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GENERAL PRINCIPLES. 1. The business of Perspective, when divested of all irrelevant conditions, is, simply, to solve the following problem :
Given the projections of a point on the horizontal and vertical planes, and likewise those of another point, on the opposite side of the vertical plane to find the intersection of the line joining them, upon the vertical plane.
Thus, let PQ be the picture-plane; E the place of the eye, given by the projectors EF, EO, or by FK, KO; G similarly the place of “ the point to be put in perspective,” given by the projectors GH, GM, or by ML, LH: then the problem is, to find on the plane PQ, considered as the plane of the drawing, the intersection of the line EG with the plane PQ.
2. The figure above given may be considered as a general picture of a model of the planes, points, and lines, drawn by parallel projectors upon a plane different from those upon which we are required to operate; in fact analogous to the figures employed in the pure theoretical geometry of three dimensions. Very simple considerations, however, enable us to make a figure more in accordance with the objects we have in view. Thus :
Let LQ denote the same line as in the former figure, viz., the intersection of the picture plane with the ground or horizontal plane; denote by e, e, the point E, so that ee, EO FK, and ee; = EF = OK; and similarly G by 9.9,. We then have the problem converted into a very simple one of Descriptive Geometry, one of the planes of projection being that on which the perspective is required to be made.
The actual construction then becomes simply this :
Join e,g, the horizontal projections of the points, meeting LQ in L; draw LN parallel to e, ee, meeting ezg, in N: then N is the perspective sought.
Viewed as a constructive process, this is direct and simple : for, either implicitly or explicitly (mostly the latter), the points 9, 9, are always given in every problem. No problem, indeed, is proposed in a complete state where this is not the case. When, however, points to be put in perspective are numerous, and when (as is mostly the case in practice) they have certain given relations to one another, some degree of modification will very much facilitate the entire process for a figure composed of many related parts. In fact, the determination on the horizontal and vertical planes, of the projections of those points when they are thus only implicitly given, takes away much from the simplicity of the whole work, notwithstanding the great simplicity of its final stages. It has, too, the practical disadvantage of requiring most of the operations to be performed on that region of the paper which is to receive the final picture—a disadvantage which a draughtsman of very moderate practice can fully estimate. This disadvantage is, however, common to most methods usually proposed, and is by no means peculiar to this ; at the same time, as a key to the modified methods, this fundamental one must be thoroughly comprehended by the student, in order to which let him construct the following problems :
(1.) Given the projections of a triangle to find its perspective, for three different positions of the eye, selected at pleasure.
(2.) Given the horizontal projection of a square, the vertical projection of one of the diagonals, and the place of the eye, to find its perspective.
(3.) Given a regular tetrahedron having one of its faces parallel to the picture, and the place of the eye, to construct it in perspective.
(4.) The eye and a cube are given, the cube having two of its diagonals perpendicular to the plane of projection ; to put the cube in perspective.
(5.) One of the diagonals of a cube is perpendicular to the picture plane, and being produced, passes through the eye: find its perspective.
(6.) Show that for all positions of the eye in the prolongation of that diagonal, the perspectives of the cube will be similar and similarly situated figures.
3. In the method just discussed, we find the perspectives of isolated given points, and also those of rectilineal figures by drawing lines to join the perspectives of their angular points. In the next, this process is reversed : viz., into finding the perspective of a point considered as the intersection of the perspectives of two lines which intersect there; and the perspectives of a line as the intersection of the perspectives of two planes which intersect in that line.
As a first consequence of this system, we shall be required to find the perspective of a finite given line, not, as in the first place, limited by its given extremities, but as infinitely produced beyond the picture, and commencing at its intersection with the picture; and, similarly, the perspective of a plane is first found, supposing that plane to be infinitely produced, and commencing at its intersection with the picture. The mutual intersections of these perspectives, taken in proper order, give conjointly the perspective of the figure itself.
4. This mode of representation has already been referred to in the pure Geometry, pp. 157, 158; and by recurring to what is there said, the circumstances just mentioned will become clear at once.
if E be the eye, AZ any straight line on the opposite side of the picture PQ: then the perspective of the infinitely produced line AZ from the picture at A will be one definite line AV, of which V is the termination.
For every line drawn from E to a point in AZ will lie in the plane drawn through AZ and E; and hence its trace on the picture will be a point in the trace of the plane EAZ upon the picture. That is, the perspective of every point in AZ is some point in the limited line AV; and the line AV will be the perspective of the entire infinite line AZ.
