WO1995003182A1

WO1995003182A1 - An angle divider

Info

Publication number: WO1995003182A1
Application number: PCT/AU1994/000402
Authority: WO
Inventors: John Trevor Lewis; Steven Michael Sutton
Original assignee: John Trevor Lewis; Steven Michael Sutton
Priority date: 1993-07-20
Filing date: 1994-07-15
Publication date: 1995-02-02
Also published as: EP0710186A4; CA2167394A1; CN1126971A; NZ268855A; JPH09500066A; EP0710186A1

Abstract

A drawing instrument (20) for dividing angles into fractional components is disclosed. In the case of bisection, a simple slot (27) and equidistant points (21, 22, 23, 24) along a perpendicular bisector is disclosed. In the case of other fractions, for example trisection and quadrisection, shaped slots (33A, 34A, 35A, 33B, 34B, 35B) are provided so as to define the locus of the division point, with the point defined by scribing an arc intersecting the loci, and joining the intersection to the vertex of the angle. The instrument may also include additional features such as rulers (29, 31) protractors (28) and measured holes (30) for drawing circles and arcs.

Description

AN ANGLE DIVIDER
Background Art
The present invention relates to drawing instruments and methods of dividing angles into fractions.

Technical Field
The most common existing method for dividing angles simply requires the use of a protractor. The angle is measured, multiplied by the fractional component required, and this angle is marked on the page. The fractional angle is then defined by drawing a line through the marked point and the vertex of the angle. Another known method, particularly for bisecting angles is by using a compass or divider to define the mid-point between the angle and draw a line through this mid-point through to the vertex to divide the angle in two.

The use of compasses in schools is becoming increasingly dangerous in view of the potential for such objects to be misused. Further, the use of such instruments and methods by professionals is time-consuming and any savings in time in such operations lead to better productivity.

It is an object of the present invention to provide an improved device and method for dividing angles.

Summary of Invention
According to one aspect the present invention provides a drawing instrument for assisting in drawing on a substrate, said instrument being formed from a material such that underlying markings on said substrate can be viewed, said instrument comprising a central indicator, at least two reference points arranged equidistant from said indicator along a line substantially perpendicular to said central indicator, the arrangement being such that the bisector of an arbitrary angle appearing on said substrate can be determined utilising said instrument.

Preferably the instrument further comprises at least one set of shaped indications arranged about said central indicator, said indicators being shaped and positioned so as to define the locus of the fractional division points of said arbitrary angle on said substrate when the vertex of said angle is placed under said central indicator, and said reference points are placed over the arms of said arbitrary angie.

Preferably said fractional division points define integral fractions of said angle.

Preferably the central indicator comprises a slot, the reference points comprise holes, and the shaped indicators comprise slots , allowing for a writing implement may be passed through each so as to mark said page.

The instrument may also include elements such as a protractor, one or more rulers, or other drawing tools.

The present invention further provides a method of dividing an arbitrary angle appearing on a substrate using a drawing instrument as described above, comprising the steps of;
positioning said drawing instrument over a drawn or defined angle such that the vertex of the angle is located on said central indicator;
positioning the lines defining the angle to be divided so as to underlie said reference points;
with said instrument being so positioned transferring an indication of the locus of the selected one or more dividers to the page;

;
pivoting said instrument about the vertex of said angle ,such that one of said reference points is moved through an arc of the same radius as the distance from the vertex to said reference point, until it meets said locus, the intersection of said arc and said one or more loci defining the line of the selected divider or dividers with the vertex of the angle.

The present invention allows for the division of angles into desired fractions using only a template type instrument and a drawing implement. It allows for the preparation of geometric figures within the classroom or drawing office without the need for pointed and potentially dangerous instruments such as compasses.

Brief Description of Drawings
The invention will be further described with reference to the accompanying drawings, in which:
Figure 1 illustrates a bisection device according to an embodiment of the present invention;
Figure 2 illustrates a second embodiment of an angle divider according to the present invention;
Figure 3 illustrates a third embodiment of an angle divider according to the present invention;
Figure 4 is a diagram illustrating the derivation of a curve to define quadrisection;
Figure 5 is a diagram illustrating the derivation of a curve to define trisection; and
Figure 6 is a diagram illustrating the derivation of a curve to define quintasection.

Detailed Description
It will be appreciated that the embodiments shown are illustrative only, and that many other implementations are possible according to the general principles described below.

Illustrated in figure 1 is a bisector instrument 1 which comprises a series of sets of points 11A, 11B through to 16A andl6 B. Preferably, the instrument is formed from a transparent plastic material. The points in each set are equidistant from and perpendicular to line 10. Line 10 can also be defined by dot 10A and 10B. The series of holes 11A and B through to 16A and B can be defined either by printing, embossing ,or by a hole through instrument 1. The line 10 can be printed or embossed or it could also be a slot or a slit. If the line 10 is not a slot or a slit then the points 10A and 10B are preferably holes, as will be indicated later.

