0.5

1 cm

42

ENGINEER AND MACHINIST’S DBA WING-BOOK/

of Figs. 3 and 6; and, having joined the contiguous points,

and the corresponding angles of the upper and lower sur-

face, we obtain the complete vertical projection of the

prism in its doubly-inclined position.

Construction of the Conic Sections.—Plate III.

Let it be required in the first place, to represent a cone

in elevation and plan. Figs. 1 and 2.

The horizontal projection of the cone is simply a circle,

described from the centre S', of a diameter equal to that

of the base. Its elevation is an isosceles triangle, ob-

tained by drawing tangents A' A, B' B, perpendicular to,

and intersecting the ground line; then set off upon the

centre line the height C S, and join S A, S B. These lines

are called the exterior generatrices of the cone.

Figs. 1 and 2.—Given the projections of a cone, and

the direction of a plane X X, cutting it perpendicularly

to the vertical, and obliquely to the horizontal plane;

required to find, first, the horizontal projection of this

section; and, secondly, the outline of the ellipse thus

formed.

Through the vertex of the cone, draw a line S E to any

point within the base A B ; this line is to be regarded as

the vertical projection of a generatrix of the cone, and the

point e, where it intersects the line X X, is the projection

of that point on the surface of the solid, where the cutting-

plane actually passes through the generatrix E S. The

point e may be projected upon the plan by letting fall a

perpendicular from E, cutting the circumference of the

base in E', and joining E' S'; then another perpendicular

let fall from e will intersect E' S' in a point e', which will

be the horizontal projection of a point in the curve

required. By drawing another line, such as S D, and

projecting its point of intersection d with the cutting

plane, to d', a second point in the curve is obtained ; and

so on for any required number of points.

The exterior generatrices A S and B S, being both

projected upon the line A' B', the extreme limits of the

curve sought will be at the points a’ and b', on that line,

which are the projections of the points of intersection

a and b, of the cutting plane with the outlines of the cone.

And, since the line a b' will obviously divide the curve

symmetrically into two equal parts, the points /', g, h',

&c., will be readily obtained by setting off above that

line, and on their respective perpendiculars, the distances

d'd?, e e2, &c. A sufficient number of points having thus

been determined, the curve drawn through them (which

will be found to be an ellipse), will be the outline of the

section required.

This curve may be obtained by another, and perhaps

simpler method, depending on the principle that all sec-

tions of a cone by planes parallel to the base are circles.

Thus, let the line F G represent a cutting plane ; the sec-

tion which it makes with the cone will be denoted, on

the horizontal projection, by a circle drawn from the

centre S' with a radius equal to half the line F G ; and,

by projecting the point of intersection H, of the horizontal

and oblique planes, by a perpendicular II IT', and noting

where this line cuts the circle above referred to, we obtain

two points IF and I' in the curve required. By a similar

construction, as exemplified in our drawings, any number

of additional points may be found.

As the projection obtained by the preceding methods

exhibits the section as fore-shortened, and not in its true

dimensions, we shall now proceed to the consideration of

the second question proposed. Let the cutting plane X X

be conceived to turn upon the point b, so as to coincide

with the vertical line b h, and (to avoid confusion of lines),

let b k be transferred to a b', which will represent, as

before, the extreme limits of the curve required. Now,

taking any point, such as d, it is obvious that, in this new

position of the cutting plane, it will be represented by d2,

and that, if we make the further supposition that the

cutting plane were turned upon a' b', as an axis, till it

should be parallel to the vertical plane, the point which

had been projected at d2 would then have described round

a1 b' an arc of a circle whose radius is the distance d'd2,

Fig. 2. This distance, therefore, being set off at d' and/',

on each side of a b', gives two points in the curve sought.

By a similar mode of operation any number of points may

be obtained, through wdiich, if we draw a curve, it will

be an ellipse, of the true form and dimensions of the sec-

tion. Or, having determined the major and minor axes,

the student may, if he deems it preferable, construct the

ellipse by any of the methods already given.

Figs. 3 and 4.—To find the horizontal projection, and

actual outline of the section of a cone, made by a plane

Y Y parallel to one side, or generatrix, and perpendi-

cular to the vertical plane.

By following the second method laid down in the pre-

ceding problem, we may readily obtain any number of

points, as F', G', J', K, &c., in the curve representing the

horizontal projection of the section specified; a simple

reference to the lines in the drawings will sufficiently

explain the application to the present case. We may only

remark that the horizontal plane passing through M gives

only one point M' (which is the vertex of the curve sought),

because the circle which denotes the section that it makes

-with the cone is a tangent to the given plane.

In order to determine the actual outline of this curve,

let us suppose the plane Y Y to turn, as upon a pivot at

M, until it has assumed the position M B, and transfer

M B parallel to itself, to M' B'. The point F will thus

have first described the arc F E till it reaches the point

E, which is then projected to E2; so that, if we conceive

the given plane, now represented by M' B', to turn upon

that line as an axis, until it assumes a position parallel to

the vertical plane, we shall find that the point E2, which

is distant from the axis M' B' by the distance F' S', Fig. 4,

will now be projected to F', Fig. 3. The same distance

F' S' set off on the other side of the axis M' B', gives ano-

ther point G' in the curve required, which is that called

the parabola.

Figs. 5 and 6.—To dravj the vertical projection of the

section of a cone made by a plane parallel to its axis

and to the vertical plane.

