0.5

1 cm

46

ENGINEER AND MACHINIST’S DRAWING-BOOK.

tical arcs should be traced at the same time, and by the

same method.

Figs. 17 and 18.—Required to draw the lines of pene-

tration of a cylinder and a sphere, the centre of the

sphere being without the axis of the cylinder.

Let the circle D' E' L' be the projection of the base of

the given cylinder, the elevation of which is shown at

Fig. 17, and let A B be the diameter of the given sphere ;

the curves formed by their intersection will present no

difficulty to the attentive student of our previous ex-

amples. For if a plane, as c d, be drawn parallel to the

vertical plane, it will evidently cut the cylinder in two

straight lines G G', H H', parallel to the axis, and pro-

jected vertically from the points G' and hi'. This plane

will also cut the sphere in a circle whose diameter is equal

to c'd, and which is to be described from the centre C

with a radius of half that line; its intersection with the

lines G G' and H H' will evidently give so many points

in the curves sought.

The planes a b' and e' f, which are tangents to the

cylinder, furnish only two points respectively in the

curves; of these points E and F alone are visible, the

other two, L and M, being concealed by the solid ; there-

fore, the planes drawn for the construction of the curves

must be all taken between c d' and e' f. The plane

which passes through the axis of the cylinder cuts the

sphere in a circle whose projection upon the vertical plane

will meet at the points D, N, and g, h, the outlines of the

cylinder, to which the curves of penetration are tangents.

Figs. 19 and 20.—To find the lines of penetration of

a truncated cone and a prism.

The straight line C D is the axis of a truncated cone,

which is represented in the plan by two circles described

from the centre O'; and the horizontal lines M N and

Mi N' are the projections of the axis of a prism of which

the base is square, and the faces respectively parallel and

perpendicular to the planes of projection.

In laying down the plan of this solid we have pur-

posely supposed it to be inverted, in order that the smaller

end of the cone, and the lines of intersection of the lower

surface F G of the prism may be exhibited. According

to this arrangement, the letters A' and B', Fig. 20, ought,

strictly speaking, to be marked at the points T and IF,

and conversely; but as it is quite obvious that the part

above M' N' is exactly symmetrical with that below it,

the distribution of the letters of reference adopted in our

figures can lead to no confusion.

The intersection of the plane F G with the cone is pro-

jected horizontally in a circle described from the centre O',

with the diameter F' G'. The arcs F F' A' and FT G' B'

are the only parts of this circle which require to be drawn.

To find the lines of intersection of the two solids in the

vertical projection, we have only to remark that, as the

perpendicular side of the prism is parallel to the axis, it

will cut the cone in a hyperbolic curve, which may be

drawn according to the method already described; the

lines K A and L B are the only parts of the curve which

are visible in the present example.

Figs. 21 and 22.—To describe the curves formed by the

intersection of a cylinder with the frustum of a cone ;

the axes of the tivo solids cutting each other at right

angles.

The axes of the solids and their projections are laid

down in the figures, precisely as in the preceding exam-

ple ; the plan being inverted, as before, the same expla-

natory remark should be kept in mind.

The intersections of the outlines of the cone in the ele-

vation, with those of the cylinder furnish, obviously, four

points in the curves of penetration; these points are all

projected horizontally upon the line A' B'. Now, suppose

a plane, as a b, Fig. 21, to pass horizontally through both

solids; its intersection with the cone will be a circle of

the diameter c d, while the cylinder will be cut in two

parallel straight lines, represented in the elevation by

a b, and whose horizontal projection may be determined

in the following manner:—Conceive a vertical plane/g,

cutting the cylinder at right angles to its axis, and let the

circle g ef thereby formed, be described from the intersec-

tion of the axes of the two solids; the line j h will now

represent, in this position of the section, the distance of

one of the lines sought from the axis of the cylinder.

Now set off this distance on both sides of the point A',

and through the points k and a, thus obtained, draw

straight lines parallel to A' B'; the intersections of these

lines with the circle drawn from the centre C' of the dia-

meter c d will give four points to, p, n and o, which being

projected vertically upon a b, determine two points to and

p in the curves required.

In order to obtain the vertices or adjacent limiting

points of the curves, draw from the vertex of the cone, a

straight line t e, touching the circle g ef and let a hori-

zontal plane be supposed to pass through the point of

contact e. If now, we proceed, according to the method

given above, to determine the intersections of this plane

with each of the solids in question, we shall find four

points i', r, q and s, which being projected vertically

upon the line e r determine the vertices i and r required.

Of the Helix.—Plate YII.

The Helix, or, as it is sometimes, though improperly

termed, the Spiral, is the curve described upon the sur-

face of a cylinder by a point revolving round it, and, at

the same time, moving parallel to its axis by a certain

invariable distance during each revolution. This distance

is called the pitch of the screw.

Figs. 1 and 2.—Required to construct the helical curve

described by the point A upon a cylinder projected hori-

zontally in the circle A' O' F', the pitch being represented

by the line A A3.

Divide the pitch A A3 into any number of equal parts,

say eight; and, through each point of division 1, 2, 3, &c.,

draw straight lines parallel to the ground line. Then

divide the circumference A' G F' into the same number of

parts ; the points of division B', C', E', F', &c., will be the

horizontal projections of the different positions of the

given point during its motion round the cylinder. Thus,

ENGINEER AND MACHINIST’S DRAWING-BOOK.

tical arcs should be traced at the same time, and by the

same method.

