0.5

1 cm

84

ENGINEER AND MACHINIST’S DRAWING-BOOK.

Supposing it were required to draw these ellipses, not

by means of their axes, but by points, any number of

these may be obtained by making horizontal sections of

the sphere. Thus, for example, if we draw the chord G H,

parallel to A' B', to represent one of these sections, and

from the point a, where it cuts the diameter E2 F2, if we

draw a perpendicular to A' B', the points a a, where it

intersects the circumference of a circle representing the

section G H in plan, will be two points in the line of

shade required. These points may be transferred to the

elevation, by supposing a section g h to be made in fig. 1,

corresponding to G H in fig. 2, and projecting the points

a' a' by perpendiculars to g h, the line representing the

cutting plane.

The outline of the shadow cast by the sphere upon the

horizontal plane is also obviously an ellipse; it may be

constructed either by means of its two axes, or by the

help of points, in the manner indicated in the figure.

Figs. 3, 4, and 5.—To draw the line of shade on the

surface of a ring of circular section, in elevation,

plan, and vertical section.

We shall first point out the mode of obtaining those

primary points in the curve which are most easily found,

and then proceed to the general case of determining any

point whatever.

If tangents be drawn to the circles represented in figs.

3 and 5, parallel to the ray of light, their points of con-

tact, a, b, c, d, will be the starting points of the required

lines of shade.

Again, the intersections of the horizontal lines ae,dg,

e f drawn through these points, with the axis of the ring,

will give so many new points in the curve. These points

are denoted in the plan (fig. 4), by setting off the distances

a e and c f upon the vertical line g' D, on both sides of

the centre C'.

Farther, the diameter F' G', drawn at an angle of 45°,

determines, by its intersections with the exterior and

interior circumferences of the ring, four other points

F', t', x, and G', in the curve in question; these points

are all to be projected vertically upon the line A B.

And, lastly, to obtain the lowest points l l, draw tan-

gents to the circles represented in figs. 3 and 5, parallel

to the line o m2 (fig. 6, representing the direction of the

ray of light referred to the vertical plane in which it lies),

and transfer the distances between the points of contact s, s,

and the axis of the ring, to the radius E' O' (fig. 4), where

they are denoted by l' X; these latter points are then to

be projected vertically to l, l, upon the horizontal lines

drawn through the same points s, s (figs. 3 and 5).

The curves sought might now, in most instances, be

traced by the points thus obtained; but should the ring

be on a large scale, and great accuracy be required, it may

be proper to determine a greater number of points. For

this purpose, draw through the centre C', a straight line

T H', in any direction, and, following the method already

explained, draw through O', one of the angular points of

the cube at fig. 6, a straight line parallel to T H', and

from the opposite point m draw a perpendicular m r’

to o' r. Then, having transferred the point r to r- by

means of a circular arc, in order to admit of this last

point being projected to r, we join o r.

Applying this construction to the figures before us, we

now draw tangents to the circles represented in. figs.

3 and 5, parallel to the line o r, and, taking as radii the

distances from their respective points of contact, h and I,

to the axis of the ring, we describe corresponding circles

about the centre C', fig. 4. We thus obtain four other

points in the curves required, namely, I', i, h, and IT,

which may also be projected upon the horizontal lines

drawn through the points h or I (figs. 3 and 4).

By drawing the straight line J' K' so as to form with

F' G' the same angle which the latter makes with the

line H' T, we obtain, by the intersection of that line with

the circles last named, four other points of the curves in

question.

Figs. 7, 8, 9, and 10.-0/ the shadows cast upon the

surfaces of grooved pulleys.

The construction of cast shadows upon surfaces of the

kind now under consideration is founded upon the principle

already announced, that %6hen a circle is parallel to a

plane, its shadoiv, cast upon that plane, is another circle

equal to the original circle.

Take, in the first place, the case of a circular-grooved

pulley (figs. 7 and 8); the cast shadow on its surface is

obviously derived from the circumference of the upper

edge A B. To determine its outline, take any horizontal

line D E upon fig. 7, and describe from the centre C' (fig. 8)

a circle with a radius equal to the half of that line; then,

draw, through the same centre, a line parallel to the ray

of light, which will intersect the plane D E in c; lastly,

describe from the point c, as a centre, an arc of a circle

with a radius equal to A C; the point of intersection, a',

of this arc, with the circumference of the plane D E, will

give when projected to a (fig. 7), one of the points in the

curve required.

To avoid unnecessary labour in drawing more lines

parallel to D E than are required, it is important, in the

first place, to ascertain the highest point in the curve sought.

This point is the shadow of that marked H on the upper

edge of the pulley, and which is determined by the inter-

section of the ray C' H' with the circumference of that

edge in the plan; and it is obtained by drawing through

the point A (fig. 7) a straight line at an angle of

35° 16' with the line A B, and through the point e, strik-

ing a horizontal line e f which by its intersection with

the line IT h, drawn at an angle of 45°, will give the

point sought.

Figs. 9, 10.—In the case of the circular moulding (fig. 7),

which is of frequent occurrence in architecture and ma-

chinery, and exactly corresponds to the lower half of the

pulley (fig. 7), the cast shadow on its surface is evidently

derived from the smaller circle I K; it will, however, be

easily constructed by the method above described, with

the assistance of fig. 8. The triangular-grooved pulley

(fig. 10), should also be treated in the same manner.

