D E S

257

DE S

the ) l0 °. ac °ver the tangent to the point d (Figure 26) ofthat it . 1Zonta l projection of the intersection, we must recollect'he tty 18 t * le projection of the tangent of the intersection ofto ,j et 0 Su rf'aces, in the point corresponding to d, and thati S> np o rm ‘ ne it) it is only necessary to find the point s, whichintersp* 1 horizontal plane, the trace of the tangent of thetoup^ < j tlon - This latter tangent is in the two planes whichfor ^ , l ® c °nic surfaces in the point of intersection ; there-t^apn, .ding the horizontal traces nr.

-mo uoii^uutai u aoco xv / , Q ^ tu tnooo m v

Point g P' an es, they will by their coincidence, determine theit i n a "® ut the plane tangent to the first surface touchesProj^J 11 * tl'at passes through its apex, whose horizontalli

of these two

' ne '4D° n ma y he obtained by drawing the indefinite rightC ' fc ular' if AD he produced till it meet the horizontal

Point ty-]! r . ace °f the surface, tqbt, in a point, q, suchh>ce he a point of the line of contact between the sur-Plttn e vn , e Pi an e ; consequently, the horizontal trace of thet] lp ' he tangent in Q to the circle T Q u v : let this tan-c ittg t jj et °re be drawn, as Q q. In like manner, by produ-hac e> ra dius b d till it meet in r the circular horizontalc ’ r cl e jp 35 ' Y z > of the second surface, and drawing to thisf t5ce 0 r e f | tangeilt i n R > the line R r will be the horizontal} r orn t [ lp , e plane tangent to the second surface.

o _ And if

> thg 6 i JOlnt s > of the intersection of the two tangents Q q,Point ^' ne s D he drawn, it will give the tangent, in“ ^ °f the horizontal projection of the intersection.Point, $ res pect to the tangent to the correspondingi obtin’ °, ^ le vertical projection, it is obvious that it maylin e hy projecting the point s in s, and by drawing‘‘It * w hich will be the tangent required.ev elorw ^ happen to be necessary to construct upon the

Patent of n„» „<• -

ha

4 . -

’WtV“ ent °f one of the conic surfaces, perhaps, even^ctir* tof ea eh of them, the effect of their mutual mter-

c <C : - '

f ^ 8 t operate for such cone in the manner we are“ Bpf Ascribe for the first.

«*rf " ? re ' v e proceed, let it be observed, that when a conic'hat ai- 18 ^ eve loped so as to become a plane, the right lines0 f U P° n h, change neither their form nor size, because' v sln t'’ em is successively the hinge upon which thehbnain , en t acts ; SO that all the points of the surface"[hen ,, a .' Va y s at the same distance from the apex. And$ th e ln this case, the conic surface is direct and circular,^tarip p f°' nts of the circular horizontal trace are at equal

^tnl a; ,' r ° m the apex ; they must, therefore, be also at anNw sta t>ce -- — - - - -*

H # - v-ciun oi mem, tne enect ot tneir mutual inter-

of *? r exam ple, if we were obliged to fabricate thevp _ exi ble substances, such as metal plates : in this

I t 0

-rnnce from the apex upon the development, anda u Pon the arc of a circle, whose radius is equal

7'1 if ';.' lr tn distance of the apex from the circular trace.

Hi ! Wx ter hav ing taken an arbitrary point to represent' tl >« J '

ft

Of

%

■"“Us ■ ^ eve '°pment, the arc of an indefinite circle,'ts p„. . equal to a c, be described, having such point

« h,

fa, liu^eiiti-e

a, -ori,, u will also be indefinitely the development; off C of tl tUl trace of the surface. Then, by measuring theK 0tn the Wrcle tq on the arc just described, beginningthe f mt T of the trace from which it is designed toNopl^oWment, the position of the point Q upon them will ’ - ‘

Nl f,. 0 1 determined; and the indefinite right6 tw** S point to the centre of the development,

O M positionit

if, A " ,J11 occupied by the right line of the surface

°ti) v re ferr f ,.] aa< f upon which the point d, d, of the.? ftitib I ' e li a i ns 0 will be found. To construct this point,a en Ure it u n ° c ^ scover its distance from the apex, andof tu° n , *h e indefinite right line, beginning from, Jh . ^opuient. In order

1 i

itie

ciit a / t ’ * rom the point d in the vertical projec-ta e side a c of the cone, in a point k, and the

3 u

line a k will be the distance sought for. By constructing,in like manner, all the other points of the intersection, suc-cessively, and passing a curve through them, the intersectionof the two surfaces described upon the development of thefirst surface will be found. Proceed in the same mannerfor the second surface.

“ Fourth Question, —To construct the intersection of twoconic surfaces, of whatever bases.

“ Solution. — Figure 30. Let a, a, be the projections ofthe apex of the first surface; cgds', its given trace uponthe horizontal plane ; b b, the projections of the apex of thesecond surface; and E H F h', its trace on the horizontalplane. Suppose a right line passing through the two apices,whose projections are the indefinite lines a b, a b, and whosetrace, i, may be easily constructed upon the horizontal plane.Along this line conceive a series of planes, each cutting thetwo conic surfaces in the system of several right lines ; andsuch of these right lines as shall be in the same plane willdetermine, by their intersections, so many points of theintersection of the two surfaces. The horizontal traces ofall the planes of this series will necessarily pass from thepoint i; and since the position of these planes is besidesarbitrary, their traces may likewise he taken arbitrarily, bydrawing from the point i, a number of lines, i k, at pleasure,by each of which the following operation may be workedfor either of them.

“ The trace, k i, of each of the planes of the series willcut the horizontal trace of the first conic surface in points,g g', which will also be the horizontal traces of the rightlines, according to which the plane cuts the conic surface :thus a G, A g', will be the indefinite horizontal projections ofthese lines, and their vertical projections will be obtainedby projecting g, g', in g, g', and drawing the indefinite linesa g, ag 1 . So, likewise, the trace, k i, of the same plane ofthe series will cut the horizontal trace of the second conicsurface in points, h, h', from which if b ii, b h', be drawnindefinitely, the horizontal projections of the lines will beobtained, according to which the same plane of the serieswill cut the second surface ; and their vertical projectionsmay be had, by projecting h, h', in h, A', and drawing theindefinite lines b h, b h'.

“ This done, for the same plane, whose trace is k i, weshall have on the horizontal projection a certain number oflines, a g, a g', b h, bh'; and the points, P, Q, r, s, in whichthose belonging to one of the surfaces, meeting those belong-ing to the other, will be the horizontal projections of so manypoints of the intersections of both surfaces. Thus, by work-ing successively, in the same manner, upon the other lines k i,we shall find new series of points, p, Q, R, s; and by after-wards passing through all the points r, a first branch of acurve, through all the points Q, a second, through all thepoints r, a third, &c., we shall have the horizontal projectionof the intersection required.

“ In like manner, for the same plane, whose trace is k i,we shall have upon the vertical projection a certain numberof lines a g, a g[, b h, b h', whose points of intersection willbe the vertical projections of as many points of the inter-section.

“ Here let it be remarked, that it is not necessary to con-struct the two projections independently of each other, andthat having constructed a single point ol one of them, it maybe projected in the other upon one of the lines that oughtto contain it: hence we acquire means of verifying theoperation, and of avoiding, in certain cases, the intersectionsof lines which cut each other in angles too oblique.

“ To find the tangents to the horizontal projection, that,for example, which touches it in the point p, we must