D E S

245

DE S

? an y d •

nori-jonf ]' nts contact > which G will be the commonWin | )e p Projection. All these imaginary points of contactUp° n ou nd in the plane of the generator, by carrying a gline p a ^ rom a to e i and drawing through the point e aCuts ji a e ' to a a ; all the points, e, c, in which this linegep e , , e curve bcdbi, will be the intersections of thepoi„ t G ln g curve with the vertical line drawn through theto ntilet - an d will indicate the altitudes of as many points ofPtojeef a j ° Ve the horizontal plane. To obtain the vertical

Poi,

8Ud;

ntg ° lls °f these points of contact, draw through all theh .’ c > indefinite horizontal lines, which will containPerp,. llt j°-1 ect i° n s ; and as they are also contained in the linetiorig „ '? U ^ ar to L m, drawn from the point g, the intersec-P r °je'cr ’ tliis line with the horizontal lines, will be the“ ^q 0118 . tbe several points of contact.

C( :iv e ,j Vr> from each point of contact, a section be con-b* c’^e by a horizontal plane, such section, which maye, ic e 0 p eie d us a second generator, will be the circumfer-Hi t ], , a circle, whose centre will be in the axis, and of®ktr eill j 16 tangent, which must be perpendicular to theve ,..? , 0 ^ the radius, will also be perpendicular to*biind • p P' ane drawn through ag, in which the radius ishiis tu„ leie f°re the tangent plane, which must pass throughWane, arid nt> " be also perpendicular to the same vertical^ r Pc>uli have, upon the horizontal plane, its trace

' :a<; l‘ of" ] t° A G. We only want, therefore, the traceq ' st aiiPQ t > 1 le tangent planes, to enable us to discover its' i ce from the

point A : now, if through the points E, C,

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t!l1 tl.GV the first generator the tangents ei, c ri, produced8( lUul p eet L M in the points i, h, the lines ai, an, will bec r l0Se distances ; therefore, if these lines be trans-* A ° tu A to *> an d from A to h, and if through the pointsperpendiculars i Q, h p, be drawn to A G, and

frodi* ,4 ’ thetill

Plan

they meet the line l m, we shall have, on the

Wan* fi, 'dto,.,, s> w e

the traces of all the tangent planes.

on the vertical plane, the traces of the same

!['® v «n t ' < !.T t tan S en t plane, a horizontal line produced tot Q, Plane of projection; this line, which is thet| 0|| gs t 0 i c ‘ rc l e > will determine, on this plane, whichlj ne ° , t l(; trace. Now, for all the points of contact,t^ 'b'utyij l , ave tbe same horizontal projection, viz. the linek 4rt >iii U ti llff 10111 l’* 16 point G, perpendicularly to A G, and; U| »t tc, # m the right line I. M. If, therefore, from the

it ^.'"definite perpendicular, K k k\ be drawn uponr, *°nt{d i. Con tain all the points of coincidence of theb|«se n ? es with the vertical plane of projection. Butth ^hve | nt . s °1' coincidence will also be found on thetj^ bite ( . Se ’P^zontal lines drawn through the points E, C ;

ling K^'° nS .^' suc b horizontal lines with the ver-^8e«t pj ’ will be each a point of the trace of one of theWl'* 5 ’ die tr eS " di’hus the line Q k will be on the verticalfSr e tl iilt ac ® of one of the tangent planes ; the line p UMother : and so of the rest, were there a

vy e “j b er.

Hu^'biig a confine ourselves for the present, to the threeof artl ples, because they are sufficient for all the

fiuj l ’ s Wc)j,] 5 08e generation we have defined. In the courseof tr ? b We s ball have occasion to investigate the gene-ra pr es ® s °f surfaces, infinitely more numerous : andthemselves, we shall apply the same method to*o ^j. a d°n of their tangent planes, and of their

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Hv,

Present we are about to propound a question,y | )f; ri °f which the consideration of the tangent

(l -m'*® appropriately and usefully applied.r ‘> by t , question..—Figure 15. Two right lines beingleir horizontal projections, ab, CD, and by their3 R

vertical projections, ab,cd; to construct the projections,p n, p n, of their shortest distance ; that is to say, of the linethat is at one and the same time perpendicular to both ; andto find the quantity of this distance.

“ Solution. —From the first of the two given right lines,conceive a plane parallel to the second; which is alwayspossible, because if from any point whatever of the first, aline be drawn parallel to the second, and if this third line beconceived to move parallel to itself along the first, it willgenerate the plane spoken of. Imagine, also, a cylindricsurface, with a circular base, having the second given rightline for its axis, and the distance required for its radius ;this surface will be touched by the plane in a line parallel tothe axis; and will cut the first right line in a point. If fromthis point, a perpendicular to the plane be drawn, it will bethe line required ; for it will pass, in fact, through a pointof the first given right line, and will be perpendicular to it,as it would to a plane passing along this right line ; it willalso intersect the second right line perpendicularly, becauseit will be a radius of the cylinder, of which such second lineis the axis.

“ It remains, then, only to construct, successively, all theparts of this solution.

“ (1.) To construct the traces of the plane drawn throughthe first right line parallel to the second, we must first findthe point, A, wherein this first line meets the horizontalplane, and which will be a point of the horizontal trace. Toeffect this, produce the vertical projection ba till it cut theline l m in the point a, draw a A perpendicular to l m, and jits intersection with the horizontal projection a b willdetermine the point A. Through the point in which the firstright line cuts the vertical plane, whose projections are b b,conceive a right line parallel to the second given right line,and construct its projections by drawing, indefinitely, B Eparallel to C i>, and b e parallel to c d. In like manner, con-struct the point, E, of coincidence of this parallel with thehorizontal plane, by drawing e e perpendicular to Li; andthe point E will be a second point of the horizontal trace ofthe plane. Then, if the right line A E be drawn, and pro-duced till it cut, in the point f, the line l m, it will give thehorizontal trace ; and it is evident, that if through the pointsf and b, the right line f b be drawn, we shall have the traceon the vertical plane.

“ (2.) To construct the line of contact of the plane withthe cylindric surface ; from any point of the second rightline, which is the axis of the cylinder (as from the point c,for example, in which it meets the horizontal plane) drop anormal, that is, a perpendicular upon the tangent plane ; andthe foot of such normal will be a point of a line of contact.

“ To find this foot, according to the method already laiddown in Figure 6, first construct the indefinite projections ofthe normal, by drawing through the point C, the line h gperpendicular to the trace A E, and through the point c,the line c k, perpendicular to the trace f b ; then havingproduced h g till it meet A e in the point g, and i. m in h, pro-ject the point G in g, and the point H in h, on the trace f b;draw the line gh, whose intersection with c K will determinethe vertical projection, i, of the foot of the normal; and weshall have, on G ii, the horizontal projection of the same point,by letting it fall perpendicularly on I. M. The projection <,t, i, of the foot of the normal being found, draw i n throughthe point I, parallel to C d, and i n, parallel to c d, and weshall have the projections of the line of contact of the planewith the cylindric surface. Lastly, the points N, n, in whichthese projections meet those of the first given right line, willbe the projections of the point of such line, through which thecommon perpendicular required will pass.