CA2441543A1 - Template and method for trisecting an angle - Google Patents
Template and method for trisecting an angle Download PDFInfo
- Publication number
- CA2441543A1 CA2441543A1 CA 2441543 CA2441543A CA2441543A1 CA 2441543 A1 CA2441543 A1 CA 2441543A1 CA 2441543 CA2441543 CA 2441543 CA 2441543 A CA2441543 A CA 2441543A CA 2441543 A1 CA2441543 A1 CA 2441543A1
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- CA
- Canada
- Prior art keywords
- template
- angle
- line
- trisecting
- pair
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
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Classifications
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B43—WRITING OR DRAWING IMPLEMENTS; BUREAU ACCESSORIES
- B43L—ARTICLES FOR WRITING OR DRAWING UPON; WRITING OR DRAWING AIDS; ACCESSORIES FOR WRITING OR DRAWING
- B43L13/00—Drawing instruments, or writing or drawing appliances or accessories not otherwise provided for
- B43L13/20—Curve rulers or templets
- B43L13/201—Stencils for drawing figures, objects
- B43L13/205—Stencils for drawing figures, objects geometrical figures
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B43—WRITING OR DRAWING IMPLEMENTS; BUREAU ACCESSORIES
- B43L—ARTICLES FOR WRITING OR DRAWING UPON; WRITING OR DRAWING AIDS; ACCESSORIES FOR WRITING OR DRAWING
- B43L13/00—Drawing instruments, or writing or drawing appliances or accessories not otherwise provided for
- B43L13/001—Mathematical drawing instruments
- B43L13/002—Angle intersecting devices
-
- B—PERFORMING OPERATIONS; TRANSPORTING
- B43—WRITING OR DRAWING IMPLEMENTS; BUREAU ACCESSORIES
- B43L—ARTICLES FOR WRITING OR DRAWING UPON; WRITING OR DRAWING AIDS; ACCESSORIES FOR WRITING OR DRAWING
- B43L9/00—Circular curve-drawing or like instruments
- B43L9/08—Proportional compasses or dividers
Abstract
A transparent plastic template permits an angle to be conveniently trisected with a pencil and straight edge. The template has three holes equally spaced apart along a hypothetical line, a central hole and a pair of outer holes. A circle is centered about one of the outer holes, and has a radius equal to one-half the spacing between the outer holes. A line is defined on the template, perpendicular to the hypothetical line and intersecting the midpoint of the hypothetical line. The template is manually positioned on the angle with its line intersecting the vertex of the angle. Simultaneously, the circle is positioned so one arm of the angle forms a tangent, and the opposing outer hole is positioned on the other arm of the angle. The central hole and the hole surrounded by the circle are then positioned to mark a pair of trisecting lines through the vertex.
Description
TEMPLATE ANA METHOD FOR TRISECTING AN ANGLE
This invention zelates to a devices and methods for trisecting an angle.
BACKGROUND OF THE INVENTIc~N
'Various devices have been proposed to enable trisectiuag of angle. Most are complex, cvznparatively expensive, difficult to use or suffer Borne significant shortcoming, as will be apparent from the prior art references below.
U.S. Patent No. 1,145,369 to Kaplan describes a device that relies on a pair of external liurtiting members pivotally connected to each other with complex links. The mechanism is cumbersome and relatively expensive.
U.S. Patent No. 1,764,5$1 to Shibuya describes angle trisector that uses a modified protractor. The device is potentially inexpensive to produce but the user rrtust select proper graduations and perform calculations, making use complex.
U.S. Patent No. 2,222,853 to Neurohr describes an angle trisector which uses a pair of side arms and a set of bars or arms pivotally connected to the side arms. The device is relatively expensive to produce and cumbersome tv use or transport.
German disclosure No. 1611808, published on February 13, 1968, describes a template for trisecting an angle. The tezrrplate has a central horizontal Iine and five holes edually spaced along the line by a distance r.
The center hole serves as the center of a circle of radius r. The periphery of the template is defined by two intersecting circles of radius 2r centered about the outermost holes. When the periphery is properly centered within angle, a pair of hypothetical lines tbxough the vertex and tangential to the circle effectively trisect the angle. Although potezrtially inexpensive, a major shortcoming is that the two trisecting lines cannot be drawn or readily mazked for drawing.
