An Improved Instrument for Drawing Curves.
Year: 1899

page_text: Date of Application, 23rd Mar., 1899-Accepted, 11th Nov., 1899
COMPLETE SPECIFICATION.
An Improved Instrument for Drawing Curves.
I, DR. LUIGI CEREBOTANI, of No. 63, Sendlinger Strasse, in the City of Munich, in
the German Empire, Professor, do hereby declare the nature of this invention and
in what manner the same is to be performed, to be particularly described and ascer-
tained in and by the following statement :-
The subject of the present invention is a process of rapidly and with mathematical
accuracy describing curves, such as closed curves (circles and ovals) and open spirals
of any pitch, circular or oval, and the apparatus used in the process. Such curves
can be described:
(.) from a point lying at any distance desired from the curve ;
(b.) without regard to the cardinal points (centre or foci etc.) of the curve to be
described;
(c.) without any point of support being necessary on the surface where the curve is
to be drawn; this being also so in the case of
(d.) those surfaces which are inaccessible at the point where the cardinal point
should be, and
(e.) where it is only desired to trace parts of curves.
On the accompanying drawing the construction of the instrument for carrying out
the process is shown.
Fig. 1 is a perspective view of the apparatus.
Fig. 2 is a plan view of an auxiliary bar turning on an axis and also sliding at m.
Figs. 3 and 4 show in perspective, and vertical section, respectively, the details of
the pivotal socket m in which the bars a b, a¹b¹ slide.
Fig. 5 illustrates the arrangement adapted for describing closed curves.
Fig. 6 is a side view of the device serving, in connection with the apparatus shown
in Fig. 5, to draw spirals of any pitch.
Fig. 7 is a vertical section of Fig. 6 and
Fig. 8 is a plan view partly in section of Fig. 6.
Fig. 9 is a view of an arrangement for drawing spirals of increasing pitch.
Fig. 10 is a diagram showing the geometrical principle involved in the process.
The auxiliary bar ab (Fig. 2) forms one of the main features of the invention.
This bar turns about the point m, which may be regarded as the starting point for
he infinitely different radii-vectors, the extremities of which form the curves to be
drawn, and which are drawn in infinitely different directions and of infinitely different
engths (starting from m). This bar is provided with a pen f sliding upon it, which
s fixed to the bar at a point depending upon the curve to be described.
The turning of the bar about this point m is such, that in taking up different
lirections, it also moves to and fro through the axis of rotation of the point (compare
Fig. 3). If the bar (Fig. 2) simply turned about the point m without sliding, e.g. by
neans of an eccentric cn, having a pin at n sliding in a slot in a b, the pen through
he motion of the bar would simply describe the arc of a circle (Fig. 2). If, however,
t is arranged to slide, so that at each moment, however small, of the rotation, the
listance of the pen f from the pivotal point changes, the pen.in moving no longer

page_text: describes simply an arc of a circle, but a closed curve (oval). The manner in which
the sliding motion is effected simultaneously with the turning motion is seen from
Fig. 5. The device consists of an eccentric r, n turning about a rotating axis, the
bar a b being hinged to it. The device embraces also a metal block m turning about
an axis ii (point of origin) which block allows the said bar to pass through it, cutting
the said axis ii at o (Figs. 3 and 4).
It is not, properly speaking, the auxiliary bar ab that carries the pen, but a
second bar ab, which stands in kinetic connection with the former, making like
oscillations about the common axis of the block m (point of origin); and at the
same time receiving from the bar a ba longitudinal movement, being a multiplied
reproduction of the latter bar.
Fig. 5 shows this arrangement. Both bars, as is seen, turn about m, that is about
the axis ii, passing through it, one above the other, through the holes op,
(Cp. Figs. 3 & 4). The first har ab receives its angular and reciprocating
movement from the eccentric and transfers this motion to the second bar a bi
in such manner that whilst the angular motion about m (i.e. about the axis i)
is a common one, the resulting reciprocating motion of the second bar ab¹-is a
multiplied reproduction of the movement of the bar a b. This can be effected
by the provision of suitable racks and toothed wheels, or the like, as shown in
Fig. 5. The transmission device must naturally be rigidly connected to the block m
and therefore must be carried by the guide-bars k.
The result of this transmission is, firstly that the lengthening and shortening of
the second bar is a continuous and constant multiple of the lengthening and shorten-
ing of the bar ab; and secondly that the width of the curve drawn by the pen f
(on the second bar) is greater, according to the degree of multiplication of the
movement.