The principles we have so fully laid down and illus-

ENGINEER AND MACHINIST’S DRAWING-BOOK.

Supposing it were required to draw these ellipses, not

by means of their axes, but by points, any number of

these may be obtained by making horizontal sections of

the sphere. Thus, for example, if we draw the chord G H,

parallel to A' B', to represent one of these sections, and

from the point a, where it cuts the diameter E2 F2, if we

draw a perpendicular to A' B', the points a a, where it

intersects the circumference of a circle representing the

section G H in plan, will be two points in the line of

shade required. These points may be transferred to the

elevation, by supposing a section g h to be made in fig. 1,

corresponding to G H in fig. 2, and projecting the points

a' a' by perpendiculars to g h, the line representing the

cutting plane.

The outline of the shadow cast by the sphere upon the

horizontal plane is also obviously an ellipse; it may be

constructed either by means of its two axes, or by the

help of points, in the manner indicated in the figure.

Figs. 3, 4, and 5.—To draw the line of shade on the

surface of a ring of circular section, in elevation,

plan, and vertical section.

We shall first point out the mode of obtaining those

primary points in the curve which are most easily found,

and then proceed to the general case of determining any

point whatever.

If tangents be drawn to the circles represented in figs.

3 and 5, parallel to the ray of light, their points of con-

tact, a, b, c, d, will be the starting points of the required

lines of shade.

Again, the intersections of the horizontal lines ae,dg,

e f drawn through these points, with the axis of the ring,

will give so many new points in the curve. These points

are denoted in the plan (fig. 4), by setting off the distances

a e and c f upon the vertical line g' D, on both sides of

the centre C'.

Farther, the diameter F' G', drawn at an angle of 45°,

determines, by its intersections with the exterior and

interior circumferences of the ring, four other points

F', t', x, and G', in the curve in question; these points

are all to be projected vertically upon the line A B.

And, lastly, to obtain the lowest points l l, draw tan-

gents to the circles represented in figs. 3 and 5, parallel

to the line o m2 (fig. 6, representing the direction of the

ray of light referred to the vertical plane in which it lies),

and transfer the distances between the points of contact s, s,

and the axis of the ring, to the radius E' O' (fig. 4), where

they are denoted by l' X; these latter points are then to

be projected vertically to l, l, upon the horizontal lines

drawn through the same points s, s (figs. 3 and 5).

The curves sought might now, in most instances, be

traced by the points thus obtained; but should the ring

be on a large scale, and great accuracy be required, it may

be proper to determine a greater number of points. For

this purpose, draw through the centre C', a straight line

T H', in any direction, and, following the method already

explained, draw through O', one of the angular points of

the cube at fig. 6, a straight line parallel to T H', and

from the opposite point m draw a perpendicular m r’

to o' r. Then, having transferred the point r to r- by

means of a circular arc, in order to admit of this last

point being projected to r, we join o r.

Applying this construction to the figures before us, we

now draw tangents to the circles represented in. figs.

3 and 5, parallel to the line o r, and, taking as radii the

distances from their respective points of contact, h and I,

to the axis of the ring, we describe corresponding circles

about the centre C', fig. 4. We thus obtain four other

points in the curves required, namely, I', i, h, and IT,

which may also be projected upon the horizontal lines

drawn through the points h or I (figs. 3 and 4).

By drawing the straight line J' K' so as to form with

F' G' the same angle which the latter makes with the

line H' T, we obtain, by the intersection of that line with

the circles last named, four other points of the curves in

question.

Figs. 7, 8, 9, and 10.-0/ the shadows cast upon the

surfaces of grooved pulleys.

The construction of cast shadows upon surfaces of the

kind now under consideration is founded upon the principle

already announced, that %6hen a circle is parallel to a

plane, its shadoiv, cast upon that plane, is another circle

equal to the original circle.

Take, in the first place, the case of a circular-grooved

pulley (figs. 7 and 8); the cast shadow on its surface is

obviously derived from the circumference of the upper

edge A B. To determine its outline, take any horizontal

line D E upon fig. 7, and describe from the centre C' (fig. 8)

a circle with a radius equal to the half of that line; then,

draw, through the same centre, a line parallel to the ray

of light, which will intersect the plane D E in c; lastly,

describe from the point c, as a centre, an arc of a circle

with a radius equal to A C; the point of intersection, a',

of this arc, with the circumference of the plane D E, will

give when projected to a (fig. 7), one of the points in the

curve required.

To avoid unnecessary labour in drawing more lines

parallel to D E than are required, it is important, in the

first place, to ascertain the highest point in the curve sought.

This point is the shadow of that marked H on the upper

edge of the pulley, and which is determined by the inter-

section of the ray C' H' with the circumference of that

edge in the plan; and it is obtained by drawing through

the point A (fig. 7) a straight line at an angle of

35° 16' with the line A B, and through the point e, strik-

ing a horizontal line e f which by its intersection with

the line IT h, drawn at an angle of 45°, will give the

point sought.

Figs. 9, 10.—In the case of the circular moulding (fig. 7),

which is of frequent occurrence in architecture and ma-

chinery, and exactly corresponds to the lower half of the

pulley (fig. 7), the cast shadow on its surface is evidently

derived from the smaller circle I K; it will, however, be

easily constructed by the method above described, with

the assistance of fig. 8. The triangular-grooved pulley

(fig. 10), should also be treated in the same manner.

The principles we have so fully laid down and illus-