CHAPTER 49 



Mathematical Methods for Dehiieating Bodies 

 and Ship Forms 



49.1 Scope of This Chapter; Definitions .... 180 



49.2 The Ueefulne-'ss of Mathematical Ship Lines . 180 

 ■19.3 E.xisting Mathematical Formulas for Deline- 

 ating Ship Lines 187 



49.4 Mathematical and Dimcnsioriless Represen- 



tation of a Ship Surface 189 



49.5 Application of the Dimensionlcss Surface 



Equation to Ship-Shaped Forms .... 191 



49.6 Summary of Dimensionlcss General Equa- 



tions for Ship Forms 192 



49.7 Limitations of Mathematical Lines .... 192 



49.8 Value and Relationship of Fairness and 



Curvature 193 



49.9 Notes on Longitudinal Curvature Analysis . 195 



49.11 



49.12 

 49.13 



49.16 

 49.17 



Graphic Determination of the Dimensionlcss 

 Longitudinal Curvature of any Ship Line . 196 



Mathematic Delineation and Fairing of a 

 Section-.\roa Curve 19S 



Ix)ngitudinal Flowplane Curvature .... 199 



Checking and Establishing Fairness of Lines 

 by Mathematical Methods 199 



Illustrative Example for Fairing the De- 

 signed Waterline of the ABC Ship .... 200 



Practical Use of Mathematical Formulas for 

 Faired Principal Lines 203 



The Geometric Variation of Ship Forms . . 204 



Selected References Relating to Mathe- 

 matical Lines for Ships 204 



49.1 Scope of This Chapter; Definitions. .M;itli- 

 omatiiul nicthod.s for doliiR'atiiig the forms of 

 bodies uiid ships are those by which the shape of 

 the outer surface, adjacent to the Uquid, may be 

 defined wholij' or in part by mathematical 

 fonnulas and equations which express the coor- 

 dinates in terms of given reference axes. These 

 may be the rectangular (x, y, z) or Cartesian 

 coordinates in one, two, or three dimensions, the 

 cylindrical coordinates about an axis, the polar 

 or spherical coordinates about a point, or whatever 

 may be convenient for the purpo.se. With a 

 selected set of numerical values a.s.signed to the 

 symbols of these equations, all or part of the 

 surface coordinates or offsets may be calculated. 



Part of a bodj' surface may be geometric, such 

 as a nose of hemispherical or ellipsoidal shape of a 

 tiixly of revolution, attached to a cylindrical 

 rniddieb(xly or circular section. Some other part 

 of the surface, such as the tail, may be highly 

 irregular, impractical for mathematic representa- 

 tion with any .set of reference axes. 



Instead of covering a whole 8-diml body .surface, 

 the mathematical formulas may be limited to 

 llio.sr> required for the delineation of 2-diml 

 ftatures. A typical ca.se is the fomiula for the 

 int<T.s<-ction of a plane with the .'5-<liml surface, 

 Hwcli U.S the designed waterline on a .ship. Hero 

 the reference,' axis almost invariably lies in the 

 inlerticcting plane. 



JOvcn for the L'-ilinil case it may be convenient 

 to divide the intersection or outhne into two or 

 more parts, with a separate origin or set of coor- 

 dinates for each part, positioned to suit the 

 mathematical formulas employed. 



The term geometric shape defines a body whose 

 outline or surface is represented by some simple 

 mathematical formula. Examples are a cube, a 

 sphere, a circular-.sectioa cylinder, a right circular 

 cone, a symmetrical pj'ramid, a parallelepiped, or 

 an ellipsoid. To achieve simplicitj' it may be 

 necessary to u.sc a particular system of coordi- 

 nates and to establish limits in one or several 

 dimensions, as for the cube iti cartesian coordi- 

 nates. For any geometric shape one such simple 

 expression is usually sufTicieiit. 



49.2 The Usefulness of Mathematical Ship 

 Lines. Before embarking on a discu.ssion of the 

 mathematical delineation of the lines or surfaces 

 of bodies and ships, it is well to answer the question 

 that arises immediately in the mind of the 

 practical naval architect and shijjbuilder: UTiy 

 bother with mathematical lines when faired lines 

 can be drawn .so quicklj' by experienced jjcrsonnel? 



It is ca.sy to give two answers to this (jucstion. 

 In the first place, analysis of the lines of many 

 actual ships, in the form available to the naval 

 archite(!t at large, imlicates that they arc not 

 strictly fair by any criteria, graphical or mathe- 

 matical. In the .second place, experienced per.son- 



186 



