DE S

255

DE S

o\v.

should

gj . ---*»• it be required to construct this intersection

at pip..,, «ists in its plane, and from one of its points, takenpto- 8Ure - tn - -_-- __ ~e

t Section h ' -o—- —.- -i-.- —

"e ttiay ° e at to ° great a distance from the curve bcde,8, ,'PP^ se another vertical plane, parallel to it, passingjjentufjj . le j n terior of the curve bcde, and whose hori-Ais V( , '•Section should be the right line e c, parallel to l m.S par. i| C f^ P^atie will intersect the cutting plane in a rightaitigp^ ^ el to its projection fg, and upon which, as upon af^'ic'al 6 suppose the cutting plane to turn, to becomer °rn as ’ an< l to present in full front the curve required. Then,t^'ti'ari] an ^ Pnints, h, as may be thought convenient, takentli e y upon bcde, suppose vertical planes perpendicularj ert ical ert, . ca l plane of projection, whose horizontal andara ' v in» Ejections will both appear at the same time, inu ‘ ou gh all the points h, the right lines ii jki i' per-ching to l m. Each of these planes will intersect the‘ n a horizontal line, perpendicular to the hinge,,hv 0 rif!| tlCa * P ro j ect ion will be the point of coincidence, i‘,^jll Uieet "i l' ne s fg,ii. In each plane this horizontal line'''H he t|. . hinge in a point, whose horizontal projection'“d trill ? ‘ntersection, j, of the two right lines e c, h j k ii ';i.° n tnl .^° ^neet the curve required in points, whose liori'C H j ,• j ec tions will be the intersections

e > to draw a tangent to it: if the vertical plane of

a ?<l w >th tl

ie curve bcde.

h, K, of the rightIn a word, this right line

,„i P ai 'ts will be equal to their horizontal projections,

J^'cal, a |'i ^‘ e cutting plane turns upon the hinge, to become•h! t>1 iiu ’p these right lines, which were at first horizontal,. l<:r efoi* |4 'Pedicular to the hinge, and do not vary in size.I, ^'culsirs ‘rough all the points i', we draw indefinite per-ia trail sfe,.»’ , I 1 ' , and if upon these perpendiculars J h

pointg i^ m ** t0 an< l J K I" 10 ™ *' to k ,we shall have as

f ei

Je, e' k"c''/i as we w l s h for, through which the required

■fhe

®ay be drawn.

united, f r U,-Ve

l^bl'Sent U)”\{ ° ne ‘ ts P°’ nts > taken arbitrarily, to draw

being constructed in its plane, if it be

till ’ by di 0 ’ - We ma y obtain the vertical projection of such!j ar h i’ . °PP'ng from the point h, upon fg, the perpendi-Pan tl,g d ats horizontal projection, by projecting i! inti lt) ' 0,) of thg UrVe B c D E ; we shall have the horizontal pro-bfj^nt i tl tan gent required, by drawing the right line f n' utitt® '“to tfi ° f' 16 curve bcde; and it will be sufficient toP°in er,t * as ,P ane of the curve any point whatever of theUp 0 1 ft ta| {e lat ’ f° r example, which is projected on theNr y g i n J 1 arbitrarily, and whose vertical projection isb e ■ as U p ' -Now, in making our deductions upon this(i ei . >v lop 8) ev ery other of the cutting plane, it cannot but'^ifiular^' through the point a, we draw to fg thelin e 0 to n ,, a n and that if upon this right line we transfer

from the point n to the right^qp.^ore i, f. 0lnt n will be the second of the tangent.

J “rawing the line h n we shall have the tangent

>1.

"'V l ' ate ve r

it s ot jon e . ^ , ve be given to bcde, we see that ther, f tligfalots t]j C ^ Possesses the property of having for either'p, ? u ^*tangent a n equal to the sub-tangent a nai .'d ollip s ' 1S P r °P er ty, which is very well known in thef a in? ' !,e3 > w'h’ w b en those two curves have a common^npt ''I'^fitiop 1 res pect to them only, from their beinga *' I., lr 'es. 8 df one cylindric surface through two dif-

(j

t e<:t iop Vl! ' <) Ptnenf a ^ P' CCUr , that we shall want to trace upon^“de hv 1-be cylindric surface, the effect of the>n „ <l«w' y ‘be

V,. a j-

u 8h

cutting plane.' For this purpose, afterUlle e curve bcde, with all its divisions,the ^ ’. ‘f indefinite perpendiculars be drawnvisions of kq, we shall have, upon the

development of the surface, the traces of the different posi-tions of the generating line, and we shall then only haveto transfer upon these perpendiculars the parts of the corres-ponding generating lines, comprised between the perpen-dicular section bcde, and the section made by the cuttingplane. Now, these parts of generating lines are equal to theirvertical projections, and these projections are all terminated,on one hand, by the right line l m, and on the other b yfg,If then the point h, for example, fall in s, upon the line r q,by transferring i i upon the perpendicular passing throughthe point s, from s to t, the point t will be, upon thedeveloped surface, that where the generating line, whichpasses through the point H, is intersected by the cuttingplane. The curve xttz, which passes through all thedeterminate points \in the same manner, will be the curverequired.

“ It is obvious, that were the tangent produced from thepoint H till it meet the horizontal trace, G F, of the cuttingplane, somewhere in a point, f ; and that if h f were trans-ferred upon R Q, from s to v, the right line t u would be tan-gent to the curve; for when the cylindric surface is developed,its elements do not alter their inclination with respect to thehorizontal plane.

“ Second Case, wherein the cylindric surface and the cut-ting plane are supposed to be in any manner whatever withrespect to the two planes of projections.

“ Solution.—Figure 28. Let a a.' and a d be the two pro-jections of the right line, to which the generating line shouldbe parallel ;CEDr, the given trace of the cylindric surface;and h g, h b, the traees of the cutting plane.

“ Conceive a series of planes parallel to the generatingline of the cylindric surface, and also perpendicular to one ofthe planes of projection, as the horizontal plane, for example:each of such planes would be projected upon a right line,O K e, parallel to A a', cutting the surface in right lines thatwould be so many positions of the generating line, and wouldmeet the horizontal plane in the points of intersection, e, F,of the line oke with the curve cede. By projecting,therefore, the points e, f, upon l m, in e, f, and drawingthrough these latter, to the right line a a’, the parallelsee',ff', the vertical projections of the intersections of thesurface with each of the planes parallel to the generatingline will be obtained.

“ The same planes will likewise intersect the cutting planein right lines, parallel to each other, all whose horizontaltraces will be upon the different points, o, of the line hg,and whose vertical projections will be also parallel to eachother. To obtain these projections, we must first find thedirection of one of them, of that, for example, which corres-ponds to the vertical plane drawn from A a'. With this view,produce A a' till it meet, on one hand, the trace of the cut-ting plane in a point N, and on the other, the right line l m,in a point b, and by projecting the point b in b on h b, thetwo points N and b will be, upon the two planes of projection,the traces of the intersection of the cutting plane with thevertical one. By projecting the point N in n upon L M, anddrawing the line nb, it will give the vertical projection ofthis intersection. By projecting on L M all the points, o, inwhich the trace gh is cut by the projections of the verticalplanes, which will give a series of points o, and drawingfrom the latter the parallels o ik, to n b, the vertical projec-tions of the intersections of the cutting plane, through thewhole series of the vertical planes, will be obtained. Lastly,the points of coincidence, i, k, of each right line oik with theprojections ee,' f f, of the sections made in the cylindric sur-face by the corresponding vertical plane, will be the verticalprojection of the intersection required; and the curve that