13. A third method of defining a line for perspective purposes is Given its plane, and its projection on the picture-plane, to find its perspective.

[Sometimes the angle which the line makes with its projection is substituted for the last datum.]

In this case we have virtually a second plane through the line given, as well as that which contains the line. This plane, moreover, being perpendicular to the picture-plane, its vanishing line passes through the centre of the picture, and is parallel to the trace, that is, to the given projection. It is therefore given in position. Whence the following construction :

P

Let PQ be the picture-plane, AB the trace, and CD the vanishing line of the plane which contains the given line; let GF be the projection of the line on the plane of the picture intersecting AB in K; and let O be the centre of the picture.

Draw OV parallel to FG meeting CD in V: then V is the vanishing point, and KV the perspective of the line.

14. Finally, a line is given by its

A

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B

containing plane, its trace, and its inclination to the picture-plane. A line drawn through the eye parallel to the given line will make an angle with the picture-plane equal to the given one; which angle will therefore be given. But the whole of the lines through the eye which do this, make equal angles with the perpendicular from the eye to the picture-plane; and hence they are situated on a right cone which has that perpendicular for its axis, and a circle on the pictureplane for its base. The centre of this circle is the centre of the picture; and the axis of the cone being given in magnitude, the radius of the circle is given in magnitude; and the circle itself is given in magnitude and position.

Whence the following construction :—

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H

Let AB be the given trace of the containing plane; K the trace of the line; CD the vanishing line of the containing plane; and O the centre of the picture. Make km the distance of the eye from the picture; mkn the complement of the given inclination of the line to the picture; and kmn a right angle. With centre O and radius OG equal to mn describe a circle cutting CD in GG'.

Then KG or KG' will be the perspective of the entire line, accord

ing as the given inclination is towards the centre of the picture or in the contrary direction.

The unfinished part of the construction of the second case of the data is identical with this: for the inclination of a line to a plane is the same thing as its inclination to its own projection on that plane.

The perspective of a point is always viewed practically (that is, the operations for finding it always imply its being so) as the intersection of the entire perspectives of two lines which pass through it. This of course renders further remark unnecessary, as it leads simply to direct constructions of those two perspectives.

SECTION II.

PRACTICAL PERSPECTIVE.

In general the application of perspective is to objects having certain features of regularity and symmetry which enable us greatly to abridge the actual work that would be required to put each angular point (as an isolated point) in perspective. For instance, nearly all the planes in which lines or points are actually given are either parallel or perpendicular to the horizon; or their relations to such planes are easily determined both in fact and in perspective. Again, most of the plane figures to be put in perspective are rectangles, or have easily assignable relations to rectangles or to parallelograms; and most of the solid angles that occur are trihedral right angles, having one of the faces horizontal. Even where these conditions are not fulfilled, the data can almost invariably be connected by subsidiary processes with data that do fulfil them; and in any particular case where even this may fail, the general methods already given will be adequate to a full and perfect construction of the perspective.

In the case of curves, as it is rarely that any one but the circle occurs in the practice of the draughtsman or engineer, it may be sufficient to state that it is deemed in practice sufficient to find the perspectives of eight points equidistant from each other (the angular points of an inscribed regular octagon), and then carefully tracing by hand an ellipse through them.

A few incidental notes, which though not geometrical perhaps as to strict form, are yet founded on geometrical and physical principles, may be given in initio. For the sake of reference, they are designated as Propositions.

PROPOSITION I.

To prepare the drawing and its scales.

(a.) To eyes of ordinary conformation, horizontal vision is painful and indistinct when the bounding line of light makes an angle of more than 30 with the optic axis. When, therefore, the breadth of the picture is fixed upon, the distance of the eye from it, to form an agreeable perspective, becomes also determined at least within very narrow limits of variation; and when the distance of the eye is first fixed upon, the breadth of the picture becomes, conversely, dependent. If this rule

be rigidly adhered to, the relation will be exhibited by the following construction; and if varied, the effect of the variation can be readily traced.

If the distance OE be given, and HL be a horizontal line through the centre of the picture; take any equal distances Oh, Ol, on each side of O, and describe the equilateral triangle ehl; draw Oe, and make OE in it equal to the given distance of the picture; and draw EH, EL parallel to eh, el. Then H, L are the lateral boundaries of the picture.*

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(b.) The height of the eye above the horizon is usually between five and six feet, and when the supposed observer stands on the same horizontal plane with the building to be put in perspective, this height becomes a fixed standard for forming all the scales of the picture. This in fact it is which forms the fundamental scale; the distance of the eye, and consequently the breadth of the picture, being regulated by it.