ENGINEER AND MACHINIST’S DBA WING-BOOK/

of Figs. 3 and 6; and, having joined the contiguous points,

and the corresponding angles of the upper and lower sur-

face, we obtain the complete vertical projection of the

prism in its doubly-inclined position.

Construction of the Conic Sections.—Plate III.

Let it be required in the first place, to represent a cone

in elevation and plan. Figs. 1 and 2.

The horizontal projection of the cone is simply a circle,

described from the centre S', of a diameter equal to that

of the base. Its elevation is an isosceles triangle, ob-

tained by drawing tangents A' A, B' B, perpendicular to,

and intersecting the ground line; then set off upon the

centre line the height C S, and join S A, S B. These lines

are called the exterior generatrices of the cone.

Figs. 1 and 2.—Given the projections of a cone, and

the direction of a plane X X, cutting it perpendicularly

to the vertical, and obliquely to the horizontal plane;

required to find, first, the horizontal projection of this

section; and, secondly, the outline of the ellipse thus

formed.

Through the vertex of the cone, draw a line S E to any

point within the base A B ; this line is to be regarded as

the vertical projection of a generatrix of the cone, and the

point e, where it intersects the line X X, is the projection

of that point on the surface of the solid, where the cutting-

plane actually passes through the generatrix E S. The

point e may be projected upon the plan by letting fall a

perpendicular from E, cutting the circumference of the

base in E', and joining E' S'; then another perpendicular

let fall from e will intersect E' S' in a point e', which will

be the horizontal projection of a point in the curve

required. By drawing another line, such as S D, and

projecting its point of intersection d with the cutting

plane, to d', a second point in the curve is obtained ; and

so on for any required number of points.

The exterior generatrices A S and B S, being both

projected upon the line A' B', the extreme limits of the

curve sought will be at the points a’ and b', on that line,

which are the projections of the points of intersection

a and b, of the cutting plane with the outlines of the cone.

And, since the line a b' will obviously divide the curve

symmetrically into two equal parts, the points /', g, h',

&c., will be readily obtained by setting off above that

line, and on their respective perpendiculars, the distances

d'd?, e e2, &c. A sufficient number of points having thus

been determined, the curve drawn through them (which

will be found to be an ellipse), will be the outline of the

section required.

This curve may be obtained by another, and perhaps

simpler method, depending on the principle that all sec-

tions of a cone by planes parallel to the base are circles.

Thus, let the line F G represent a cutting plane ; the sec-

tion which it makes with the cone will be denoted, on

the horizontal projection, by a circle drawn from the

centre S' with a radius equal to half the line F G ; and,

by projecting the point of intersection H, of the horizontal

and oblique planes, by a perpendicular II IT', and noting

where this line cuts the circle above referred to, we obtain

two points IF and I' in the curve required. By a similar

construction, as exemplified in our drawings, any number

of additional points may be found.

As the projection obtained by the preceding methods

exhibits the section as fore-shortened, and not in its true

dimensions, we shall now proceed to the consideration of

the second question proposed. Let the cutting plane X X

be conceived to turn upon the point b, so as to coincide

with the vertical line b h, and (to avoid confusion of lines),

let b k be transferred to a b', which will represent, as

before, the extreme limits of the curve required. Now,

taking any point, such as d, it is obvious that, in this new

position of the cutting plane, it will be represented by d2,

and that, if we make the further supposition that the

cutting plane were turned upon a' b', as an axis, till it

should be parallel to the vertical plane, the point which

had been projected at d2 would then have described round

a1 b' an arc of a circle whose radius is the distance d'd2,

Fig. 2. This distance, therefore, being set off at d' and/',

on each side of a b', gives two points in the curve sought.

By a similar mode of operation any number of points may

be obtained, through wdiich, if we draw a curve, it will

be an ellipse, of the true form and dimensions of the sec-

tion. Or, having determined the major and minor axes,

the student may, if he deems it preferable, construct the

ellipse by any of the methods already given.

Figs. 3 and 4.—To find the horizontal projection, and

actual outline of the section of a cone, made by a plane

Y Y parallel to one side, or generatrix, and perpendi-

cular to the vertical plane.

By following the second method laid down in the pre-

ceding problem, we may readily obtain any number of

points, as F', G', J', K, &c., in the curve representing the

horizontal projection of the section specified; a simple

reference to the lines in the drawings will sufficiently

explain the application to the present case. We may only

remark that the horizontal plane passing through M gives

only one point M' (which is the vertex of the curve sought),

because the circle which denotes the section that it makes

-with the cone is a tangent to the given plane.

In order to determine the actual outline of this curve,

let us suppose the plane Y Y to turn, as upon a pivot at

M, until it has assumed the position M B, and transfer

M B parallel to itself, to M' B'. The point F will thus

have first described the arc F E till it reaches the point

E, which is then projected to E2; so that, if we conceive

the given plane, now represented by M' B', to turn upon

that line as an axis, until it assumes a position parallel to

the vertical plane, we shall find that the point E2, which

is distant from the axis M' B' by the distance F' S', Fig. 4,

will now be projected to F', Fig. 3. The same distance

F' S' set off on the other side of the axis M' B', gives ano-

ther point G' in the curve required, which is that called

the parabola.

Figs. 5 and 6.—To dravj the vertical projection of the

section of a cone made by a plane parallel to its axis

and to the vertical plane.