0.5

1 cm

THE PROJECTION OF SHADOWS.

83

through the opposite extremity to, draw a perpendicular,

r s, to this line, and set off, on the perpendicular, the dis-

tance r s, equal to the side of the square,

and join s o. Now, draw through the

point A', in the original figure, a line

A' a, parallel to s o, intersecting the

circle A' a B' in the point a, which

being projected by a line parallel to the r

axis of the cylinder, and meeting the

line A a, drawn at an angle of 45°, gives

the first point a in the curve C da.

The other points will be obtained in like

manner, by drawing, at pleasure, other lines, such as D'd',

parallel to A' a.

Figs. 9, 10.—For defining the shadow on an ogee sur-

face, the construction is explained by the diagrams. The

points C and D are formed by radii 0 C, E D, drawn at

right angles to the direction of the light.

To find the outline of the shadow cast into the interior

of a hollow hemisphere.

Let A B C D (fig. 184, annexed) represent the horizontal

projection of a concave hemisphere. Here it is sufficiently

Tig. 184.

C'

obvious that, if we draw, through the centre of the sphere,

a line perpendicular to the ray of light A C, the points

B and D will at once give the extremities of the curve

sought. Take, now, upon the prolongation of the line B D,

any point O', from which, as a centre, describe a semi-

circle with the radius A O, and from the point A' draw the

straight line A' c e' parallel to a line found in the same

manner as oto, in fig. 183; the point a' of its intersec-

tion with the circle A' a O', projected to a, will be another

principal point in the outline of the shadow.

By imagining similar sections, such as E F, parallel to

the former A C, and laying down in the same way semi-

circles corresponding to them, the remaining points in the

curve sought may be obtained. But, as this curve is an

ellipse, of which the diameter D B is the major axis, and

the line 0 a the half minor axis, it follows that this last

line being determined, the curve may be constructed bv

the ordinary methods for ellipses.

We shall now find no difficulty in constructing the

cast shadow in the interior of a concave surface, fig. 185,

formed by the combination of a hollow semi-cylinder

and a quadrant of a hollow sphere, called a niche, as we

Fig. 185.

IS

know the mode of tracing the shadows upon each of

these figures separately. Thus, the shadow of the sphe-

rical portion is part of an ellipse i cD, whose semi-axis

major is OD; the semi-axis minor is obtained by describing

the semicircle B2 % E, with the radius 0 B, drawing from

the point B2 the straight line B2 i', parallel to a line

found in the same way as o to, fig. 183, and finally project-

ing the point of intersection i' to i on the straight line

B O. The point e, where this ellipse cuts the horizontal

diameter A F, limits the cast shadow due to the two

distinct surfaces; therefore, all the points beneath it must

be determined upon the cylindrical part.

Plate LI V. Figs. 1, 2.—To find the line of shade in a

sphere, and the outline of its shadow cast upon the hori-

zontal plane.

The line of shade in a sphere is simply the circumference

of a great circle of which the plane is perpendicular to the

direction of the luminous ray, and consequently inclined

to the two planes of projection. This line will, therefore,

be represented in elevation and plan by two equal ellipses,

the major axes of which are obviously the diameters

C D and C' D', drawn at an angle of 45°.

To find the minor axes of these curves, assume any

point O2, upon the prolongation of the diameter of the

perpendicular C' D', (fig. 2), draw through this point

the straight line O2 O', inclined at an angle of 35° 16', to

A' B' or its parallel, and erect upon it the perpendicular

E2 F2. The projection of the two extremities E2 and F2

upon the line A' B', will give, in the plan, the line E' F'

for the length of the required minor axis of the ellipse;

and this line being again transferred to the elevation,

determines the minor axis of the line of shade in it.

83

through the opposite extremity to, draw a perpendicular,

r s, to this line, and set off, on the perpendicular, the dis-

tance r s, equal to the side of the square,

and join s o. Now, draw through the

point A', in the original figure, a line

A' a, parallel to s o, intersecting the

circle A' a B' in the point a, which

being projected by a line parallel to the r

axis of the cylinder, and meeting the

line A a, drawn at an angle of 45°, gives

the first point a in the curve C da.

The other points will be obtained in like

manner, by drawing, at pleasure, other lines, such as D'd',

parallel to A' a.

Figs. 9, 10.—For defining the shadow on an ogee sur-

face, the construction is explained by the diagrams. The

points C and D are formed by radii 0 C, E D, drawn at

right angles to the direction of the light.

To find the outline of the shadow cast into the interior

of a hollow hemisphere.

Let A B C D (fig. 184, annexed) represent the horizontal

projection of a concave hemisphere. Here it is sufficiently

Tig. 184.

C'

obvious that, if we draw, through the centre of the sphere,

a line perpendicular to the ray of light A C, the points

B and D will at once give the extremities of the curve

sought. Take, now, upon the prolongation of the line B D,

any point O', from which, as a centre, describe a semi-

circle with the radius A O, and from the point A' draw the

straight line A' c e' parallel to a line found in the same

manner as oto, in fig. 183; the point a' of its intersec-

tion with the circle A' a O', projected to a, will be another

principal point in the outline of the shadow.

By imagining similar sections, such as E F, parallel to

the former A C, and laying down in the same way semi-

circles corresponding to them, the remaining points in the

curve sought may be obtained. But, as this curve is an

ellipse, of which the diameter D B is the major axis, and

the line 0 a the half minor axis, it follows that this last

line being determined, the curve may be constructed bv

the ordinary methods for ellipses.

We shall now find no difficulty in constructing the

cast shadow in the interior of a concave surface, fig. 185,

formed by the combination of a hollow semi-cylinder

and a quadrant of a hollow sphere, called a niche, as we

Fig. 185.

IS

know the mode of tracing the shadows upon each of

these figures separately. Thus, the shadow of the sphe-

rical portion is part of an ellipse i cD, whose semi-axis

major is OD; the semi-axis minor is obtained by describing

the semicircle B2 % E, with the radius 0 B, drawing from

the point B2 the straight line B2 i', parallel to a line

found in the same way as o to, fig. 183, and finally project-

ing the point of intersection i' to i on the straight line

B O. The point e, where this ellipse cuts the horizontal

diameter A F, limits the cast shadow due to the two

distinct surfaces; therefore, all the points beneath it must

be determined upon the cylindrical part.

Plate LI V. Figs. 1, 2.—To find the line of shade in a

sphere, and the outline of its shadow cast upon the hori-

zontal plane.

The line of shade in a sphere is simply the circumference

of a great circle of which the plane is perpendicular to the

direction of the luminous ray, and consequently inclined

to the two planes of projection. This line will, therefore,

be represented in elevation and plan by two equal ellipses,

the major axes of which are obviously the diameters

C D and C' D', drawn at an angle of 45°.

To find the minor axes of these curves, assume any

point O2, upon the prolongation of the diameter of the

perpendicular C' D', (fig. 2), draw through this point

the straight line O2 O', inclined at an angle of 35° 16', to

A' B' or its parallel, and erect upon it the perpendicular

E2 F2. The projection of the two extremities E2 and F2

upon the line A' B', will give, in the plan, the line E' F'

for the length of the required minor axis of the ellipse;

and this line being again transferred to the elevation,

determines the minor axis of the line of shade in it.