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Again, since AB, GD represent intersecting lines, the originals are in one plane; and the line DB through their vanishing points is the vanishing line of that plane.
The point C being in the trace of that plane, and CF drawn parallel to the vanishing line, it is the trace of the plane itself.
Wherefore F is the trace of the line GD in that plane.
Through a given point to draw a line perpendicular to a giren plane,
and to find the point of intersection.
(2.) Find the trace of the perpendicular through the given point (Prop. vi.).
(3.) Find the intersection of the perpendicular and plane (Prop. iv.).
The construction is then complete, and composed of steps which have been already demonstrated.
To find the shadow of a given line on a given plane, the light ema
nating from a given point. 'Let P be the luminous point, and AB the given line. Draw perpen
diculars Pp, Aa, Bb to the given plane (Prop. vii.). Draw PB, pb to mcet in B', which will be the shadow of B; and PA, pa to meet in A', which will be the shadow of A. Join A'B', which will be the shadow required.
This construction is adapted to all cases; but it becomes something less complex as to operation when the intersection of the given line with the given plane is also given. For then B and B' coalesce, as in the right-hand figure.
In this way the shadows of the angular points of any polygon or polyhedron may be found. When the line or surface is curved, recourse is generally had to finding the shadows of prominent points of the figure, and tracing a curve through them by hand.
PROPOSITION IX. To find the shadow cast by a giren line on a giren plane, the rays
(like sun-light) being taken as parallel.
This problem differs but little from the preceding in its general principle of construction, though in some respects the operations are rather
more complicated, whilst in others they are somewhat simpler. The former construction was decribed in detail ; but it will be easy for the student now to follow out the details from a general description of the operations to be performed. At the same time it will be desirable to notice particularly one or two special cases.
The problem divested of technicalities becomes the successive applications to different points of the figure of the following operation : Given a point, to find the intersection with a given plane of a line
drawn through that point, parallel to a giren line.
This problem requires the vanishing point of the pattern-ray to be substituted for that of the perpendiculars in Prop. v.
It then becomes precisely identical with Prop. vi., as far as determining the trace of the line so drawn is concerned. Having, then, this line, its intersection with the given plane is found as in Prop. iv. The point thus found is the shadow sought.
The question as to whether the light comes in a direction on the same or a different side of the picture-plane only varies the question so far as the perspective of the pattern-line is concerned : though writers often make two separate cases of it, which is perfectly unnecessary. There is, however, one case, namely, that when the pattern-ray is parallel to the picture-plane, that requires a separate consideration; as the trace and vanishing points have here no existence at a finite (and therefore available) distance. With that case we shall close the subject.
Now all lines (as these rays) parallel to the picture and to one another are represented on the picture by parallel lines. through which any one of these passes being therefore given, the direction of its representation on the picture is also given. The process, therefore, differs from the general case already described in this particular only: that the rays, instead of emanating on the picture from the vanishing point, are all drawn parallel to one another.
SCHOLIUM. The inverse class of perspective problems has attracted little attention from writers on perspective in this country. Still when even practically viewed, this is quite as important as the direct one. The two classes stand in a relation to each other analogous to that of the direct orthographic projection, and the operations known as Descriptive Geometry; and for completeness of system both classes of the problems are as indispensable as in the orthographic. The student who wishes to pursue the subject further, is referred to Cousinery, Géométrie Perspective, 1828 ; and to a work under the same title by Dufour, of Geneva, 1827.
Conclusion. It is not irrelevant to caution the student against indulging in a course of speculative discussions that seldom fail to be brought before his attention, respecting the “fallacies” or “illusions” of perspective. These are, one and all, only so many mental delusions, arising out of a confused or erroneous view of the character of perspective, and the geometrical properties involved in the formation of a picture. Most books contain some or other of these; often, too, in a very plausible form : and they are in each age revived as so many new discoveries, to be again refuted as they had been twenty times before.
These chiefly turn upon the following mistakes :
(1.) Confounding the picture on the retina with that on the canvas. This gives rise to the sophisms respecting straight lines being represented on the picture by curves, and many others.
(2.) The supposition that we have the power in the same picture to move the axis of the eye to the right or left. This is a fertile source of fallacy, and contrary to the fundamental principle of perspective, that a picture is only composed of such scenery as can be taken in with the eye in one fixed position. This supposition is the foundation of the panorama, but not consistent with the idea of a picture.
(3.) Speculations respecting the diminution of the optic angle often lead to very absurd fallacies, especially where the plane of the objects is parallel to the picture-plane. If the student should encounter the oft-quoted example of a row of equal columns parallel to the picture, or two flag-staves stuck upright in the ground : then let him remember that the optic angle is diminished in the picture and in the objects precisely alike; and that in order to this being the case, the most distant pillar must have the same linear diameter on the picture that the nearest has, and that the flag-staves must be equidistant at top and bottom on the picture as well as in the actual position which they occupy. The whole confusion arises from an imperfect conception of the problem, and of the distinction always to be made between linear and angular magnitude, as things per se different from each other.
