41* 



PBRSPKCTIVE. 



PERSPECTIVE. 



If B be at A, the intersecting point of B B" (74) ; R and iU image 6* 

 will ooineide. If B lie in that part of A B which U on the contrary >de 

 of the plane of the picture to that on which the vertex U situated, it* 

 image o will lie between the intersecting and vanishing poinU, A, P, of 

 the line : and if B be supposed to recede farther and farther from the 

 former, the nearer to the latter will its image approach ; so that the 

 vanishing point is the limit of the successive images of points, farther 

 and farther distant from the vertex, or it may be considered as the 

 image of an infinitely distant point in the original line. 



79. If the point M be situated between the intersecting and station 

 points of the line, its image 6' will be on the contrary side of the inter- 

 acting point to that on which the vanishing point P is situated ; and 

 if B be the station point D of the original line, it can have no image, or 

 it* image may be considered at an infinite distance from the vanishing 

 point in either direction. 



80. If the point B" lie on the contrary tide of the vertical plane to 

 that on which the plane of the picture is situated, its image A will lie 

 on the contrary side of the vanishing point to that on which the inter- 

 secting point is situated ; and, as before, the vanishing point may be 

 considered as the limit of the images in this direction ; or as the image 

 of a point in the original line at an infinite distance from the station 

 point in either direction. 



81. Let two or more original lines be conceived as lyingin an original 

 plane T Z, and suppose a plane w, which will be termed the ranitliimj 

 plane of the original one, to pass through the vertex parallel to that 

 original plane. The lines T z, D E, in which an original plane cuts the 

 plane of the picture and the vertical plane, are called the intersecting 

 and Italian Una, respectively, of that plane ; and the lines w r, T v T, 

 in which the vanishing piano cuts the same two planes, are called the 

 rani*hin;i line and parallel of the vertex to that original plane. 



82. The intersecting, station, and vanishing lines, and the parallel of 

 the vertex, are all parallel to each other, these four lines being the 

 mutual intersections of two parallel planes by two other parallel 

 planes. 



83. The intersecting and station points (74) of any original lines, 

 lying in one plane, are points in the intersecting and station lines of 

 that plane : and the vanishing points of the same original lines lie in 

 the vanishing line of that plane ; for the radiols of the originals must 

 lie in the vanishing plane of that in which the original lines lie : and 

 these radiols must form with each other, and with the parallel of the 

 vertex, angles respectively equal to those which the original lines form 

 with each other and with the intersecting or station lines. 



84. The vanishing line and parallel of the vertex to any plane will 

 be those also of all planes whatever which are parallel to the first : and 

 the radial and vanishing point of on original line will be those also of 

 all lines parallel to the first, whether they lie in one plane, or in dif- 

 ferent ones. 



85. The perspective projections, or images, of any number of original 

 parallel lines, will be either parallel lines, parallel to the originals (73), 

 or will be lines passing through the respective intersecting points of 

 those originals (77), and through their common vanishing point; and 

 the points in which the indefinite images of original lines, not parallel, 

 out one another, will be those of the points in which the originals cut 

 one another. 



80 It has been shown that the image of a line is parallel to that 

 line's director ; if, therefore, two or mure lines have a common station 

 point, and consequently a common director, or if the station points of 

 two or more lines lie in one director, the imsges of those lines in either 

 case will be parallel lines ; and in these coses only can original lines, 

 not parallel, have parallel images. 



87. The ratios which exist between the definitive images and the 

 original segments of lines are easily deducible, cither geometrically or 

 analytically ; but as these theorems do not lead to rules of frequent 

 practical utility, we forbear, with one or two exceptions, entering into 

 them. Let B* B represent a finite portion of an original line, bisected 

 by the point B', then the rays y B', v B', v B, and the radial v p of B" B, 

 will be harmonica! lines ; the definite image of the original line will 

 consequently be harmonically divided by the images b', b, of B' and B, 

 and by the vanishing point p. Conversely, if any segment of an 

 indefinite image of a line be bisected by a point, the segment of the 

 original line between its station point and the original of the image 

 farthest from that station point will be harmonically divided by the 

 originals of the other two points. If the point which bisects a finite 

 line A B be the station point of that line, the image a 6 of A B will be 

 bisected by the vanishing point p. 



88. If an original finite line A B be parallel to the plane of the 

 picture, its image at will be to AB in the ratio of the dittanct of the 

 piettu-t (70) to the perpendicular distance of the plane, parallel to the 

 picture in which A B lies from the vertex ; and if A B be divided by a 

 Itoint D in any ratio, the image o 6 will be divided by </, that of D in 

 the same ratio. 



