0.5

1 cm

\

PERSPECTIVE.

103

picture; let ab (Fig. 204) be the side of the square, O the

point of sight, d the point of distance. Draw a 0, b O,

for the indefinite per-

spectives of the side, and

a d for the perspective of

the diagonal, and where

it intersects 6 0 in c,

draw c e parallel to a 6,

and the perspective of

the square is completed.

2d. When the diagonal

of the square is perpendicular to the base of the picture,

let a (Fig. 205) be the hither or nearest angle of the

square, 0 the point of sight, d d' the points of distance.

The diagonal of the square being perpendicular to the

picture, will have the point of sight for its vanishing point.

Draw, therefore, a 0 as its indefinite perspective. Set

(Fig 205 )

off from a to 6 the length of the diagonal, and draw

6 d intersecting a 0 in c, which is the perspective of

the farther extremity of the diagonal, then draw the

sides (which make angles of 45° with the picture) to the

distance points d d'.

2d. When one side of a square is given, making any

angle with the picture; let a b (Fig. 206) be the given

side, produce it to the horizon in c. Set off the distance

of the eye on the central line, from / to E, and draw c F,

which will be a parallel to the given side a 6, and c is con-

sequently the vanishing point for all lines parallel to a b.

(Fig. 206.)

The point a belongs equally to two sides of the square,

one the given side a 6, the other the hither side of the

square. Let us see how this latter is to be found. As

this side makes a right angle with a b, from F draw

F gr at right angles to F c, and continue it till it meets

the horizon. In this case the lines would extend beyond

the limits of the paper, and to avoid that inconvenience

(Fig. 207.)

we may adopt another method of working, viz., by the

diagonal. Therefore from F draw F d, making an angle

of 45° with F c, and its intersection d with the horizon

will be the vanishing point of one diagonal of the square*

From F also draw F h, at an angle of 45°, with F c for the

vanishing point h of the other diagonal.

Now, to complete the square, draw from the extremities

of the given side the perspectives of the diagonals a d, b h,

and produce the latter indefinitely; and to find the length

of the sides take the following means, founded on what

has previously been learned.

Through the intersection of the diagonals at o draw a

straight line m n, parallel to the base of the picture, and

make o n equal to o m, and n will be a point in the side

of the square, through which draw r s, intersecting the

diagonals, and join 6 s, a r, and the square is completed.

Problem II.—To divide a line given in perspective in

any proportion; let A B or Be (Fig. 207) be the given line,

and let it be required to divide it into two equal parts.

Now, A B being parallel to the base of the picture, will

have its perspective exactly the same as the original, and if

divided into equal parts, its perspective will also be divided

into equal parts. But B c being oblique to the base of

the picture will, in reality, be divided into two equal

parts in the point e, although these parts appear unequal.

If through A and c we draw a line produced to the

horizon, the point O will be the vanishing point of all

lines parallel to A O.

Hence, if' we draw D O,

B O, these three lines

will be parallel among

themselves. And as the

lines A B, Be, are com-

prised within the parallels

AO, BO, and are cut by

the third parallel D O, it

follows that their parts

are proportional among

themselves. And, therefore, if A D is the half of A B, c e

will be the half of B c.

The same will follow if the lines are in the vertical

plane, as kg divided into two equal parts in o.

From what has been stated it will be seen that the pro-

blem reduces itself to these:—1. When the line is situated

in a plane parallel to the picture, the divisions are made

as in any other straight line. 2. When it makes any

angle whatever with the picture, we draw through one of

its extremities a line, either parallel to the base, or to

the side of the picture, accordingly as the given line is

situated in a horizontal or vertical plane, and we divide

this line into the number of equal parts demanded, then

through the last division, and through the other extre-

mity of the given line we draw a line produced to the

horizon, and from all the points of division draw parallels

to this line, which will, by intersection, give the perspec-

tive divisions of the line required. For example, let it be

required to divide the line m n (Fig. 208) into three equal

parts, through n draw the horizontal line n 3 indefinitely,

PERSPECTIVE.

