DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0122
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.