89. If an original plane figure be parallel to the plane of the picture 

 the image of that figure will be similar to the original ; its periphery 

 will be to that of the original, in the ratio of the dutance o) the 



Ws hall continue to employ tbt urn* conventional not. lion a. before 

 i!mlmuilnf u oriel... I point and line by c.piui letter., sad their taunt bv 

 .null, thui a i It the Image of A s, snd to on. 



to the perpendicular distance of the plane of the original figure from 

 the vertex ; and the area of the image will be to that of the original, as 

 the squares of these lines. If, therefore, the distance of the picture be 

 equal to that of the original parallel plane from the vertex, the image 

 of an original figure in that plane will be equal, as well as similar, to 

 the original : this may occur if the original plane coincide with that of 

 the picture, or if the vertex be at equal distances from both, and lie 

 between them ; or if the vertex be infinitely and therefore equally 

 distant from both on the same aide.* 



00. If an original plane, or planes, be parallel to the plane of the 

 picture, their vanishing plane will coincide with the vertical plane : no 

 such planes can therefore have any vanishing line. 



81. If an original plane, or planes, be perpendicular to the plane of 

 the picture, their vanishing plane will pass through the distance of the 

 picture (70) ; consequently the vanishing line of such plane, or planes, 

 will pass through the centre of the picture. 



92. If an original plane pass through the vertex, its vanishing plane 

 will coincide with it ; the intersecting and vanishing lines will there- 

 fore coincide in on*, as will also the station lines, and ]rallel of the 

 vertex ; and the images of all lines and plane figures, in such an 

 original plane, will coincide in one line, that in which the plane itself 

 cuts the plane of the picture. 



93. The vanishing planes of two original planes will form the same 

 dihedral angle that the original planes form with each other, and the 

 line in which the vanishing planes intersect will pass through the 

 vertex and be parallel to that in which the original planes intersect 

 each other ; it will, therefore, be the radial of this latter-named inter- 

 section. The intersection of the two vanishing planes, or this radial, 

 will out the plane of the picture in the vanishing point of the inter- 

 sections of the original planes, which vanishing point will obviously be 

 the intersection of the two vanishing lines determined by the two 

 vanishing planes. 



94. It follows, therefore, that the line in which two original planes 

 cut one another will have for its vanishing point that in which the two 

 vanishing linee of the original planes cut each other, and that the 

 intersecting point of the common intersection of two original planes 

 will be that in which the intersecting lines of those planes cut each 

 other. 



95. Every vanishing plane is supposed to have on auxiliary one, per- 

 pendicular both to it and to the plane of the picture, and therefore 

 passing through the distance of the picture : this auxiliary vanishing 

 plane will cut the plane of the picture in an auxiliary ranithing line 

 perpendicular to the principal one, and passing through ilt centre, and 

 also perpendicular to the intersecting lines of the original planes. The 

 line in which the auxiliary vanishing plane cuts the principal vanishing 

 plane is termed the principal radial of the original plane, or planes, to 

 which the vanishing plane pertains. This principal radial is obviously 

 perpendicular to the principal vanishing line, and meets it in its centre, 

 which will consequently be the vanishing point of all lines in the 

 original planes perpendicular to their intersecting lines. 



96. The principal radial will form, with the distance of the picture 

 and with the auxiliary vanishing line, angles equal respectively to the 

 complement of the angle, and to the angle itself, which the original 

 planes make with the plane of the picture. 



97. The auxiliary radial of any vanishing plane is one lying in the 

 auxiliary vanishing plane, and perpendicular to the princi[l radial : 

 this auxiliary radial is that of all lines perpendicular to the original 

 planes, the common vanishing point of which is the point in which the 

 auxiliary radial meets the auxiliary vanishing line. This auxiliary 

 vanishing point is the image, or projection of the points in which the 

 auxiliary radial intersects all the original planes to which the principal 

 vanishing plane is common. 



98. The auxiliary vanishing plane, being perpendicular to the 

 original planes, as well as to their vainishing plane, and to the plane of 

 the picture, will intersect those original planes in lines pcrpondk-nl.ir 

 to their intersecting lines, and parallel to their principal radial. 



99. It follows from these definitions, that the vanishing lines of all 

 planes perpendicular to one or more parallel original planes will pass 

 through the auxiliary vanishing point of those planes. 



100. If the original plane, or parallel planes, be perpendicular t,> th,. 

 plane of the picture, their principal radial will coincide with the 

 distance of the picture. Their auxiliary radial will bo parallel to the 

 plane of the picture j and the vanishing lines of all planes, perpen- 

 dicular to the original planes, will be parallel to each other, and 

 perpendicular to the vanishing line of the original planes. 



101. If a circle be supposed, described in an original plane on the 

 point, as a centre, ill which the auxiliary radial cuts that plane, all 

 lines touching that circle will be intersections with that original plane, 

 of vanishing planes of other original planes, inclined to the first in a' 

 certain angle. The point in which any one of these tangents to the 

 circle cuts the intersecting line of the first original plane will there- 

 fore be a point in the vanishing line, to be determined by each such 



* From Ihin theorem the relation between perpcctirc projection, nml pro. 

 jection on a co-ordinate plane by parallel linen, ciihcr pcrpendicuUr or obliyue to 

 that plane, will be at once perceived j the latter being the limit of th 

 image, ai the orte of the converging ray. may be auppoeed to become more 

 and more dlatant from the original : or the former may be considered ai a 

 peripeeUre projection, the vertex being at an infinite distance. 