In use, instrument 1 is placed over an angle, such that the vertex lies upon or underneath the line 10 and the rays or arms of the angle pass through any one of the sets of holes 11 A, 11B through to 16A, 16B. As instrument 1 is transparent, it can be positioned very accurately so as to overlie the required points. Upon the correct positioning of instrument 1, if line 10 is a slot, a line can be drawn through the slot which is the bisection of the angle. Alternatively, a point can be transferred to the drawing paper through hole 10A or 10B. The draftsperson then need only connect this point with the vertex of the angle to define the bisection of the angle.

Illustrated in figure 2 is another drawing instrument 20 embodying the principles of the present invention. It will be appreciated that instrument 20 includes certain features known from standard drawing templates, such as a protractor arrangement 28, and ruler elements 29, 31, in this case measuring inches on element 29 and centimetres on element 31. It will be appreciated that additional geometric elements could be incorporated as desired within instrument 20 without detracting from the operation of the instrument to perform the operations described below.

Points 21 and 22, and 23 and 24, each form a set of points relative to slot 27 to enable bisection as described with reference to figure 1. Slot 27 is placed so as to overlie the vertex of an angle to be bisected, and the rays or arms of the angle to be bisected pass through both of either points 23, 24 or 21, 22.

Instrument 20 includes a set of curved slots 33A , 33B, 35A and 35B, which define the fifths of an angle. Sets of slots 34A and 34B , and 36A and 36B, each define the trisections of an angle.

In use, these slots are used in a similar way to bisection points as described above. Instrument 20 is placed over an angle to be divided with the vertex of the angle underlying slot 27. If, say, a trisection is required using the curved slots 36A and 36B, with the vertex along slot 27, the arms or rays of the angle to be trisected are positioned through points 23 and 24. A drawing instrument, such as a pencil, is used to lightly mark the page through slots 36A and 36B. Then using a pencil or similar object inserted onto the vertex through slot 27, the instrument is pivoted about that vertex with another pencil or the like inserted through hole 23 or 24 to define an arc. The intersections of the curves drawn through slots 36A and 36B define the trisections of the angle.

It is noted that the trisection points will only be defined if the arc has the same radius as the distance from the vertex to points 23 and 24. The instrument is preferably sized with the intended type of writing instrument in mind, so that the slot will accommodate the writing tip while the nearby part of the pencil or the like holds the template material, so that pivoting can occur about the correct point.

Alternatively, a compass could be used to trace this arc. Similar procedures are used to define the trisections with slots 34A and 34B, using holes 21 and 22 rather than 23 and 24.

To perform a division of an angle into fifths, again the instrument 20 is placed with slot 27 overlying the vertex of the angle to be divided into five, and points 22 and 21 located along the rays of the angle. A drawing instrument is used to mark curves on the page using slots 33A , 33B, 35A and 35B. By placing a pencil or the like in slot 27 of the instrument onto the vertex of the angle to be divided, and another pencil in hole 21 or 22 and pivoting about the vertex of the angle to be divided, a further arc can be drawn. The intersections of the arc and the curves define the division points of the angle into fifths.

It will be appreciated that it is generally considered that it is not possible to trisect, or divide into fifths, an angle. This is only true if only lines and arcs can be used. The following equations demonstrate the derivation of the shape of curves (i.e. the shape to be defined by a pencil when placed in the curved slots) for trisection.

It will also be understood that the curve shapes can be derived in other ways, for example parametrically, and the present explanation is provided only by way of example. Similarly, other fractional dividers can be derived by similar means, as will be apparent.

The following discussion refers to figure 5. The problem is to define the trisection of arc AB. Points A and B are fixed, and lie along the x-axis. The perpendicular bisector of AB is the y axis. The locus of point P , which is the trisector of arc AB, where x and y are positive, can be derived as follows:
PB =2CP=2x
PB2= y2+(B x)2
4X2 = y2+B2+x2-2Bx hence, 3X2 = y2 + B2 - 2Bx
y2 = 3x2-B2 +2Bx
y2 = (3X - B) ( x + B)
Let B = 1 unit, then
y2 = (3x-1)(x+1)
which defines the locus of the trisector of AB. This is therefore the shape of the curve required for the slots for trisection.

With reference to figure 4, a derivation of the quadrisection of a curve will be discussed. The arc AB is required to be quadrisected. As in figure 5, A and B are fixed points on the y axis, and the y axis is the perpendicular bisector of AB.

The locus of point P is the quadrisector when x and y are positive may be derived as follows:
CO = c = y + I
CB = 2x OB = b
PB =CP2 = y2+ (b-X)2
CB 2 =Cê + b2=(y+I)2+x2 Iê = CPê -xê =y2 +(b-x)2-x2
=y2+ b2 2bx CE =(y+I)2+b2
4x2=y2+12+2y1+ b2
EMI6.1

Hence,
2 = 4x4+4x3-3x2-2x+1
(4xê-1)
Thus, the locus of point P being the first quadrisector is defined. The central quadrisector is defined by the bisector, and the third is defined by the mirror image, or negative x axis shape, of the locus defined above.

