0.5

1 cm

44

ENGINEER AND MACHINIST’S DRAWING-BOOK.

Penetrations of Cylinders, Cones, and Spheres.—

Plate Y.

Figs. 9 and 10.—To find the curves resulting from the

intersection of two cylinders of unequal diameters, meet-

ing at any angle.

For the sake of simplicity, let ns suppose the axes of

both cylinders to be parallel to the vertical plane, and let

ABED and N 0 Q P be their projections upon that plane.

In constructing, in the first place, their horizontal projec-

tion, we shall observe that the upper end A B, of the

larger cylinder, is represented by an ellipse, A' K' B' M',

which may easily be drawn by the help of the major axis

K M equal to the diameter of the cylinder, and of the

minor A' B', the projection of the diameter. The visible

portion of the base of the cylinder being similarly repre-

sented by the semi-ellipse L' D' II', its entire outline will

be completed by drawing tangents L' M' and H' K'. The

upper extremity P N of the smaller cylinder will also be

projected in the ellipse P i N.

Now, suppose a plane, as a g, Fig. 10, to pass through

both cylinders parallel to their axes ; it will cut the sur-

face of the larger cylinder in two straight lines passing

through the points f and g', on the upper end of the

cylinder; these lines will be represented in the elevation,

by projecting the points/' and g to f g, and drawing a f

and c g parallel to the axis. The plane a' g' will, in like

manner, cut the smaller cylinder in two straight lines which

will be represented in the vertical projection by cl h and

e i, and the intersections of these lines with af and c g

will give four points l, k, m and n in the curves of pene-

tration. Of these points one only, that marked l, is visible

in the plan, where it is denoted by V.

Second Method. Fig. 9.—To find the curves of 'pene-

tration in the elevat ion, without the aid of the plan.

Let the bases D E and Q 0 of both cylinders be conceived

to be turned over into the vertical plane after being trans-

ferred to any convenient distance, as D2 E2 and Q2 O2,

from the principal figure ; they will then be represented by

the circles D2 H2 E2, and Q2 G' O2. Now draw a2 c2

parallel to D E, and at any suitable distance from the

centre I ; this line will represent the intersection of the

base of the cylinder with a plane parallel to the axes of

both, as before. The intersection of this plane with the

base of the smaller cylinder will be found by setting off

from R a distance R p, equal to I o2, and drawing through

the point p a straight line parallel to Q 0. It is obvious

that the intersection of the supposed plane with the con-

vex surfaces of the cylinders will be represented by the

lines a f, c g and d li, e i, drawn parallel to the axes of the

respective cylinders, through the points where the chords

a1 c2 and d1 e2 cut the circles of their bases ; and that,

consequently, the intersections of these lines indicate points

in the curves sought. These points may be multiplied

indefinitely by conceiving other planes to pass through the

cylinders, and operating as before.

The vertices r and s of the curves are determined by

the intersection of the axis of the smaller cylinder with

the lines r r2, s s2, drawn through the extremities of a

chord r2 s2 passing through a point q, at a distance from

the centre I equal to the radius of the smaller cylinder.

Figs. 11 and 12.—To find the curves of penetration of

a cone and sphere.

Let D S be the axis of the cone, A' L' B' the circle of

its base, and the triangle A B S its projection on the

vertical plane: and let C, C' be the projections of the centre,

and the circles E' K' F' and E G F those of the circum-

ferences of the sphere.

This problem, like most others similar to it, can be solved

only by the aid of imaginary intersecting planes. Let

a b, Fig. 11, represent the projection of a horizontal plane;

it will obviously cut the sphere in a circle whose diameter

is a b, and which is to be drawn from the centre C', in

the plan. Its intersection with the cone is also a circle

described from the centre S' with the diameter c d; the

points e and/' where these two circles cut each other are

the horizontal projections of two points in the lower curve,

which is evidently entirely hidden by the sphere. The

points referred to are projected vertically upon the line a b,

at e and /. The upper curve, which is seen in both projec-

tions, is obtained by a similar process; but it is to be ob-

served that the horizontal cutting planes must be taken

in such positions as to pass through both solids in circles

which shall intersect each other. For our guidance in

this respect, it will be necessary, first, to determine the ver-

tices to and n of the curves of penetration.

For this purpose, conceive a vertical plane passing

through the axis of the cone and the centre of the sphere;

its horizontal projection will be the straight line C' L',

joining the centres of the two bodies. Let us also make the

supposition that this plane is turned upon the line C C', as

on an axis, until it becomes parallel to the vertical plane ;

the points S' and L' will now have assumed the positions

S2 and L2, and consequently the axis of the cone will be

projected vertically in the line D' S3, and its side in S3 L3,

cutting the sphere at the points p and r. If we now con-

ceive the solids to have resumed their original relative

positions, it is clear that the vertices or adjacent limiting

points of the curves of penetration must be in the hori-

zontal lines p o and r q, drawn through the points de-

termined as above ; their exact positions on these lines may

be ascertained by projecting vertically the points to' and

n, where the arcs described by the points p and r in re-

storing the cone to its first position, intersect the fine S L.

It is of importance further, to ascertain the points at

which the curves of penetration meet the outlines A S and

S B of the cone. The plane which passes through these

lines, being projected horizontally in A' B', will cut the

sphere in a circle whose diameter is i j; this circle, de-

scribed in the elevation from the centre C, will cut the

sides A S and S B in four points at which the curves of

penetration are tangents to the outlines of the cone.

Figs 13 and 14.—To find the lines of penetration of a

cylinder and a cylindrical ring or torus.

Let the circles A' E' B', F' G' K' represent the hori-

zontal, and the figure A C B D the vertical projection of

ENGINEER AND MACHINIST’S DRAWING-BOOK.

