PAN
257
PAN
of cont.° Ve tbe bne AB > produced, and consequently a curve1'he la ''^ flexui- e.
V a nl ^ anoratn ' < ‘ curve takes place when the cylinder is cutP^ ane at oblique angles to the axis.
.-To find the indefinite representation of
Tin 17 — fvUKX uic ut
r 'fs 0 f r , ( !' the original plane,
°f th t,te UTl gmai piane, in a plane parallel to the
(i 0 )i n f e picture; given the height of the eye, the intersec-Pliine frorn °j r ^ na ^ P^ ane , and the distance of the original
'nt,
lr J'Ur
er sei
—^ et p be tbe stat ‘ on poi nt ; g K hi thePlane „ 10,1 °f the picture ; and A B a line in the original
1 Vj 0n 1*1 t ’ O
In ft vvlu i ch ‘he plane parallel to the axis stands.
«o. ] ^l (re 2, No. 5, draw P K, which make equal to PI,Nke k „ ^° - draw p o and k w perpendicular to p k ;
toj> K. to the height of the eye; draw o w parallel
to Aa ) ’ c l n ? c l uce PK to r; in No. 1, draw P r perpendicular
Plane’ vv -,| ''I 1 ® A B at r, and the intersection of the originalto j. _ „ ul1 toe
ft--,
(> *nt B ^t>ce o w, meeting rs at s ; produce kz to meet
i ui'a*" ~
P l£ to r; in No. 1, draw P r perpendicularB at r, and the intersection of the original1 1“ r, 1 " e Panoramic surface at k ; make p r, No. 5, equalt. tu °' I : draw rs perpendicular to K P : in No. 5, make'di.f,.’• fi 3 ’ ec l""l to the height of the several lines, whoserepresentations are required, above the original
lit w ,* draw -° .-
L *, v ., , 0 r ' 0 ° M > an d o v, cutting k tv respectively
‘ n d z. I n jio. 2, make G h equal to the semi-cir-
, ...... ....... draw w v perpen-
tII; make wv,wx,wy, and w z, respectively
to a sec-visual
v, x, y, z, will be ther, t, u, v ; and since r, t, u, v, may be con-
2. In No. 2, make G H equal to the semi-cir-
'’'ciik> ence GKH > No. 1 ; bisect G H at w
to G- ...
I«n* tl> 0 ,0 w x, w y, and to z, No. 5 ; then with the commontins’ G u > and the deflections w v, w x, w y, and to z, describet'el„ ft 1V(;s G v h, G x ii, G y H, GZ it, which will be the de-a "d it tbe representation of the lines required, and G
'«! c (m '^ 1 be their vanishing points. For O p r s, No. 5, mayder; 0 S !; f:reti as a plane passing along the axis of the eylm-tW 0 f,, the axis, o the eye, p the station point, Etc " ~~’ays . oylindric surface, and O r, O t, o u, and O v,^PfesoJl”’ Conse quently, the points
lta «°n8 of .....
tat '0iis as tlle centres of the original lines, whose represen-t'd tl, e ,..paired, and since the optic planes of these lines^"neoi, 0 ^ lr "^ er obliquely, the sections will be elliptical, and,W« a mly ’ tll «ir envelope will be the figure of the sines,
lW eSerib ed.
n We 111 — To describe the representation of a line in“‘"t of J^fcndicular to the axis of the cylinder, giving theIn h; {| e „ < We °n the original plane.
‘ tli e ^ draw g d perpendicular to Git ; make G d equalt or deflection of the curve, and describe the
•t'on of ®’nical curve dw, and die will be the represen-l 'i °f G ® ' ne Pcpendicular to the plane, in which the origi-V^aoe, ® n ; G a- ii, Gy II, and G 2 n, are situated.
- Given the indefinite ^er^-ece^fntie
v>, 2 ,'*" A 1V /~t• “ ' ' - 7 -- -
S( Q , °f a S f 'xiven the indefinite representation, G z H,
]j| s A i>, l d ,lt fine, to determine the finite portion, whose
vi a"*!* ^°‘ I’ outing the intersection at a and
< i a i
4 . 1 '•«* P• " a:
1 N 0 . 2^1°' e< l l, al to the extension of K a, No. 1 ;
t0 t * 16 extens ' on K No. 1 ; draw
Ut
Nion;
yb'i
> perpendicular to G II, cutting the curveaB ,d a> z b 1 will be the finite portion re-
mill __ 7 111 - ... .. . . r .
h.