The height of the top of the picture above the horizontal line is a question less easy of decision. Of course it may include an angle of 30 on the ground of our distinct vision being limited by a right cone of rays, the axis of which cone is the perpendicular from the eye to the picture. But in fact, from our being so habituated to elevate the optic axis itself that we do it unconsciously to some extent, we can take in without pain a more extended space vertically than laterally.

(c.) Having fixed upon the boundaries and marked the centre of the picture, draw a line through that centre horizontally. This is called the horizontal line.

(d.) To form a scale of heights.

Let HR be taken of the height of the eye, and divide the entire line

PR in conformity with it; draw lines to any point G in HQL: then the lines ab, cd, ef, etc., are the perspective heights of the line PR when removed backwards from the picture, so that the perspective of R shall be distant from the picture so far as to be represented on the horizontal lines bb', cc', dd', etc.

*If the horizontal breadth be taken much more than this, it cannot be seen in all its parts without moving the axis of the eye so as to see one part separately from another (which would be tantamount to looking at the perspective of a perspective), or else distressing the eye to attempt to take it all in at one view. If, on the contrary, it be much less, the picture has precisely the same effect as an agreeable picture would be made to have by cutting off a portion at each end-a mutilated appearance.

(e.) To form a scale of distances.

Let the figure be supposed to be a section of the entire system by a

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R

plane perpendicular to the ground plane, passing through the eye. It may be considered, then, as a subsidiary figure in a plane at right angles to the picture and to the horizon.

Set off upon the line RQ in which this plane intersects the horizontal plane the several distances Q1, Q2, Q3, etc. Make QO (from the same scale) equal to the height of the eye; and OE parallel to QR equal to the distance of the picture. Then E1, E2, E3, etc., will cut from QO segments Q1, Q2, Q3, etc., which represent on the picture the corresponding distances on the horizontal plane.

The proof is obvious.

(f) To form a scale of breadths.

Divide the ground line RQ per scale; draw lines RO, aO, bO, cO,

etc., to the centre O (or any point in HL): then if 1, 2, 3, etc., be the divisions for distances, and parallels be drawn to RQ, as in the figure, a b1, or b1 c1, etc., will be the perspective breadth at the distance 1. Similarly a, b, or b, c, etc., will be that at the distance 2; and so on.

These scales, though often used by practical draughtsmen, are not essential. Though given here in accordance with such frequent usage, they are not recommended to the draughtsman, who is also a geometer. When used, it is better to keep them as subsidiary figures than to encumber the actual drawing with them.

PROPOSITION II.

To put a plane figure in perspective.

As a general statement of the rule, it may be directed to put each entire component line of a rectilnear figure in perspective. These perspectives, by their mutual intersections, give the perspective required. In the case of curves, set a series of points sufficiently near for the

purpose, and having put these in perspective, trace a curve through them by hand or by instruments, as the case may admit.

In practical perspective, however, we have seldom to operate upon figures which do not possess some kind of symmetry, which greatly simplifies the entire work by diminishing the number of independent steps. Thus the ground-plan of a building is mostly composed of rectangles, or of figures somehow related to rectangles; the walls are invariably rectangular, and the tops and bottoms of a range of windows are each in line; the roofs are rectangles, or occasionally isosceles triangles; and so on. It is the same in machinery, in a great degree, for even wheels do not often occur except in horizontal or vertical positions. On this account directions somewhat more special may be given for operating. (a.) Let the plane of the figure be parallel to the picture.

This, on the general principles of perspective, is an isolated or exceptive case; since the vanishing line of the plane itself being infinitely distant, cannot be brought into use. It is usually waved in practice by making it dependent on the construction of another case; viz., by finding the perspectives of some of the points by feigning them to be the intersections of the trace of the plane on the ground-line and some other lines. The following construction is more simple in principle as well as in practice.

ANALYSIS.-Let PQ be the picture-plane, RS a plane parallel to it, containing the figure ABC to be put in perspective: draw lines, AE, BE, CE to the eye E, meeting the picture in abc. Then the figure abc will be the perspective of ABC, by the definition.

Draw perpendiculars AA', BB', CC' to the picture, and likewise the perpendicular EON to the picture, meeting the picture at O, and the parallel plane at N. Draw also the planes ANE, BNE, CNE, cutting PQ in A'O, B’O, C’O.

Then the lines AN, A'O, AA', NOE, AaE are in one plane; and hence since this plane cuts the parallel planes, each line in one is equal and parallel to the corresponding line in the other, namely, A'B' to AB, A'O to AN, etc. Also since AE, A'O are in one plane, the point a, which is the perspective of A, is in the line A'O; and similarly b, c, etc, are in B'O, C'O, etc.

Again, since aO is parallel to AN, the triangles EOa, ENA are similar, and hence

AN: aO:: EN : EO, a given ratio.

Now O is the projection of the eye upon the picture and A'B'C' is

VOL II.

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