A similar calculation with respect to figure 6 can be used to define the quintasection, or division into fifths, of an angle.

PK = QK = BK
KO=k AO=BO=b
For locus point P (x,y) rê =bê + kê = xê +(y+k)2
2 2 2 - k=
PBê = yê+(b-x)ê
PMê = 4xê -MQê =rê-(r-MQ)2 MQ 2x2
r
It can be shown that, where b = 1 unit, y6+3y4+3y2-15x6-2x5+31x4+4x3-17x2-2x + 35X4y2-4X3y2-30X2y2-13X2Y4 2xy4 4xy2 +1=0
It will be understood that there are 4 curves required for division into 5 equal divisions - the loci of points P and Q as shown, and the mirror images of these loci on the negative x side of the y axis. The derivation of P is shown, however, those skilled in the art will appreciate that a similar calculation will yield Q. It will be understood that the shape of the locus, and accordingly the shape of the corresponding slot in the instrument, may be similarly derived for any desired fractional section.

Referring to figure 2, it is noted that the instrument is also adapted to produce circles and arcs of various sizes. By inserting a pencil or the like through one of the holes in the surface, and a further pencil through another hole, rotation about the first point will inscribe a circle. In particular, use of the marked holes 30 on each ruler section 29, 31 enables circles and arcs of measured sizes to be drawn. Use of such an instrument further reduces or removes the necessity for students to use pointed compasses.

Another feature of the instrument 20 is the provision of holes at fixed points around protractor 28, at 300 25, and at 450 26. Hole 25 also defines 600 when taken from the other side. These enable easy division into commonly used fractions of 900. It can be seen that the entire device is drawn to an enlarged scale, with the entire device being approximately credit card sized and correspondingly easy to store.

Figure 3 shows a larger device, more appropriate for professional use.

Instrument 40 has many features in common with the instrument 20 shown in figure 2, including ruler portions 41, 42; marked holes in the ruler portion 43, and central line 47. The protractor element 48 in this case describes a full half circle, with marker points 54 at various angles relative to point 55. Marker points are also provided at various angles to vertex hole 49 - hole 45 at 300, hole 44 at 450, and hole 46 at 600. In this case, 4 sets 50, 51, 52, 53 of angle dividing slots and corresponding locating holes are disposed on instrument 40, with operation in each case substantially as described with reference to figure 2. The different height sets of dividing curves are provided to facilitate angles with rays of different heights and sizes of angle.

The instrument according to the present invention may be used for various other drawing operations, using the facilities described, as will be apparent to the reader. It may be formed from, for example, a transparent or translucent plastics material. It will also be understood that where reference is made to holes in a substrate, a transparent substrate through which pressure could be transferred to the paper beneath, or any other arrangement allowing for the point to be aligned and marked beneath may be employed.

It will be appreciated that a large range of embodiments are possible within the general principles described. Modifications made by those skilled in the art can be made to this invention without departing from the scope of this invention.

Claims

1. A drawing instrument for assisting in drawing on a substrate, said instrument being formed from a material such that underlying markings on said substrate can be viewed, said instrument comprising a central indicator, at least two reference points arranged equidistant from said indicator along a line substantially perpendicular to said central indicator, the arrangement being such that the bisector of an arbitrary angle appearing on said substrate can be determined utilising said instrument.

2. An instrument according to claim 1, further comprising at least one set of shaped indications arranged about said central indicator, said indicators being shaped and positioned so as to define the locus of the fractional division points of an angle on said substrate when the vertex of said angle is placed under said central indicator, and said reference points are placed over the arms of said arbitrary angle.

3. An instrument according to claim 1 or claim 2, wherein said fractional division points define integral fractions of said angle.

4. An instrument according to any one of the preceding claims, wherein said central indicator comprises a slot through which a writing implement may be passed so as to mark said page.

5. An instrument according to any one of the preceding claims, wherein said reference points comprise holes through which a writing implement may be passed so as to mark said page.

6. An instrument according to any one of the preceding claims, wherein said shaped indicators comprise slots through which a writing implement may be passed so as to mark said page.

7. An instrument according to any one of the preceding claims, wherein said instrument includes an integral protractor.

8. An instrument according to any one of the preceding claims, wherein said instrument includes means for scribing an arc or circle.

9. A method of dividing an arbitrary angle appearing on a substrate using a drawing instrument as defined in any of the preceding claims, comprising the steps of; positioning said drawing instrument over a drawn or defined angle such that the vertex of the angle is located on said central indicator; positioning the lines defining the angle to be divided so as to underlie said reference points; with said instrument being so positioned transferring an indication of the locus of the selected one or more dividers to the page; pivoting said instrument about the vertex of said angle such that one of said reference points is moved through an arc of the same radius as the distance from the vertex to said reference point until it meets said locus, the intersection of said arc and said one or more loci defining the line of the selected divider or dividers with the vertex of the angle.