(4.) Much confusion has arisen, too, from overlooking the fact that whether it be a scene or the picture of that scene that is before us, we actually see with the same instrument, the eye, in both cases. then suppose, or even admit, that the eye does in any specific way modify the sensible form of the scene, it must also modify the picture (or perspective) in precisely the same manner.
Other sources of fallacy also present themselves occasionally ; but the cautions above given will be sufficient for most cases, and moderate care will suffice for all others,
“A bird's-eye view of a scene" is a common and well understood expression for a view obtained from a point considerably elevated above the plane of the scene itself, so as to include also the interior (as well as the exterior) of the works which were concealed from the eye at a less elevation. Such a view, however, if represented on a plane, would be a perspective view; and though it would give a good general notion of the interior structure, would be too much distorted to be of use in reference to construction, where any close degree of accuracy was required. Like perspective under ordinary circumstances, it has suggested the employment of orthographic projection as a substitute ; for the isometric projection is, in reality, only a special case of the orthographic, the plane of projection itself being restricted to one single position.
If the dihedral angles of a given trihedral right angle be bisected, the bisecting planes will pass through the same line. A plane perpendicular to this line is taken as the plane of projection, the projectors being parallel to the line itself. If the three faces of the trihedral angle be the ground and the two walls which form the angle of the building, the line in question will rise obliquely above the ground; and an eye placed anywhere in it to view the interior of the walls and ground may be said to have a bird's-eye view of the structure. Any point in either of the planes projected on the forenamed plane by lines parallel to the line instead of by lines radiating from the eye, will change the perspective into an isometric projection of the point.
The axis of projection is the line in which the three bisecting planes intersect; and it is easy to prove that it is equally inclined to each of the edges of the trihedral angle; and likewise equally inclined to each of the faces. We have now to ascertain what each of these inclinations is.
Let the right angles ASB, BSC, CSA of the trihedral angle be denoted as in the figure (they are seen obliquely, as they could not be otherwise exhibited without a model); make AS, BS, CS all equal, and denote their common length by a; draw the plane ABC, and the perpendicular AD to it from S; and draw the plane ASD, cutting ABC in AE, and BSC in SE.
Then ABC is evidently an equilateral triangle, whose centre is D, and whose side BC is bisected in E. Moreover the plane ABC is parallel to the plane of projection, and SD is the axis of projection. Then, AB = BC = CA = a v 2;
13 = 3V6;
DE = 4 AD = ÖV6;
DS NS A' – AD
fav These elements serve for the determination of everything relating to the subject. Thus, for instance, we have
Which are the inclinations of an edge and a face respectively to the plane of projection ; and agreeing with the fact that the triangle ASE is right angled at S, so as to jender it necessary only to calculate one of these inclinations from a formula.
To construct the isometric angle is, however, more compatible with
the geometrical character of the operations than to compute it; the more especially as we at the same time obtain the lengths of all the lines concerned in reference to any assumed magnitude of the edge SA (or SB, or SC). This is effected very simply in the following manner.
In as, sb taken at right angles, make as, sb equal to AS, SB of the former figure; join ab, and describe the equilateral triangle abc upon it. This is equal to the triangle ABC.
Find the centre d of the triangle abc, and join ad, bd, cd. Then these are equal to AD, BD, CD.
On as describe a semicircle, and in it set as off ad' equal to ad, and join sd'. Then, sad' is equal to SAD, asd' to SED, and sd' to SD.
For, since sd'a is in a semicircle, it is a right angle; and we have sa = SA, and ad' ad = AD: wherefore the two rightangled triangles SAD, sad' have two sides of the one equal to two sides of the other, and hence the third sides SD, sd' are also equal, and the angles opposite to these equal, viz., sad' to SAD.
Also SED being the complement of SAD, and asd' of sad', these are also equal.
We are now able to construct the isometric scale.
Make the angle saa, equal to the isometric angle by the preceding ; on sia set off the units of the natural scale (suppose inches); and draw s,a,, 804, etc., perpendicular to adj. When ad, a,ag, etc. will be isometric inches :that is, on an isometric drawing those lines will represent real inches in the model or other machinery delineated. They will also represent such parts of a building or range of works as are delineated to the scale of one inch.
If there be frequent repetitions of this class of operations, this scale should be carefully made of wood or brass, with a sliding bar perpendicular to ad,, perhaps in the manner of the T-square. This being so moved as to pass over any division on casi, as s, it would mark the distance ad, to be taken as its correspondent isometric distance.
For subdivisions, as eighths or tenths, a smaller bar, limited to pass over only as far as as, might be also used. Both bars may be fixed so as to move in grooves on the triangular plate which forms the scale.
The properties of this projection which (as far as is yet known) can be brought into practical use are few and simple; and it would be out of place here to give those which are merely curious. To obtain those few let us recur to the first figure, page 380.
1. The three equal edges SA, SB, SC, being equally inclined to the plane of projection, are projected into equal lines AD, BD, CD;