Russian Patent No. SU1735061-A1 to Memyrin describes a mechanism compzisi.ng czanl~s and slides linked to trisect an angle. "fhe mechanism is complex, expensive and cuzzrbezsorxre.
U.S. Patent No. 5,210,951 to Chen describes an instrument for trisecting an angle that has two circular plates and foot pointers. Two pointers are intended to define either an acute or obtuse angle. Two other pointezs ate intended to divide the angle into three equal angles when oriented pezpendicular to one other. The device is relatively expensive but awkward to use.
BRIEF SUMMARY OF TIxE INVENTION
In general, the invention provides a template that in effect identifies three congruent triangles between an angle to be tzisected. The terrrplate zelies on aligning and marking points (prefezably defined by sir~aple clearance holes) and a circular arc, all observing specific geometric relationships. This arzangerzxent allows the template to be positiozred to r~aazl~
trisecting lines, effectively along certain sides of the hypothetical triangles. This operating principle need not be understood to use the template but will be described in greater detail below with reference to the dzawings.
In one aspect, the invention provides a template fvz trisecting an angle displayed on a flat surface. The template comprises a first aligning point adapted foz visual location over one arm of the angle, a second point where a marking hole is located, a line intended to be placed into intersecta~ag relationship with the vertex of the angle, and a circular arc intezrded tv be aligned in tangential relationship with the other arm of the angle. The lure is perpendicular to a hypothetical line between the first and second points and alig~aed axially with the midpoint of the hypothetical litre. The cizcular arc adapted for visual location in tangential relationship with the othez azzo,.
The czzcular arc is centered about the second point and has a radius equal to one half of the distance between the fiarst and second points.
Xn one e~oabodiznent,.the invention provides a transparent angle-xrisecting template that compzises a pain of clearance holes, and a pair of circles of equal radius, each centered about a different clearance holes. The circles have a radius equal to one-half of the distance between the clearance holes, and consequently side-by-side forzniz~g a point contact. A central clearance hole is located at the midpoint of a hypothetical line between the pair of clearance holes, substantially at the point of contact between the two cizcles.
A line extends perpendicular to the hypothetical line in aligned with the midpoint of the hypothetical line. This embodiment is symmetric and gives the user the option of using eithez circle to orient the template against one arm of the angle.
The cenkral clearance hole of the ether cizcle can be located over the other arm.
In this embodiment, the peripheral shape of the template is immaterial.
In another embodiment, the invention provides a template whose periphery is instrumental to trisecting an angle. The template coznprises a pair of clearance holes, a lateral side edge perpendicular to a hypothetical line between tl~e clearance holes and aligned with the midpoint of the line, and a projection that extends laterally relative to the side edge. The projection has a part--circular periphery centered about one of the clearance holes and has a radius equal to one-half the distance between the clearance holes. The template is positioned zelative to an angle in substantially the same manner as discussed above. However, o~ace oriented, the latezal side edge can be used to immediately draw one trisecting line. The other trisecting line is marked through the clearance hole centered in the part-circular peziphery, and can be drawn with a straight edge extended thzough the vertex of the angle to the zxaazl~.
Other aspects 4f the invention will be apparent from a description below of preferred embodiments and will be nnore speci:fzcally defined in the appended claims.
BRIEF DESCRIPTION Q,~I; '~'~~~~AWINGS
The invention will be better understood with respect to drawings ~, which:
Fig. 1 is a plan view of a template for trisecting angles;
Fig. 2a shows how the tezx~,plate is positioned for marking the position of trisecting lines in an acute angle;
Fig. 2b shows how trisecting lanes are drawn after marking;
Fig. 2c illustrates the priztciples underlying the invention;
Fig. Zd illustrates how the trisecting lines can be verified with the template;
Fig. 3 shows how the template can be used to trisect a right angle;
Fig. .4 shows how the template can be used to trisect an obtuse angle;
Fig. 5 shows how the template can be used to trisect a 180 degree angle;
Fig. 6 shows how the tezr~plate can be used to trisect a reflective angle;
Fig. 7 shows how the template can be used to bisect an angle;
Fig. 8 shows a method of using the temple to trisect a very small angle;
Fig. 9 is a plant view of an alternative template;
Fig. 10 shOwS how the template of fig. 9 is positioned fox marking of trisecting lines izt an acute angle;
Fig. 11 and atad 12 are elevations! views showing how a particular application of the template, specifically trisecting an angle on a vertical surface adjacent a flat horizontal member that can support the template for sliding movement.