Penetrations of Cylinders, Cones, and Spheres.—

Plate Y.

Figs. 9 and 10.—To find the curves resulting from the

intersection of two cylinders of unequal diameters, meet-

ing at any angle.

For the sake of simplicity, let ns suppose the axes of

both cylinders to be parallel to the vertical plane, and let

ABED and N 0 Q P be their projections upon that plane.

In constructing, in the first place, their horizontal projec-

tion, we shall observe that the upper end A B, of the

larger cylinder, is represented by an ellipse, A' K' B' M',

which may easily be drawn by the help of the major axis

K M equal to the diameter of the cylinder, and of the

minor A' B', the projection of the diameter. The visible

portion of the base of the cylinder being similarly repre-

sented by the semi-ellipse L' D' II', its entire outline will

be completed by drawing tangents L' M' and H' K'. The

upper extremity P N of the smaller cylinder will also be

projected in the ellipse P i N.

Now, suppose a plane, as a g, Fig. 10, to pass through

both cylinders parallel to their axes ; it will cut the sur-

face of the larger cylinder in two straight lines passing

through the points f and g', on the upper end of the

cylinder; these lines will be represented in the elevation,

by projecting the points/' and g to f g, and drawing a f

and c g parallel to the axis. The plane a' g' will, in like

manner, cut the smaller cylinder in two straight lines which

will be represented in the vertical projection by cl h and

e i, and the intersections of these lines with af and c g

will give four points l, k, m and n in the curves of pene-

tration. Of these points one only, that marked l, is visible

in the plan, where it is denoted by V.

Second Method. Fig. 9.—To find the curves of 'pene-

tration in the elevat ion, without the aid of the plan.

Let the bases D E and Q 0 of both cylinders be conceived

to be turned over into the vertical plane after being trans-

ferred to any convenient distance, as D2 E2 and Q2 O2,

from the principal figure ; they will then be represented by

the circles D2 H2 E2, and Q2 G' O2. Now draw a2 c2

parallel to D E, and at any suitable distance from the

centre I ; this line will represent the intersection of the

base of the cylinder with a plane parallel to the axes of

both, as before. The intersection of this plane with the

base of the smaller cylinder will be found by setting off

from R a distance R p, equal to I o2, and drawing through

the point p a straight line parallel to Q 0. It is obvious

that the intersection of the supposed plane with the con-

vex surfaces of the cylinders will be represented by the

lines a f, c g and d li, e i, drawn parallel to the axes of the

respective cylinders, through the points where the chords

a1 c2 and d1 e2 cut the circles of their bases ; and that,

consequently, the intersections of these lines indicate points

in the curves sought. These points may be multiplied

indefinitely by conceiving other planes to pass through the

cylinders, and operating as before.

The vertices r and s of the curves are determined by

the intersection of the axis of the smaller cylinder with

the lines r r2, s s2, drawn through the extremities of a

chord r2 s2 passing through a point q, at a distance from

the centre I equal to the radius of the smaller cylinder.

Figs. 11 and 12.—To find the curves of penetration of

a cone and sphere.

Let D S be the axis of the cone, A' L' B' the circle of

its base, and the triangle A B S its projection on the

vertical plane: and let C, C' be the projections of the centre,

and the circles E' K' F' and E G F those of the circum-

ferences of the sphere.

This problem, like most others similar to it, can be solved

only by the aid of imaginary intersecting planes. Let

a b, Fig. 11, represent the projection of a horizontal plane;

it will obviously cut the sphere in a circle whose diameter

is a b, and which is to be drawn from the centre C', in

the plan. Its intersection with the cone is also a circle

described from the centre S' with the diameter c d; the

points e and/' where these two circles cut each other are

the horizontal projections of two points in the lower curve,

which is evidently entirely hidden by the sphere. The

points referred to are projected vertically upon the line a b,

at e and /. The upper curve, which is seen in both projec-

tions, is obtained by a similar process; but it is to be ob-

served that the horizontal cutting planes must be taken

in such positions as to pass through both solids in circles

which shall intersect each other. For our guidance in

this respect, it will be necessary, first, to determine the ver-

tices to and n of the curves of penetration.

For this purpose, conceive a vertical plane passing

through the axis of the cone and the centre of the sphere;

its horizontal projection will be the straight line C' L',

joining the centres of the two bodies. Let us also make the

supposition that this plane is turned upon the line C C', as

on an axis, until it becomes parallel to the vertical plane ;

the points S' and L' will now have assumed the positions

S2 and L2, and consequently the axis of the cone will be

projected vertically in the line D' S3, and its side in S3 L3,

cutting the sphere at the points p and r. If we now con-

ceive the solids to have resumed their original relative

positions, it is clear that the vertices or adjacent limiting

points of the curves of penetration must be in the hori-

zontal lines p o and r q, drawn through the points de-

termined as above ; their exact positions on these lines may

be ascertained by projecting vertically the points to' and

n, where the arcs described by the points p and r in re-

storing the cone to its first position, intersect the fine S L.

It is of importance further, to ascertain the points at

which the curves of penetration meet the outlines A S and

S B of the cone. The plane which passes through these

lines, being projected horizontally in A' B', will cut the

sphere in a circle whose diameter is i j; this circle, de-

scribed in the elevation from the centre C, will cut the

sides A S and S B in four points at which the curves of

penetration are tangents to the outlines of the cone.

Figs 13 and 14.—To find the lines of penetration of a

cylinder and a cylindrical ring or torus.

Let the circles A' E' B', F' G' K' represent the hori-

zontal, and the figure A C B D the vertical projection of