With such an arrangement there must be a point in the second bar a¹b¹, at which
the curve drawn is a complete circle, and this point will be at that place where the
distance from the pivotal point (point of origin) to the pen, bears the same propor-
tion to the distance from the pivotal point to the eccentric point, as the relation of
the transmission, that is, as 3:1, 4:1 and so on.
When the circle-point in any angular position of the said rods has been ascer-
tained according to the relation stated, it will be found that for any angular position,
there is always the same relation as regards the circle-point once found; and that is
the reason why with this point (circle point) a mathematical circle is described
through the movement effected by the eccentric on the one hand, and by the trans-
mission device on the other. The geometrical principles upon which this is based
will be clear from Fig. 10.
m is the pivotal point (point of origin); ba, ba, ba represent three different
positions of the upper bar; b'a', b'a', b¹¹ represent the corresponding positions of
the lower bar; fff are the corresponding three points on the circle which is to be
described; 8 is the supposed centre of this circle; a is the axis of the eccentric;
nnn are the three corresponding positions of the point of eccentricity, the eccen-
tricity (xn) being constant.
Suppose, now, that the points fff are situated on the periphery of the circle to
be described, the centre of which is the point 8; furthermore, that these three points
have resulted from the rotation of the eccentric on into the three positions xn,
xn, xn2: it is clear then that three triangles will have formed, viz. 8 mf, 8m f¹,
8 m, similar to the triangles & m n, xm n¹, x m 12, the first three triangles having
the common side 8 m, and the latter three the common side x m.
It is thus obvious that

page_text: 1, for instance, and mf = 3; then
Assume, now, by way of example, that in =
the relation between m a¹ and mf¹ must be m n¹ : m f¹= 1 : 3, similarly
By reason of the change of position, m n will have increased by a certain amount,
5 so that m n = 1 + 2, and in the same way m n² = 1 + y.
If, now, the proportions stated above are correct, then the increase in the
lower bar a'b' (that is, the line prolonged by dots in Fig. 10) must be three times that
just mentioned; in other words, the increase in the line mf must be three times that
in the line m n, so that if
5
Above and below the circle-point a curve is described resulting from those lengths
which are composed of the lengths corresponding at the circle-point to the proportion,
plus or minus, (according as to whether the pen moves above or below the circle-
point,) a constant dimension. This constant dimension is the distance the point is
removed from the circle-point.
From this it will be seen that the curve can be an oval, but cannot be a mathe-
matically uniform one. To attain the uniformity of this oval it would suffice either
to make the surface to be passed over by the pen inclined, instead of parallel, to the
plane of the movements; or to make the eccentricity of the said eccentric xn
diminishing and increasing, which can be done by, for instance, a further excentricity
at the axis.
It will thus be understood how a circular line etc., may be described by means of
the instrument shown in Fig. 5. And it will be equally clear to see how spiral curves
also may be described by means of the same process, and the peculiar device, if the
following points are considered.
The reason why the curve which is described is closed (circular or oval) is that the
eccentricity is a constant one. Should the eccentricity, however, be subject to a
continual change, so that at each infinitely small moment of the motion it either
becomes always shorter or longer, the resulting curve can obviously not be a closed
one. From this it follows:
(a.) that in the case of a circle point, the curve which is described is a circular
spiral;
(b.) that above or below this circle point oval spirals are formed;
(c.) that the pitch is in proportion to the increase or decrease of eccentricity;
(a.) that in the case of a regular increase or decrease of eccentricity, the circular or
oval spiral always has a like pitch, and
(e.) that in the case of an irregular increase or decrease of the eccentricity by a
constantly progressive coefficient, the circular or oval spiral has a pitch which is
uniformly increasing.
In Figs. 6, 7 and 8 the arrangement is shown which takes the place of the said
eccentric xn, and the purpose of which is to effect a change of the eccentricity.
Fig. 6 is a side elevation of this arrangement.
Fig. 7 is a cross section of Fig. 6 and Fig. 8 is a plan view of the lower part of
ig. 6.
