0.5

1 cm

106

ENGINEER AND MACHINIST’S DRAWING-BOOK.

circular, and its summit in the eye of the spectator. Let

us conceive this cone cut by the plane of the picture,

and its section in the picture will be the perspective

of its base, or of the given circle. This operation can

be performed by the rules of descriptive geometry, and

the result will be the same as by the problem above. The

perspective of this circle is, then, necessarily an ellipse,

since it is the result of the section of a cone by a plane

which passes through both sides, and is not parallel to its

base. It is further to be observed, that in the ellipse the

principal axis m n does not pass through the point o, the

perspective of the original centre of the circle, but is the per-

spective of a chord M N, determined by the tangents M cNc.

Pkoblem IX.—To inscribe a circle in a square given in

perspective, and of which one side is parallel to the base

of the picture.

We know by the preceding pro-

blem that the ellipse should touch the

four sides of the square in their appa-

rent or perspective centres at the

points e,f, g,h, {Fig. 216). We can,

therefore, consider the line ef as the

minor axis of the ellipse which we

seek; and as we know that the major

axis should cross it perpendicularly

in its centre, we divide ef into two

equal parts, and through i, the centre of the ellipse, draw

perpendicularly to of an indefinite line, which will be the

direction of the major axis, of which we have to de-

termine the length. Take fi or i e, and carry it from h

to j, and draw through h and j a straight line to the minor

axis at h, and hk will be equal to half the major axis

which we set off from i to l and m. Having now the

major and minor axes of the ellipse, it is easy to draw it

with the aid of a slip of paper, or in any other way.

(Figs. 216 and 2170

(Fig. 218.)

Through the points of division, on the geometrical plan

of the circle {Fig. 218), draw radii, and produce them to

intersect the sides of the circumscribing square. Then

from the intersections visual

rays may be drawn, and the

corresponding points obtained

in the perspective square,

from which radii drawn to

the perspective centre will

cut the perspective circle in

the points required.

But we may in most cases

dispense with the visual rays,

and obtain the perspective

divisions of the square by Problem II., thus:—

From the hither angle A of the square {Fig. 219) draw

I1 Fig. 2190

w

In the next figure the same method may be thus ap-

plied:—

Let a c {Fig. 217) be the given square, through the in-

tersection of its diagonals draw/e, and divide it into two

equal parts in i, and through i draw an indefinite line

parallel to g h. Through q, the perspective centre of the

circle, draw a line perpendicular to g h, and produce it

both ways, when it will cut the side dcins, and the line

l to in r. Carry the length s r from g or h upon l m to

p or y, and draw gp or hy to cut s q in the point o, and

either of the lines g p o or h y o will be the rule with

which to operate in describing the ellipse as before.

If it is required to divide the periphery of the circle

into any number of parts, equal or otherwise, it may be

thus performed:—

any line, as A B, A G, equal to the side of the square on

the plan, and on them set off the intersections of the radii.

Then from B draw through C a line cutting the horizon

in D, and from G through F a line cutting the horizon in

E, and from the divisions of the line A B, AG, draw lines

to these points, which will divide AC, AF, perspectively

in the same ratios, and from these divisions in A C, A F,

draw radii to the perspective centre, which will divide

the quadrants of the periphery of the circle, as required.

Repeat the operation for the other sides.

There is yet another method, which is of very great

applicability.

(Fig. 220.)

Through the divi-

sions in the plan of

the circle {Fig. 220),

draw lines parallel to

one of the sides of

the square, and pro-

duce them to intersect

any of its sides, and

from the perspectives

of these intersections

draw lines to the va-

nishing point of the

side to which the lines

drawn through the divisions of the circle are parallel.

ENGINEER AND MACHINIST’S DRAWING-BOOK.

circular, and its summit in the eye of the spectator. Let

us conceive this cone cut by the plane of the picture,

and its section in the picture will be the perspective

of its base, or of the given circle. This operation can

be performed by the rules of descriptive geometry, and

the result will be the same as by the problem above. The

perspective of this circle is, then, necessarily an ellipse,

since it is the result of the section of a cone by a plane

which passes through both sides, and is not parallel to its

base. It is further to be observed, that in the ellipse the

principal axis m n does not pass through the point o, the

perspective of the original centre of the circle, but is the per-

spective of a chord M N, determined by the tangents M cNc.

Pkoblem IX.—To inscribe a circle in a square given in

perspective, and of which one side is parallel to the base

of the picture.

We know by the preceding pro-

blem that the ellipse should touch the

four sides of the square in their appa-

rent or perspective centres at the

points e,f, g,h, {Fig. 216). We can,

therefore, consider the line ef as the

minor axis of the ellipse which we

seek; and as we know that the major

axis should cross it perpendicularly

in its centre, we divide ef into two

equal parts, and through i, the centre of the ellipse, draw

perpendicularly to of an indefinite line, which will be the

direction of the major axis, of which we have to de-

termine the length. Take fi or i e, and carry it from h

to j, and draw through h and j a straight line to the minor

axis at h, and hk will be equal to half the major axis

which we set off from i to l and m. Having now the

major and minor axes of the ellipse, it is easy to draw it

with the aid of a slip of paper, or in any other way.

(Figs. 216 and 2170

(Fig. 218.)

Through the points of division, on the geometrical plan

of the circle {Fig. 218), draw radii, and produce them to

intersect the sides of the circumscribing square. Then

from the intersections visual

rays may be drawn, and the

corresponding points obtained

in the perspective square,

from which radii drawn to

the perspective centre will

cut the perspective circle in

the points required.

But we may in most cases

dispense with the visual rays,

and obtain the perspective

divisions of the square by Problem II., thus:—

From the hither angle A of the square {Fig. 219) draw

I1 Fig. 2190

w

In the next figure the same method may be thus ap-

plied:—

Let a c {Fig. 217) be the given square, through the in-

tersection of its diagonals draw/e, and divide it into two

equal parts in i, and through i draw an indefinite line

parallel to g h. Through q, the perspective centre of the

circle, draw a line perpendicular to g h, and produce it

both ways, when it will cut the side dcins, and the line

l to in r. Carry the length s r from g or h upon l m to

p or y, and draw gp or hy to cut s q in the point o, and

either of the lines g p o or h y o will be the rule with

which to operate in describing the ellipse as before.

If it is required to divide the periphery of the circle

into any number of parts, equal or otherwise, it may be

thus performed:—

any line, as A B, A G, equal to the side of the square on

the plan, and on them set off the intersections of the radii.

Then from B draw through C a line cutting the horizon

in D, and from G through F a line cutting the horizon in

E, and from the divisions of the line A B, AG, draw lines

to these points, which will divide AC, AF, perspectively

in the same ratios, and from these divisions in A C, A F,

draw radii to the perspective centre, which will divide

the quadrants of the periphery of the circle, as required.

Repeat the operation for the other sides.

There is yet another method, which is of very great

applicability.

(Fig. 220.)

Through the divi-

sions in the plan of

the circle {Fig. 220),

draw lines parallel to

one of the sides of

the square, and pro-

duce them to intersect

any of its sides, and

from the perspectives

of these intersections

draw lines to the va-

nishing point of the

side to which the lines

drawn through the divisions of the circle are parallel.