Fig. 1 shows a template 10, a flat menczber foamed of transparent plastic. The ter~aplate 10 lass a zectangular periphezy 12 and a thickness between one-sixteenth and one-eighth inches. The exact dimensions, shape and thickness of this template 10 are not czitical. The template 10 need only be shaped to seat against a surface on which an angle has been drawn, such as a sheet of drawing paper.
Viewed in the orientation of fig. l, the template 10 cozzrpzi.ses identical left and right ci.zcles 14, 16 with a conazazorx zadius r. The cizcles 14, 16 may be defined on the upper dace 18 of the template 10 by scoring oz with conventional marking material. The left circle 14 is centered about a left marking hole 20 (effectively defining a left marking or sighting point), and the tight circle 16, about another right marking hole 22 (defining a right marking or sighting point). In fzg. l, a hypothetical line 24 has been shown in phantozza outline, extending between the marking holes 20, 22. The holes 20, 22 ate spaced apazt by a distance equal to twice the radius r, and the circles 14, 16 are consequently side-by-side forming a point contact at the midpoint (not specifically indicated) of the hypothetical line 24. A central marking hole 26 is fozrmed in the template 10 midway between the outer holes 20, 22, at the midpoint of the hypothetical line 24. A Line 30 is formed on the upper face 1$, perpendicular to the hypothetical line 24 and aligned with the midpoint of the hypothetical line 24.
Fig 2a and 2b 5hOW5 how the template 10 can be used to mark trisecting lines 32, 34 on an acute angle 36. The angle 36 is defined by a pair of arms 3$, 40 that intersect to define a vertex 42, and zaaay have been drawn previously on a drawing sheet (not illustrated). The template 10 is displaced by hand until the following conditions are simultaneously met: the hole 20 is located over the angle arm 38; the circle 16 is located in tartgezttial relationship with the angle arrn 40; and the line 30 intersects the vertex 42 of the angle 36, au as shown in fig. 2a. A pencil or other mazkiztg instntment may then be used to place marks 44, 46 on the drawing sheet through the central marking hole 26 and xnarking holes 22. These marks 44, 46 are shown as filled circles in fi.g.
2a.
The template 10 is then removed, and trisecting lines 32, 34 as shown in fig. Zb are drawn. Each of the trisecting lines 32, 34 is formed by aligzzing a ruler or other straight edge edge with the vertex 42 and one of the marks 44, 46. The general principle of operation inherent in the invention will be apparent in fig. 2b. The geometry of the tepnplate 10 effectively identifies three triangles 48, 50, 52 between the angle arms 38, 40 of the angle 36, when the template IO is positioned as required relative to the angle 36. Elementary geometry indicates that the three triangles 48, 50, 52 are congruent, and that their angles (not numbered) proximate to the vertex 42 are identical. The central marking hale 26 and the marking hole 22 are consequently oriented to fall on trisecting lines 32, 34. This template 10 is symmetric and consequently the hole 22 can be used as a sighting point to be positioned on the angle arm 40, and the hole 20 may then be used to mark a trisecting line.
Fig. 2d ialustrates another aspect of the template 10, naz~oely, detecting human erzox and verifying that trisec'g lines 32, 34 do in fact trisect the angle 36. Fig. 2d shows the same angle 36 and the same trisecting lines 32, 34 produced in by the steps apparent in figs. 2a-Zc_ zn this instance, the template 10 has been rotated so that the hole 22 can be overlayed with the lower angle arm 40 and the upper circle 14 can be placed in tangential relationship with the upper angle arm 38. 'this could serve as an alternative arrangement in which the trisecting lines are marked for drawing through upper hole 20 and the central hole 2b. In this instance, the user simply sights through the upper hole 20 and the central hole 26 to confirm that the previously drawn lines 32, 34 pass through the holes 20, 32, confitxning that they do trisect the angle 36.
The versatility of the trisecting template 10 will be zzzoze apparent from figs. 3-6 which illustrate how the template can be used respectively to trisect a angle, an obtuse angle, a 180 degree angle, and a reflective angle.
