0.5

1 cm

102

THE ENGINEER AND MACHINIST’S DRAWING-BOOK.

their perspectives will be a B c D; the direction of which

will be to the vanishing point E, situated at the inter-

section of the picture with a plane E f, passing through

the eye F, parallel to the planes of the triangles A a B

C c D, passing through the given original lines.

The line F E is a line parallel to the given lines drawn

through the eye to meet the picture. As it is in the

plane E /. its intersection with the picture determines the

vanishing points of the lines a B, cD.

[Fig. 200.)

E

This problem may be considered in a different manner.

Let AB C D (Fig. 201) be the original lines as before;

but in place of supposing them situated in two triangles,

let us suppose them situated in a plane B C, inclined to

[Fig. 201.)

the original plane. Let the plane G E, passing through

the eye F, be parallel to B C, and let it cut the picture

in the line H E, which, as we know, will then be the

vanishing line for all planes parallel to B C. The line

F E is drawn through the eye parallel to the original

lines given, it lies in the plane G E, and cuts the picture

in E. It lies also, however, in the plane / E, and there-

fore E is the vanishing point sought.

Let us consider the practical application of this problem,

with the view to its more perfect elucidation :•—

Let A B (Fig. 202) be the plan of a cube, and C D the

elevation of one of its sides, with diagonal lines drawn on

it. Draw the visual rays O A, O B, the central plane O M,

the picture line H L, and the lines O H, O L, to determine

[Figs. 202 and 203,)

/\

' m

f/

the vanishing points of the sides. Then, to find the vanish-

ing points of the oblique lines; from O in the ground-plan,

on the line O L, construct a right angled triangle O L N,

of which the angle N O L is equal to G C D, and set up

the height L N, from l to n, in Fig. 203, which gives n as

the vanishing point for C G and all lines parallel to it.

Set the same length off downwards, from l to to, and

to is the vanishing point for F D, and all lines parallel

to it. In the same way find the vanishing points on the

left hand side, by drawing the triangle O H K, and set

off the length H K in Jc and r above and below the vanish-

ing point s of the horizontal lines of the cube. The

reason of this process is obvious, for we have only to

imagine the triangles OLN,OHK, revolved round O L

and O H until their plane is at right angles to the paper,

and we then perceive that N and K are the heights

over the horizontal vanishing points, that a plane passing

through O at the height of the eye of the spectator, and

at 45° with the horizontal plane, would intersect the

plane of the picture.

The drawing of the figure is esplained by the dotted

lines.

In what we have hitherto advanced are comprehended

all the principles of perspective, and we shall now pro-

ceed to apply these principles in the solution of various

problems.

Problem I.—The distance of the picture and the per-

spective of the side of a square being given, to complete

the square, without having recourse to a plan.

1st. When the given side is parallel to the base of the

THE ENGINEER AND MACHINIST’S DRAWING-BOOK.

their perspectives will be a B c D; the direction of which

will be to the vanishing point E, situated at the inter-

section of the picture with a plane E f, passing through

the eye F, parallel to the planes of the triangles A a B

C c D, passing through the given original lines.

The line F E is a line parallel to the given lines drawn

through the eye to meet the picture. As it is in the

plane E /. its intersection with the picture determines the

vanishing points of the lines a B, cD.

[Fig. 200.)

E

This problem may be considered in a different manner.

Let AB C D (Fig. 201) be the original lines as before;

but in place of supposing them situated in two triangles,

let us suppose them situated in a plane B C, inclined to

[Fig. 201.)

the original plane. Let the plane G E, passing through

the eye F, be parallel to B C, and let it cut the picture

in the line H E, which, as we know, will then be the

vanishing line for all planes parallel to B C. The line

F E is drawn through the eye parallel to the original

lines given, it lies in the plane G E, and cuts the picture

in E. It lies also, however, in the plane / E, and there-

fore E is the vanishing point sought.

Let us consider the practical application of this problem,

with the view to its more perfect elucidation :•—

Let A B (Fig. 202) be the plan of a cube, and C D the

elevation of one of its sides, with diagonal lines drawn on

it. Draw the visual rays O A, O B, the central plane O M,

the picture line H L, and the lines O H, O L, to determine

[Figs. 202 and 203,)

/\

' m

f/

the vanishing points of the sides. Then, to find the vanish-

ing points of the oblique lines; from O in the ground-plan,

on the line O L, construct a right angled triangle O L N,

of which the angle N O L is equal to G C D, and set up

the height L N, from l to n, in Fig. 203, which gives n as

the vanishing point for C G and all lines parallel to it.

Set the same length off downwards, from l to to, and

to is the vanishing point for F D, and all lines parallel

to it. In the same way find the vanishing points on the

left hand side, by drawing the triangle O H K, and set

off the length H K in Jc and r above and below the vanish-

ing point s of the horizontal lines of the cube. The

reason of this process is obvious, for we have only to

imagine the triangles OLN,OHK, revolved round O L

and O H until their plane is at right angles to the paper,

and we then perceive that N and K are the heights

over the horizontal vanishing points, that a plane passing

through O at the height of the eye of the spectator, and

at 45° with the horizontal plane, would intersect the

plane of the picture.

The drawing of the figure is esplained by the dotted

lines.

In what we have hitherto advanced are comprehended

all the principles of perspective, and we shall now pro-

ceed to apply these principles in the solution of various

problems.

Problem I.—The distance of the picture and the per-

spective of the side of a square being given, to complete

the square, without having recourse to a plan.

1st. When the given side is parallel to the base of the