103

picture; let ab (Fig. 204) be the side of the square, O the

point of sight, d the point of distance. Draw a 0, b O,

for the indefinite per-

spectives of the side, and

a d for the perspective of

the diagonal, and where

it intersects 6 0 in c,

draw c e parallel to a 6,

and the perspective of

the square is completed.

2d. When the diagonal

of the square is perpendicular to the base of the picture,

let a (Fig. 205) be the hither or nearest angle of the

square, 0 the point of sight, d d' the points of distance.

The diagonal of the square being perpendicular to the

picture, will have the point of sight for its vanishing point.

Draw, therefore, a 0 as its indefinite perspective. Set

(Fig 205 )

off from a to 6 the length of the diagonal, and draw

6 d intersecting a 0 in c, which is the perspective of

the farther extremity of the diagonal, then draw the

sides (which make angles of 45° with the picture) to the

distance points d d'.

2d. When one side of a square is given, making any

angle with the picture; let a b (Fig. 206) be the given

side, produce it to the horizon in c. Set off the distance

of the eye on the central line, from / to E, and draw c F,

which will be a parallel to the given side a 6, and c is con-

sequently the vanishing point for all lines parallel to a b.

(Fig. 206.)

The point a belongs equally to two sides of the square,

one the given side a 6, the other the hither side of the

square. Let us see how this latter is to be found. As

this side makes a right angle with a b, from F draw

F gr at right angles to F c, and continue it till it meets

the horizon. In this case the lines would extend beyond

the limits of the paper, and to avoid that inconvenience

(Fig. 207.)

we may adopt another method of working, viz., by the

diagonal. Therefore from F draw F d, making an angle

of 45° with F c, and its intersection d with the horizon

will be the vanishing point of one diagonal of the square*

From F also draw F h, at an angle of 45°, with F c for the

vanishing point h of the other diagonal.

Now, to complete the square, draw from the extremities

of the given side the perspectives of the diagonals a d, b h,

and produce the latter indefinitely; and to find the length

of the sides take the following means, founded on what

has previously been learned.

Through the intersection of the diagonals at o draw a

straight line m n, parallel to the base of the picture, and

make o n equal to o m, and n will be a point in the side

of the square, through which draw r s, intersecting the

diagonals, and join 6 s, a r, and the square is completed.

Problem II.—To divide a line given in perspective in

any proportion; let A B or Be (Fig. 207) be the given line,

and let it be required to divide it into two equal parts.

Now, A B being parallel to the base of the picture, will

have its perspective exactly the same as the original, and if

divided into equal parts, its perspective will also be divided

into equal parts. But B c being oblique to the base of

the picture will, in reality, be divided into two equal

parts in the point e, although these parts appear unequal.

If through A and c we draw a line produced to the

horizon, the point O will be the vanishing point of all

lines parallel to A O.

Hence, if' we draw D O,

B O, these three lines

will be parallel among

themselves. And as the

lines A B, Be, are com-

prised within the parallels

AO, BO, and are cut by

the third parallel D O, it

follows that their parts

are proportional among

themselves. And, therefore, if A D is the half of A B, c e

will be the half of B c.

The same will follow if the lines are in the vertical

plane, as kg divided into two equal parts in o.

From what has been stated it will be seen that the pro-

blem reduces itself to these:—1. When the line is situated

in a plane parallel to the picture, the divisions are made

as in any other straight line. 2. When it makes any

angle whatever with the picture, we draw through one of

its extremities a line, either parallel to the base, or to

the side of the picture, accordingly as the given line is

situated in a horizontal or vertical plane, and we divide

this line into the number of equal parts demanded, then

through the last division, and through the other extre-

mity of the given line we draw a line produced to the

horizon, and from all the points of division draw parallels

to this line, which will, by intersection, give the perspec-

tive divisions of the line required. For example, let it be

required to divide the line m n (Fig. 208) into three equal

parts, through n draw the horizontal line n 3 indefinitely,