SH A

427

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i Be ,.^ a t plane ; y becomes the intersection of the vanish-hor - ltle ^e plane of shade with the vanishing-line of thethe V° n ’ therefore y is the vanishing-point of the shadow ofon la . e W x upon the plane of the horizon. The next planeten’ the shadow is thrown is dbhm; now l ii is thethe 8 .S'hne of the plane dihm, and u is the point wheren i 8 ^. an 'ahing-line of the plane of shade cuts l 1 v ; thereforehe vanishing point of the shadow upon the plane de jin.Co t^tp'aneon which the shadow is projected is the planeanj . now vb is the vanishing-line of the plane c d mo,H. ,,'t intersects the vanishing-line of the plane of shade inpl’ a herefore e is the vanishing-point of the shadow upon thei®ct a C . D11 °’ r -i'he next surface on which the shadow is pro-»ow i . 3 the plane, mnpo, of the roof of the penthouse:of 7 " ‘ ...

th er *i!^ e with the vanishing-line t i. 1 of the plane mnpo;Itjj 3 ° re z 1 is the vanishing-point of the shadow on the plane0> The next surface on which the shadow is projectede nlnna - - - - of the wall:

•s the intersection of the vanishing-line of the plane

18 th°-

e plane, bfgi, of the wall; but u has already beenof t i 11 to be the vanishing-point of the shadow. The planeje c . f r °°f is the last surface on which the shadow is pro-' n 8T : n °' V S 1,1 ’ 8 ' ts van i s l 1 i n S"l* ne > an( l it meets the vanish-itig.' n . e °f the plane of shade in z, therefore z is the vanish-j^POint of the shadow upon the roof,tyjjj Carr ying the shadow of a line across several planes, itsurprising if some little inaccuracy takes placech a j, tl * e oW iquity of intersections; it might be a greatfali s Ce ’ whether, when the part of the shadow, d l, whichp°j tl t ll P 0n the plane b f g l, is drawn- from the vanishing-o Ugh ’ through the point d, it will meet the pole at l, as itto do. To remedy this, begin with the shadow, e d,ct ^" r °ceed in the reverse order, until it meets the line W x1 Il'ich it must in principle, and will not be liable to

v ar-

-it’t'Uchthe

in practice.

its which direct the shadows upon the severaltrtiei also be found by the methods shown in the

The f ° jecti °n-

°f at, Allowing observations will be useful in the practiceHy h a< Ws.

bi, e f n . a ®traight li ne, that throws a shadow, is parallel tol liis n Ur ®» ^ is then represented parallel to the original. In. e Jt has no .vanishing-point; or, in other words, the

'll;

lst arw^ P°* nt t ' ie l‘ ne ma 7 b® sa 'd to be at an infinite

t^e lip e : and, therefore, instead of tiie vanishing-point of'Ifilty e ^eing joined to the vanishing-point of the sun’s rays,(1 , stra ight line from the vanishing-point of those raysto the projection of the line which throws the sha-

s

‘hi

cr ft p a it Will be the vanishing-line of the plane of shade;° f shad tlie intersection of the vanishing-line of the planewith the vanishing-line of the plane on which the^To 18 to he thrown, will give the vanishing-point of the. w on th ’ -

!" . 5 r d

1 ” v ar| ‘ ra )’ s being parallel to the picture: for here, also,tl le ^{‘fog-point of the rays is at an infinite distance; but> w;i?. e °f shade will still have a vanishing-line, this

ap a ir>Tif n *' la t plane, after the same analogy as lines whichto the picture. This case is similar to that of

n 8 ° ,l )01n t of the line that throws the shadow parallelv s4„ 8 r ays, as shown in a former example.

0l vs projected from a given point; as by the light of^ It e a candle or lamp.

* Siven 1 , n * : > if the representation of the luminous pointResent’ * ts 8eat u P on an y plane, together with the(I. atlon of any point in space, and its representation>fi nB . 1 Pi ane > the shadow of the point will be found byPoint .o a .. ... . .

in 8tr aight line from the luminous point through the8 Pace, and by drawing another straight line from

the seat of the luminous point through the seat of the pointin space; and the intersection of the two lines thus drawnwill represent the shadow of the point upon the plane. Butwhen the relation of several planes represented in a picture,the representation of the light with its seat, and the repre-sentation of a point in space with its seat, are given, to pro-ject the shadow of the point on the other planes, other con-siderations become necessary.

Figure 2.—For this purpose, let A b c d be the inside of aroom, consisting internally of the vertical planes A h, e i,f K, G C, and of the horizontal planes aefgb and D iiikc :also, let l be the luminous point, and m its seat in the planeaefgb. In order to form an idea of the point L, in respectof the other planes, it is necessary to have the intersection ofa line drawn through l, in a given position with one of theplanes. Thus, if it is known that the straight line l a,parallel to the picture, cuts the plane of the wall, b k, in thepoint a ; the position of the point l to any of the other planesmay be easily determined, as follows:

Through a draw a b parallel to the vanishing-line, N O, ofthe plane b k, cutting b g, the intersection of the planes b kand A G, in b ; through b draw b m parallel to p Q, the vanish-ing-line of the floor, cutting a e, the intersection of the planesA G and a h, in c; also f e, the intersection of the planes A Gand E I, in d. Draw c Q parallel to N O, the vanishing-line ofthe plane A h ; and d f parallel to e s, the vanishing-line of theplane E I. Then, because the intersecting and vanishing-linesof any plane are parallel to each other, and because a linedrawn parallel to the intersecting line is parallel to the pic-ture ; therefore the representations of all the lines, ab, b c, orb d, c Q, and df are all parallel to the picture, and in a planepassing through the luminous point L.

Figure 3.—Given the vanishing-lines, A b, c d, e f, of threeplanes, ghik,lmno, and M N i Q e, the common intersec-tion, n o, of the planes ghik and lnno; also the inter-sections, n i, and Mn, of the planes giiik and imno withthe plane mniqb; the representation, ab, of a line in theplane L M N o ; the point of light, c ; c d, a line parallel tothe picture ; and d, the point where it intersects the planem n i Q e : to find the shadow of the line on the plane ghik.

First, find the representation of a ray of light parallel tothe picture, thus : draw d e parallel to A b, cutting M n at e ;draw ef parallel to E f : then if a b be not parallel to ef pro-duce b a tof and join f c, which is the ray required. Secondly,find the vanishing-line of a plane of shade passing throughthe line a b, and the ray f c, thus : produce a b to meet CDin d, which is tne vanishing-point of a b ; through d drawd f parallel to f c ; and d f will be the vanishing-line of theplane required. And, lastly, find the shadow of ab uponthe plane ghik, thus: produce on and ab to meet in g;from f, through g, draw the lineF/ii; and from the point jof light, c, draw c b h and c a i ; then h i will be the shadowof the line, as required.

For d e being parallel to A b. the vanishing-line of the planemniqe, de will be parallel to the picture ; and since ef isdrawn parallel to E f, the vanishing-line of the plane i,mno,ef will be parallel to the picture ; and because b a meets ef !in ffc is a ray of light parallel to the picture, meeting :the line a b ; and because c d is the vanishing-line of the plane jl m n O, and ab is in the planeL mno, therefore the vanishing- \point of a b is in C D, and consequently at d, where a b pro-duced meets c d : and because d is the vanishing-point of ab,the vanishing-line of the plane of shade will pass through dparallel toyc ; but f is the intersection of the vanishing-lineof the plane of shade, with the vanishing-line e f of the planeghik, on which the shadow is projected, therefore f is the jvanishing-point of the shadow on the plane ghik; and j