101

more or less indirect; it is now necessary to show the

direct method of solving the same problems.

Let it be required to draw in perspective a square the

same as the last. From the station point 0 (Fig. 198),

draw to the angles of the square a bed visual rays Oa,Ob,

&c., and find the vanish-

ing points m k' as before.

Draw in the vertical pro-

jection the ground-line

m k, and the horizon

np, and transfer to the

ground-line the inter-

section of the visual rays

with the picture ; draw

from a the lines am', a k',

and their intersections

with the perpendiculars

1,2,8, will give the limits

of the sides; then draw

a k', b k', intersecting at

c, which completes the

square. If, as in the last

figure, another square be

drawn at a height above

the given plane, equal to

the side of the original

square, we shall obtain

the representation of a cube. *

Let A B C D (Fig. 199) be the horizontal projections

of four straight lines perpendicular to the picture B D, b d.

The perspective of these lines b a, d c, f e, h g, will con-

verge to the point of sight i', and if the original lines were

infinite, they would appear to meet in the point of sight.

These lines maybe con-

sidered to be the boun-

daries of four planes,

two horizontal, onefg

above, and the other

b c below the horizon-

tal plane, and two

vertical, b c, d g, and

as the boundary lines

of these planes may

be considered as of

infinite extent, so

may also the planes

themselves. It is ob-

vious from the figure

that the perspectives

of all lines, situated

in a horizontal plane,

can never, in any case,

pass the horizon; that

the horizon is in like

manner the vanishing

line of all horizontal

planes, and that the

trace of the centra]

plane is the vanishing

line for all vertical

planes parallel to it.

In the example, the vertical planes are shown to be

squares, and their diagonals are consequently inclined to

the picture in an angle of 45°. The perspectives of these

diagonals will have their vanishing points in the vanishing

lines of the planes which contain them, or in the trace of

the central plane. The distance ik,il of these points k l,

will be equal to the distance I i of the eye from the picture.

The diagonals of the horizontal planes will have their va-

nishing points in the point of distance, as we have already

seen. The plane A C being parallel to the picture, will

have for its perspective the similar square aeg c.

The line b c is evidently inclined to the ground plane, and

it is also situated in a vertical plane perpendicular to the

picture. Its perspective will therefore have its vanishing

point in the vertical of the picture, and in a point m which

will be more or less high, as the inclination of the line

is greater or less. The perspectives of the diagonals of

the side d g will have their vanishing points in n m.

Rule VII.—Hence all lines inclined to the horizon,

parallel among themselves, andinclined invertical planes

perpendicular to the picture, have their vanishing points

in the vanishing line of the planes which contain them.

Hence the perspectives of planes parallel to the picture

cannot have vanishing points, but will always be of the

same figures as their originals.

Further to illustrate the vanishing points of inclined

lines let A B C D (Fig. 200) be two original parallel lines ;

o

{Fig. 199.)