'
The principal part of the arrangement is formed by the part 2.
urned by the handle at the top towards the left or right hand, the whole piece and
If this part is
ll that is connected with it turns also. The wheel e is mounted on a bracket, which

page_text: is rigidly connected to the part x. The slide y reciprocates in the part x, and turns
with the latter. A toothed rack h, the teeth of which engage with those of the
wheel e, is rigidly connected with the slide y. If the device only consisted of these
parts, it would simply turn about the axis of x, but have no further movement. The
wheel e, however, whilst turning with the whole system about the axis x, also turns
on its own axis, whence it is clear that the slide y during rotation about x, also
experiences a sliding motion. The rotation of the wheel on its axis during the rota-
tion at is effected by the endless screw w, which is rigidly connected with the
sleeve vand the toothed wheel. It surrounds the axis x and may be secured to it,
or be independent of it, so that during the rotation of the part x, it is either at rest,
or in motion, moving either in the saine direction as x, or in an opposite direction.
In the following, three cases are considered:
(1.) That of rotation of the sleeve with the part x;
(2.) That of the sleeve being at rest, whilst a rotates; and
(3.) That of rotation of the sleeve, but not in the same manner as the part 2.
The teeth of the wheel e engage both the rack h and the thread of the screw w.
If the sleeve v and thus the thread w is affixed to the part x, the thread w receives
only the movement of the part , and if the latter turns, not only does the slide y
with the rack and sleeve v turn with like speed, but also the screw c. In this
case the wheel e would not rotate on its axis, because its teeth would not ascend the
thread w. The sleeve v can be secured to the part by the screw (Fig. 6), the
bracket of which is rigidly affixed to the part x (Cp. Fig. 1), if this screw is screwed
into one or other space between the teeth of the wheel. The system vw can,
by turning the screw c (Fig. 6) be united with the immovable bridge it to a rigid
whole, which remains stationary upon any rotation of the part with its adjacent
parts, hereby forming the second of the three cases above mentioned. In this second
case the wheel e, whilst the part is rotating on its axis, simultaneously rotates on
its own axis. The reason of this motion is clear, for the teeth e whilst engaging with
the thread w rest on au endless inclined plane and ascend uninterruptedly. By this
is effected not only the uniform turning of the wheel e about its axis, but also the
desired engagement with the teeth of the rack h, and thus the regular sliding of the
party (simultaneously with rotation about the axis r). In other words, by this
process a uniform alteration of the eccentricity is produced, as already indicated,
since the bar ab is pivotted to the slide y (Figs. 6, 7 and 8) by means of the pin n
(point of eccentricity) and the slide g noving above the rails kk.
As, now, at every infinitely small moment of the rotary motion of the eccentric
about the part (i.e. by every swing of the bars a b, a¹b about the axis i, Fig. 5)
the eccentricity increases or decreases by an equal coefficient, that which has been
set forth under (d.) now takes places, viz. the formation of a spiral which has
always the same pitch.
The third of the above mentioned cases, in which the sleeve v etc., does not rotate
like the part x, can be a twofold one. Either the movement of the sleeve v has
the same direction as the part x, but an unlike speed; or the movement of the
sleeve is opposite to that of the part 2.
In that case in which the rotation of both parts is in the same direction, but
unequal, the following arrangement is adopted.
The wheel r is rigidly connected with the part z. Similarly with 2, i.e. the
latter can have a common axis q in the stationary bearing t. If now we remember,
firstly that the wheel r engages with and with ; and that, secondly, these
wheels are not of equal size, it is obvious how the rotation of results in rotation of
the sleeve, and with it the worm w, the speed of which is greater or smaller according
to the speed of the part x.
If then there is a supply of toothed, wheels of various sizes, which can be
placed upon the adjustable axis q, various different speeds of rotation of the thread. w
on the sleeve v can be effected.
In the case of rotation of the sleeve in opposite direction to the part x, it is
sufficient to remark that the assistance of a third toothed wheel, in addition to
.

page_text: between » and ™ is all that is necessary. Also, in this case, the speed is wholly
dependent upon the dimensions of the one or other of the wheels.
The result of the different rotation of the sleeve v and worm w, simultaneously
5 with the part x, is the same as was referred to under (c.) viz.: that the pitch of the
resulting spirals can be any desired. For in the case of the sleeve v remaining at
rest, whilst the part with adjacent parts rotates, the decrease and increase of the
eccentricity, i.e. the pitch of the resulting spiral, corresponds to the pitch of the
thread w.
In order to attain another pitch of the resulting spiral, there would have to be, if
at the sleeve v no other arrangement could be made than stopping it during the
rotation of x, another thread wo together with a suitable wheel e and rack, h applied,
which would take time and cause much trouble.
If, however, the sleeve is so applied that it also receives a motion of its own, not
the same as that of 2, through the rotation of x, the effect will be the same
as if another thread had been applied to w. For if the sleeve moves in like
direction, hut less rapidly than the part x, the effect on the wheel e and conse-
quently on the rail h and slide y, is exactly as if a worm w of smaller pitch had been
provided.