Moreover, as all the lines drawn from E to AZ must lie between the parallels AZ and EV, we see how V is to be determined ; viz., by drawing EV parallel to AZ to meet the picture plane in V. This point, V, is called the vanishing point of the line AZ, from the perspective of AZ terminating or vanishing at that point.
It has been shown in the Geometry (Pls. 1. 26) that all lines parallel to AZ have the same vanishing point V; and hence when the inclination (or direction) of a line is given to the picture and the place of of the eye also given, then one point in its perspective representation is given, whatever in other respects may be its situation. The trace of the particular line upon the picture plane gives another point A; and hence the entire perspective of the indefinitely extended line is given when its trace and vanishing point are given or found.
5. The case of the perspective of a plane is precisely similar in its general character.
Thus, let PQ be the picture, BH the plane to be put in perspective (supposed to be infinitely extended on the side of PQ opposite to the eye E) and AB its trace on the picture plane. Then if a plane CL be drawn through E parallel to HB, the trace of which on the picture is CD; the band of the picture plane, infinitely produced both ways, is the perspective of the infinitely extended plane of which HABG is a part.
For, as before, the perspective of every point in HABG is situated somewhere in that band; and there is no point in that band which is not the perspective of some point in HABG.
The line CD is called the ranishing line of the plane HABG, from the perspective of the plane terminating or vanishing in that line.
It follows, that all parallel planes have the same vanishing line.
Moreover, all lines situated in the same plane, as MN, etc., will have their vanishing points, as V, etc., in the vanishing line of that plane.
For HB, CL being parallel planes, and EV, MN parallel lines, one of which MN is (Hypoth.) in one of the planes HB, the other must be in the other (Pls. 1. 4). Whence since EV is in the plane CL, its trace V must be in the trace CD of the plane CL. That is, the vanishing point of MN in the plane HB is situated in the vanishing line CD of the plane HB.
And again, all lines anyhow situated in any number of parallel planes, have their vanishing points in the common vanishing line of those planes.
These simple theorems will find important applications presently.
6. It only remains, in this part of the subject, to justify the remarks made at the opening of the section, by showing :
(1.) The intersection of the perspectives of two lines is the perspective of the intersection of the two lines themselves.
Let AC, BC be two lines which intersect in C, and have A, B for their traces on the picture plane; and let V, W be their vanishing points respectively. Then AV, BW will be their indefinite perspectives, intersecting in c; and it is alleged that c is the perspective of C, or, in geometrical language, that C, C, E are in one straight line.
This has been already proved at p. 157.
(2.) The intersection of the perspectives of two planes is the perspective of the line in which those planes intersect.
Let MA, NA be two planes which intersect.in CA, and their traces on the picture-plane be GH, KL; and let VS, VR be planes through the eye E parallel to them, having EV for their intersection, and BC, DF for their traces on the picture-plane. Then the bands BCHG, DFLK are the perspectives of those planes, having the line AV for their intersection.
Now, since the planes MA, AN which intersect in AC are respectively parallel to the planes SV, VR which intersect in EV, the line EV is parallel to AC (Pls. 1. 16). Whence V is the vanishing point of AC, and AV is the perspective of the line AC. Wherefore, the intersection of the perspectives of the two planes MA, AN is the perspective of their intersection AC, as stated in the enunciation.
We have next to show the method of defining the data of a perspective problem, considering all the lines employed to be traced upon the picture-plane of the paper. We cannot, indeed, exhibit the immediate data in the direct manner that we are able to do in the commencement of a problem in the Descriptive Geometry; but with very little subsidiary construction we are able to render the exhibition of a perspective problem in a light quite as simple and intelligible.
7. In all direct problems in perspective, the position of the eye with respect to the picture forms part of the data. A perpendicular from the eye to the picture meets it in a specific point, called the centre of the picture; and a line through it parallel to the horizon (or the top and bottom of the picture itself) is called the horizontal line. When this point is not taken at, or very near to, the middle of the horizontal line, the effect of the picture is that of a painful distortion on account of the eye being
* In fact, all things considered, the Descriptive Geometry itself involves subsidiary cousiderations and constructions not less operose than those required by Perspective. Indeed, Descriptive Geometry is mainly conversant with inverse operations : operations upon the projections of the direct figures, so as to get the projections of the quæsita of the problem, and thence the quæsita themselves. English writers have confiued themselves almost entirely to the projections of the figures which constitute the data of a problem : the French to the processes which are requisite for the determination of the quæsita. The French begin where the English writers leave off; the one is the complement of the other: neither is perfect and complete in itself; together they form a complete and perfect system. What the English “ Orthographic Projection” is to the Descriptive Geometry, this determination of the working data from the absolute conditions of the problem is to the practical operations of perspective solution.