forms may be limited by imposing the restrictions that the 

 sectional-area curve have no maximum or minimum other than at x = m, or that the 

 body have no inflection points. These criteria are formulated mathematically and a 

 method of computing boundary curves delineating permissible ranges of parameters is 

 developed. 



Formulas for generating seventh- degree polynomial forms are also derived and 

 applied to compare sixth- and seventh-degree forms with the same values of r , r , C , 

 and m. It is found that practical seventh- degree forms with the same values of those 

 parameters may differ appreciably from the sixth-degree form. Thus these parameters 

 do not suffice to fix a form, although they serve to develop the entire class of sixth- 

 degree polynomial forms. 



Bodies of revolution with useful application derived from polynomials not of the 

 sixth degree may be fitted (by the method of least squares) very closely by means of 

 sixth-degree forms. From this point of view the usefulness of a series of sixth-degree 

 polynomial forms is greatly enhanced. 