- , u -xb'", and a>'vb'\ ^ ^Tnd
°f the indefinite representations J >
J^ing examples show the application of the pun
M^ Qn *pt\ reP F eSenta ^ 0 V 0li Ltc C DEr, be the plan ofk" au&e ; ® 1 Figure 2, No. 1.—Let d , 0U „ u t i ie s ta-
S Point cont act Willi the picture at C- 1 , ° intersec-
of ,0 P ’ draw, g h, parallel to D C, cu » a ^ n . ft \ s0f< ' u §l\ n Paaoramic picture in tbe pom 8 ! t j ie pic-
Ver p draw ki, parallel tod), or r C, cutting 13 u
ture in I and K; then G and h are the vanishing points of alllines parallel to E F or D C, and I and K tlie vanishing pointsof all lines parallel to C f or D e ; draw F p and d p, cuttingthe picture at f and d. Let IVY be the ridge of the build-ing, bisecting ed at w, and EC at v ; produce w V to y, cut-ting i k at u ; make UY equal to the height of the house,Nos. 3 and 4 ; and make the angle iiyz equal to half thevertical angle of the roof, and let r z cut i k at z ; draw x qparallel to yz, cutting ki, or ki produced at q ; draw zxparallel tovw; produce e d to x, and F c to meet x z at t ;join Q x and Q t, which are the intersections of the optic planesformed by the inclined side of the roof, and the verticalplanes standing upon de and cr ; draw in parallel to PO,and on parallel to ip. To find the inclination of the opticplanes ; draw p m perpendicular to x Q, cutting x Q at u, andthe panoramic intersection at M; from P, with the distancep u, describe the arc mj, cutting pi at J ; join o J ; produceNI and o J to meet each other in s. In like manner, draw p tperpendicular to TQ, cutting TQ at ti, and the panoramic in-tersection at t; from P, as a centre, with the radius tv,describe the arc v to, cutting ipattr; join ok; produce N Iand O w to meet each other in R; then n o S is the inclinationof the optic plane, whose intersection is Q x; and n o r is theinclination of the optic plane, whose intersection is Q t.
Figure 2, No. 6.—Upon any convenient line, i n, extendthe panoramic intersection ; according to the correspondingplaces of No. I, as shown by similar letters ; that is, 1 1, t G,G d’, d o', o'f-,f k, and k h, No. 6, would cover 1 1, t g, g d’,d! o', C’ f,f k, and K ii, No. 1. In No. 6, draw t it, g o, d d,o' o,f f, k k perpendicular to i n.
In No. 5, produce kp to a; make pa equal to the radiusof the cylinder ; through a draw i o parallel to OP; on pamake Tj equal to P j, No. 1 , and P q, No. 5, equal to p q, No. 1 ;draw j g and q h, No. 5, perpendicular to P a; make q hand j g each equal to the height of the walls, Nos. 3 and 4 ;join og and oh; produce o g to l, and oh to i, meeting oi;produce o j and o q to meet g o at o and k.
In No. 6, make <r equal to nr, No 1; Go equal to go,No. 5 ; g l equal to G l; k h equal to G k ; k i equal to G i;describe the panoramic curves 1 1 it, i o k, g k ii, G i h ; thenthe curves i o k and G k h will cut each other at c, and thecurves ilk, gjh, will cut eacli other at c, so that the pointsc', C, c, will be in the same perpendicular; C c will thusrepresent the angular line of the building. Let the perpen-dicular d'd cut the curve G i h at d, and the curve G k H at d;and the intercepted portion d d is the angular line at one end;in like manner, let the perpendicular f f cut the curve ilkat f, and the curve iok at f; and the intercepted portion f fof the perpendicular f f is the representation of the angularline at the other end ; therefore c c D d represent the front,and c c F f the end, exclusive of the triangular part adjoiningthe roof, which will be formed in the following manner :make tx and xu equal to GK each; describe the samepanoramic curve, rcj; and if the other semi-panoramiccurve, x v w, is described, and if k K be produced to v, v willbe the vanishing point for the gable top ; and if K n be madeequal to k v, n will be the vanishing point of the otherinclined plane of the roof; and thus the representation oflines and planes will have vanishing points and vanishinglines, as in the methods of describing the perspective repre-sentation of objects upon a plane surface ; and if the pointsd, d, c, o,f, f, are found, and the lines c d, CD, and cf, c f,are made straight instead of being curved and produced, theywill find their own vanishing points in the line I H; but moreremote from each other than the points G and K.
What is here observed, is exemplified in No. 7, as willappear sufficiently clear by a little reflection.