Figs. 17 and 18.—Required to draw the lines of pene-

tration of a cylinder and a sphere, the centre of the

sphere being without the axis of the cylinder.

Let the circle D' E' L' be the projection of the base of

the given cylinder, the elevation of which is shown at

Fig. 17, and let A B be the diameter of the given sphere ;

the curves formed by their intersection will present no

difficulty to the attentive student of our previous ex-

amples. For if a plane, as c d, be drawn parallel to the

vertical plane, it will evidently cut the cylinder in two

straight lines G G', H H', parallel to the axis, and pro-

jected vertically from the points G' and hi'. This plane

will also cut the sphere in a circle whose diameter is equal

to c'd, and which is to be described from the centre C

with a radius of half that line; its intersection with the

lines G G' and H H' will evidently give so many points

in the curves sought.

The planes a b' and e' f, which are tangents to the

cylinder, furnish only two points respectively in the

curves; of these points E and F alone are visible, the

other two, L and M, being concealed by the solid ; there-

fore, the planes drawn for the construction of the curves

must be all taken between c d' and e' f. The plane

which passes through the axis of the cylinder cuts the

sphere in a circle whose projection upon the vertical plane

will meet at the points D, N, and g, h, the outlines of the

cylinder, to which the curves of penetration are tangents.

Figs. 19 and 20.—To find the lines of penetration of

a truncated cone and a prism.

The straight line C D is the axis of a truncated cone,

which is represented in the plan by two circles described

from the centre O'; and the horizontal lines M N and

Mi N' are the projections of the axis of a prism of which

the base is square, and the faces respectively parallel and

perpendicular to the planes of projection.

In laying down the plan of this solid we have pur-

posely supposed it to be inverted, in order that the smaller

end of the cone, and the lines of intersection of the lower

surface F G of the prism may be exhibited. According

to this arrangement, the letters A' and B', Fig. 20, ought,

strictly speaking, to be marked at the points T and IF,

and conversely; but as it is quite obvious that the part

above M' N' is exactly symmetrical with that below it,

the distribution of the letters of reference adopted in our

figures can lead to no confusion.

The intersection of the plane F G with the cone is pro-

jected horizontally in a circle described from the centre O',

with the diameter F' G'. The arcs F F' A' and FT G' B'

are the only parts of this circle which require to be drawn.

To find the lines of intersection of the two solids in the

vertical projection, we have only to remark that, as the

perpendicular side of the prism is parallel to the axis, it

will cut the cone in a hyperbolic curve, which may be

drawn according to the method already described; the

lines K A and L B are the only parts of the curve which

are visible in the present example.

Figs. 21 and 22.—To describe the curves formed by the

intersection of a cylinder with the frustum of a cone ;

the axes of the tivo solids cutting each other at right

angles.

The axes of the solids and their projections are laid

down in the figures, precisely as in the preceding exam-

ple ; the plan being inverted, as before, the same expla-

natory remark should be kept in mind.

The intersections of the outlines of the cone in the ele-

vation, with those of the cylinder furnish, obviously, four

points in the curves of penetration; these points are all

projected horizontally upon the line A' B'. Now, suppose

a plane, as a b, Fig. 21, to pass horizontally through both

solids; its intersection with the cone will be a circle of

the diameter c d, while the cylinder will be cut in two

parallel straight lines, represented in the elevation by

a b, and whose horizontal projection may be determined

in the following manner:—Conceive a vertical plane/g,

cutting the cylinder at right angles to its axis, and let the

circle g ef thereby formed, be described from the intersec-

tion of the axes of the two solids; the line j h will now

represent, in this position of the section, the distance of

one of the lines sought from the axis of the cylinder.

Now set off this distance on both sides of the point A',

and through the points k and a, thus obtained, draw

straight lines parallel to A' B'; the intersections of these

lines with the circle drawn from the centre C' of the dia-

meter c d will give four points to, p, n and o, which being

projected vertically upon a b, determine two points to and

p in the curves required.

In order to obtain the vertices or adjacent limiting

points of the curves, draw from the vertex of the cone, a

straight line t e, touching the circle g ef and let a hori-

zontal plane be supposed to pass through the point of

contact e. If now, we proceed, according to the method

given above, to determine the intersections of this plane

with each of the solids in question, we shall find four

points i', r, q and s, which being projected vertically

upon the line e r determine the vertices i and r required.

Of the Helix.—Plate YII.

The Helix, or, as it is sometimes, though improperly

termed, the Spiral, is the curve described upon the sur-

face of a cylinder by a point revolving round it, and, at

the same time, moving parallel to its axis by a certain

invariable distance during each revolution. This distance

is called the pitch of the screw.

Figs. 1 and 2.—Required to construct the helical curve

described by the point A upon a cylinder projected hori-

zontally in the circle A' O' F', the pitch being represented

by the line A A3.

Divide the pitch A A3 into any number of equal parts,

say eight; and, through each point of division 1, 2, 3, &c.,

draw straight lines parallel to the ground line. Then

divide the circumference A' G F' into the same number of

parts ; the points of division B', C', E', F', &c., will be the

horizontal projections of the different positions of the

given point during its motion round the cylinder. Thus,