MECHANICAL DRAWING CUHVED SURFACES.] APPLIED MECHANICS. 



789 



through this example, will master most others without 

 great difficulty. 



UNFOLDED SURFACES. Besides the method of 

 drawing objects by projection, it is sometimes useful to 

 represent surfaces developed or unfolded. This may be 

 clearly understood by conceiving that a pattern of some 

 kind say a series of squares, as a in Fig. 26, is wrapped 

 or folded round the roller 4. We shall suppose that the 

 Fig. 26. 



piece of a flat paper, or fabric, containing the pattern, 

 will fold exactly round the roller so that the edges meet. 

 The projection of this pattern, as it would appear in a 

 side elevation of the roller, may be drawn as in Fig. 27. 

 Observing that the pattern is divided into three parts in 

 height and five in width, we draw an elevation and a 

 plan of the roller, as in Fig. 27, dividing the elevation 

 into three equal parts in height, and dividing the cir- 

 nce of the circular plan into five equal parts by 

 the points a, b, e, d, e. Tracing up lines from such of 

 these points as would be visible, looking on one side of 

 the roller, v'z. , a and 6, so as to cross the horizontal 

 Hues on the elevation, we get at once the elevation of 

 the pattern as it would appear in projection when folded 

 round the roller. This would really be a process of en- 

 velopment, for we have first drawn the pattern on a 

 flat surface and then enveloped the roller in it. The 

 process of development is just the converse ; for it con- 

 sists in developing or unfolding a surface already on the 

 body, and then spreading it out flat. Suppose, for 

 instance, that we had the elevation of a roller, such as 

 Fig. 28, and we desired to develop the pattern which is 



Fig. 27. 



Fig. 28. 



plex, or made up of curved lines, we should have to trace 

 the development of a greater number of points in order 

 to get a correct drawing of it as it would appear when 

 unfolded. 



We may here observe that a conical surface is develop- 

 able as well as that of a cylinder or roller ; and the me- 

 chanical draughtsman has seldom occasion to deal with 

 the development of any surfaces except these. 



Fig. 29. 



The development of a conical surface 

 is effected thus (Fig. 29). Since all the 

 straight lines drawn from a, the apex of 

 the cone, to the circular base are exactly 

 equal, we have only to draw a circle from 

 any centre c with a radius c d equal to 

 ab, the length of any one of those lines. 

 Then measuring round the circular plan 

 of the base from any point e, and 

 setting off the same measurement 

 from a point d to / on the circle we have described 

 from the centre c, and drawing the straight lines c d and 

 c /> we 8 et the development of the conical surface. A 

 piece of paper cut to the shape of the circular sector cdf 

 would exactly fold round the cone, the edges c d and c f 

 meeting. To develop any pattern, we should proceed, 

 as in the case of the cylindrical surface, to find the de- 

 velopment of a number of points. We shall show the 

 method of proceeding with one point marked g in the 

 elevation. Through the point g draw a parallel to the 

 base g k, and a line a g from the apex meeting the base 

 in 1. Project I down to I, on the plan, and measuring 

 round the circumference from e to J, , set off the same 

 measurement from e, to I, on the development, and 

 draw the line c Z, , which will be the development of the 

 line a I. Then from the centre c with a radius a k, draw 

 a portion of the circle cutting c l t in g^, and g 1 will be 

 the development of the point </, for it is in the developed 

 line c l t , and at the proper distance from the apex. A 

 similar process may be adopted with respect to any 

 other points, and thus a pattern of any kind might be 

 developed. 



To develop the fnistrum of a cone that is, a cone with 

 a portion of it towards the apex cut off, as in Fig. 30 

 wo imagine the cone completed, as indicated by the 



Fig. 30. 



represented in projection on its surface. We 

 draw any required number of horizontal lines, 

 at a, b, c, d, e, through the marked points 

 of the pattern, and trace perpendicular lines 

 from the same points till they meet the circum- 

 f'-rciice of the circular plan in the points 

 /, fj, h, k, I. AVe then draw a straight line of the length 

 of half the circumference of the circle, and make it the 

 base of an oblong of the same height as the roller. 

 Dividing the base by the points f 3 g 3 h s , into parts of the 

 game lengths respectively as the parts of the half circum- 

 f'Ti'iice separated by/ j h in the plan, and dividing the 

 height, a e, by the points 6, c. rf,, into parts correspond- 

 ing with the divisions of the roller in height ; and 

 through all these points, drawing parallels and perpen- 

 flirulnrs to the base ; we get the positions of the different 

 points/, 0, /i |, <tc., in the elevation, as they should be 

 in tlie development, anrl cm trace, through these points, 

 the lines of the paUern. Should the pattern be coin- 



dotted lines in the elevation, and have only to develop 

 the complete cone and cut off the wanting part, leav- 

 ing the development of the actual surface of the frus- 

 trum. 



INTERSECTIONS. In consequence of the facility 

 with which materials can be shaped into coues, cylinders, 

 or any forms which have circular sections, and for other 

 reasons connected with strength and fitness, these forms 

 are generally used in practical mechanics. It is there- 

 fore most important for the draughtsman to be expert 

 in the projections and developments of such forms. 

 Among the numerous problems in drawing them, there 

 is an interesting, and often useful, class depending on 



