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254

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the traces themselves, and which it would be necessary toconstruct from points. But were choice to he made of thesystem of planes parallel at the same time to the generatinglines of the two surfaces, each of such planes would cut thetwo surfaces in right lines, and these lines would cut eachother in points appertaining to the intersection of the twosurfaces. By this means, the points of the horizontal projec-tion would be constructed by the intersections of right lines.Indeed, this is but a necessary consequence of what has beensaid relative to the case of two conic surfaces.

“ For two revolving surfaces, having their axes in thesame plane, but not parallel to each other, the system ofspherical surfaces having their common centre in the pointof coincidence of the two axes, would be preferable to that ofplanes ; for each of the spherical surfaces would cut the tworevolving surfaces in the circumferences of two circles, hav-ing their centres upon the respective axes, and their planesperpendicular to the plane drawn along the two axes •. theintersecting points of these two circumferences, which wouldbe at the same time on the spherical surface and upon thetwo revolving surfaces, would belong to the intersectionrequired. Thus the points of the projection of the intersec-tion would be constructed by the coincidence of the circleswith the right lines; in which case the most advanta-geous position for the two planes of projection is to haveone perpendicular to one of the axes, and the otherparallel to both.

“ These few observations, with respect to such curved sur-faces as most frequently meet together, will suffice to showhow the general method should be employed, and how, by aknowledge of the generation of curved surfaces, that speciesof sections may be adopted which will yield the most readyconstructions.

“ When the respective form and positions of two curvedsurfaces are defined, not only is the curve of their intersec-tion in space determined, but also all the affections of thesecurves immediately follow. Thus, for example, in each oftheir points the direction of their tangent is determined :it is the same as to their normal plane, viz., of the plane thatcuts the curve in a right angle, and which is, consequently,perpendicular to the tangent at the point of intersection. Aswe shall have frequent occasion to consider planes normalto curves of double curvature, we shall not here enter intoany detail as to their determination ; for as they are alwaysperpendicular to tangents, it is enougli to have given themode of constructing projections of tangents to the intersec-tions of curved surfaces.

“ Second general Problem .—From a point taken at plea-sure upon the intersection of two curved surfaces, to drawa tangent to sucli intersection.

“ Solution. The point chosen upon the intersection of twocurved surfaces, is at the same time upon both such surfaces.If then, from this point, as considered on the first surface, atangent plane be drawn to such surface, it will touch theintersection in the point in question. So, likewise, if fromthe same point, as considered on the second surface, a tangentplane be drawn to sucli surface, the plane will touch the inter-section in the point under consideration. The two planes,then, will touch the intersection in the same point, whichwill also be one of their common points, and consequently oneof those of the right line in which they intersect each other:therefore, the intersection of the two tangent planes will bethe tangent required.

“ From this problem arises the following observation, whichis of great utility in descriptive geometry.

“ The projection of the tangent of a curve of double cur-vature is itself tangent to the projection of a curve; and its

the projection of that of the cur

of

cuf'

point of contact

double curvature. . , g

“ Thus, if from all the points of the curve of dou fit®vature, perpendiculars be supposed to fall upon oneplanes of projection, as upon the horizontal plane, jo li>!pie, all such perpendiculars will be upon a vertical v er/surface, which will be cut by the horizontal plane in jj th®projection of the curve. In like manner, if, f 1-0111 e vet'points of the tangent to the curve of double eurvatu^jcnjtical lines be supposed to drop, they will be m ® . 0 „ ol

plane, intersected by the horizontal plane in the p r0 J j p|aO ethe tangent. Now, the cylindric surface and the vei ve r*evidently touch each other in the whole extent ot i«

tical line dropped from the point of contact, and ^ tb ecommon to them both: the intersections, there to > | 8 ne,

cylindric surface, and of the plane along the hor

rizontal p

oftl*

touch each other in a point that will be the intersecti ] #n 0.line of contact of the cylindric surface and the ver j 10 c urV 9 'Consequently, the projections of a curve of n° u otherture aqd of one of its tangents will touch each ^ tl>®point that is the projection of the point of concurve. g# id 10

“ We shall now proceed to apply what has bsome particular cases ; and to begin with simple ^ a pi# 11 ® 1tions, we shall first suppose one of two surfaces towhose intersection we would determine. . 0 fag* vet>

“ First Question .—To construct the intersectioncylindric surface with a plane of a given position. ^ t0

“ The position of planes of projections being a* ^ t"will first suppose, what is always possible, ^!npndic ul , lir ,l!rplanes have been so chosen as to have one p el P e . neI i<l ,cU . 1the generating line of the surface, and the other P^gtruc'^fto the cutting plane ; for in this supposition the c j^pit 0 .is much more easy ; and then to give to studen s ^ of P r °making projections, we will suppose the two pjection in any other manner. . jj n e 0 „(

“ Solution. First case, in which the generating pla 11 ® 9 ^surface is supposed to be perpendicular to one o a udprojection, as to the horizontal plane, for exa r

right line, to which the generating line of the cj ^jeeti®^

.ivlii® 11 ^'

- ef

as to the horizontal plane, for

cutting plane perpendicular to the other. . ,j 0 n

“ Figure 27.—Let A be the horizontal proje ■ . .j,. t

a . . -C .i.„ cvi 111 . ,

nTuTti«*!"' !>yS be a a" its verticil

the horizontal 'nr t,le c / in(i ric surface,

quentlv that of H° JeCtl ° n oft,,e indefinite surface,

vertical pfoLetio^ T',? ° f "“action ; let/.? ft, b®that of *1) * le Cl *tting plane, which

trace of the ’ ntei s ® ction requi,^ and F o the ho* 0 t0the curve bo nt P, “f If is evident, that iftangents e e" C P er P er, dicular to l m, the ’ t l] e

vertical Jd J' ’ ‘ he ri « ht lines « e», c c", w’ 11, ! 0f "“tfroming line, in

the verticil m’ *• ® cu , ttln g plane, will terminate P“Now '. P.oj'-Ction of the required intersection- . te rs® <r- ■ . Vom a P 0l,| t taken arbitrarily upon the « jl,

' • • will be

tion (a point whose horizontal projection "taken at pleasure on the curve B c u E ’ anl j n t H’It th* 1 !projection may be had by projecting the p°

vvliOi

se

up 1

, 0 ®

.aUoev' 11 ^

f g) we would draw to this intersection, it /pla'/.jsOi

such tangent should he comprised in the ^ i; \\eJ9*..a

.1 F lie right It' 1 /pliiti®.^

that its vertical projection would be the 1 I s "’ n ,rent P’ ^lii® 11that it would he comprised in the vertica | . ( ,j'ectio"’j )g tl>®the cylindric surface, and that its horizonta P^ vV ,,n Id p# 1will he the same with that of the tangent p ^ * cU rV^right line f 11 N, tangent in H to theThus all is determined as to the intersection