The spiral which results would in this case have a pitch which is less than that
which results in the case of the sleeve v remaining at rest. If the sleeve v
furthermore rotates in opposite direction, the effect on the wheel e and rails y
is the same as if a screw of greater pitch were introduced, so that the spiral resulting
and h
has a pitch which is greater than that when the sleeve v etc. remains at rest.
These unequal movements in different directions are, as already mentioned, effected
by the arrangement of the toothed wheels.
The spirals above referred to are of uniform pitch. If, however, a spiral with
increasing or decreasing pitch is to be drawn, it is only necessary instead of the
toothed wheels r¹ (Fig. 7) to provide toothed snails 88² (Fig. 9). The latter are
rigid on their axes with the toothed wheels 73,2 (see dotted lines, Fig. 9). This
arrangement is used when it is required to give the screw w (Fig. 6) a movement
simultaneous with that of the block x. The teeth of these toothed snails engage
in such manner that the sum of their distances from the point of rotation is always
constant; i.e that the decrease and increase of the distances mutually complete each
other. The consequence is that whilst the one snails at the commencement travels
a distance in the direction of the arrow of a certain angle, at the other snail 8¹ a
movement (in the direction of the arrow) takes place, which is equal to a fraction of
this angle. Thus the rotation of 2 compared with that of is a very slow one;
further, the result is, that whilst the first snails at the last space of its periphery
with which it engages with the snail 8¹, travels a distance which corresponds to a
small angle, at the snail s¹ there is a movement which corresponds to a far larger
angle, whereby the movement of the wheel 72 compared with that of the wheel,
is a very rapid one.
From this it follows that the change of the eccentricity is not an equal one, but
one increasing or decreasing by an always progressive amount. Furthermore, that
the resulting spiral has a pitch which is always an increasing or decreasing one.
With reference to Fig. 9 it is to be noted, that it shows how it is possible not
nerely to produce a single complete revolution increasing in spirality, but also
hrough suitable transmission gear (whilst the snails 88' complete a single revolution)
o produce several revolutions at the spiral to be described.
It is to be noted that the spirals drawn by the above described process are not
nerely composed of portions of circular curves, but are true spiral curves.
Having now particularly described and ascertained the nature of my said inven-
ion, and in what manner the same is to be performed, I declare that what I
laim is:
1. A process for rapidly describing closed and spiral curves of any pitch: (a.) the
arting point for the radii-vectors formed lying outside the curve, (6.) no attention

page_text: having to be given to the cardinal points of the curve (circle, ellipse or spiral) i.e.
centre or foci, all substantially as hereinbefore set forth.
2. An apparatus for effecting the process as claimed in Claim 1, comprising a
bar (b) turning through an angle about an axis (i) and having a reciprocal move-
ment in the direction of its length, and mechanism for transmitting this motion to
a second bar (a¹¹), whereby the latter receives a sitnilar longitudinal motion, which
is a multiple by a constant coefficient of the former, all substantially as herein before
set forth.
3. An apparatus is claimed in Claim 2, in combination with a device for altering
the angle of oscillation at each infinitely small moment, so that the said angle may
become constantly smaller or greater, said device comprising a slide (3) hinged by
means of a pin (a) and slide (1) or the like to the frame carrying the bars (ab, a¹ b¹)
said slide (3) having a guide-block upon which is rigidly mounted an axis (2) having
a sleeve (v) with thread (w), which engages with a toothed wheel (e) rotating upon an
axis mounted in a bracket upon the guide-block and operating a rack (h) affixed to
the said slide (y); the said sleeve (v) also having rigidly affixed to it a toothed
wheel (1) engaging with a stop-screw (1) carried rigidly from the guide-block, and a
stop-screw (c) carried by a rigid frame (t), all operating substantially as hereinbefore
set forth.
4. An apparatus as claimed in Claims 2 and 3, whereby the rotary motion of the
axis (x) is communicated to the sleeve (u) loosely mounted on it, by means of a
system of toothed wheels (r 2 etc.,) in such manner that the motion of the sleeve t
varies from that of the axis (e), all substantially as hereinbefore set forth.
5. In an apparatus as claimed in Claims 2, 3 and 4 the replacing of the foothed
wheels (r) by toothed snails (881), for the purpose of attaining an increasing or
decreasing movement of the sleeve (v) corresponding to a uniform movement of the
'part (a), all substantially as herein before set forth. :