Common reference numbers have been used to identify angle arms, vertices, and trisecting ?tines in figs. 3-6, the same reference numbers used in figs.
2a--c.
The same general method for positioning the template 10 relative to a displayed angle, marking the drawing sheet to identify points on trisecting lines 32, 34, and actually drawing trisecting lines 32, 34, is used in each instance and may be understood with reference to the description of figs. 2a and 2b. The template 10 would normally be removed befvze drawing trisecting lines 32, 34, but has been illustrated together with the resulting trisecting lines 32, 34 as this arrangement better illustrates the geometric relationship between features on the tez~aplate 10, the angle in issue, and the trisecting lines ultimately draw~a.
Reference is made tv fig. 7 which illustrates how the te~onplate 10 can also be used tv bisect an angle 36. The template 10 is displaced zelative to the angle 36 until each of the holes 20, 22 are located over the az~xts 38, 40 of the angle 36. Tlxe line 30 of the template is simultaneously positioned to intersect the vertex 42 of the angle 36. It will be recalled that the line 30 is perpendicular to the hypothetical line 84 (not shown in fig. 7) extending between the centers of the pair of holes 20, 22, and essentially bisects the line 74, thus being equidistant from the holes 20, 22. Accordingly, when placed in the suggested orientation, the bisector 53 can be znazl~ed through the central hole 26, and then drawn through the mark and the vertex 42. The steps of znarlsSng and drawing the bisector 53 have been omitted to avoid needless reproduction _7_ of drawings, and the template 10 is effectively shown repositioned over the angle 36 and the bisecting line 53 after drawing.
How the template 10 may be used to trisect a very small angle will explained with reference to fig. 8. The angle requiring trisecting is identified as s2 is in fig. 8, preferably ~orrrxed on a transparency, and for purposes o~ exemplification may be 3 degrees. The object is not to apply the tezx~plate 10 directly to the small angle SZ, which is not feasible, but to produce another trae,sparency (assumed to be the plane of the drawing page) that cac~
be used afterward to mark and draw trisecting lines 32, 34. A line 56 is drawn ~rozn the vertex 42 of angle s2 to define with the angle arm 40 a larger angle of 60 degrees (arbitrarily selected). The template 10 is then used to trisect the 60--degree angle to produce trisecting lip,es 58, 60. The template 10 zs then used to trisect the larger combined 63-degree angle to produce trisecting lines 62, 64. As apparent ~rozzr fig. $, the Lines 58, 62 form an angle of one-third SZ
or 1 degree, and the lines 60, 64 ~orzz~. an angle of two-thirds S2 or 2 degrees.
The transparency so produced can be overlaid by the transparency bearing the angle S2, and rotated to position the trisecting lines 58, 62 relative to the angle Sa iuato alignment with the upper angle arm 38 to mark or draw the trisecting line 32. 'The Litaes 60, 64 can then be rotated into alignrne~ut with the upper arm to z~aark or draw the trisecting line 34. Since the working triseetors 58-64 represent one-third S2 and twv-thirds SZ only a single pair is required to draw both trisectors 32, 34.
Several points should be noted regarding trisection of the angle SZ. First, angle values have been indicated in fig. 8 solely for exemplification. .
Second, since the trisector 10 does not rely on mathematical calculations or use of protractors, the value o~ S~ and the value of the large angle used to facilitate trisection o~ angle ~2 are immaterial and need not be known.
Reference is ~oaade to fig. 9 which illustrates another template 70 _g_ for trisecting angles. This template 70 is a thin planar nrAember formed of opaque plastic. In the orientation of fig. 9, the template 70 is seen to comprise left and right circular holes 72, 74, the left hole 72 serving as a sighting hole and the right hole 74 as a marking hole. The template 70 has a lateral side edge 76 oriented perpendicular to a hypothetical line 78 between the left and right holes 72, 74 and aligned with the midpoint of the hypothetical line 78. The side edge 76 effectively defines a line corresponding in essence to the line 30 associated with the terzlplate 10 of fig. 1, and performing the same function.
A
central marking hole 80 is located midway between the left and right holes 72, 74 at the rnidpoint of the hypothetical line 78. The template 70 has a projection 82 that extends laterally relative to the side edge 76 and has a part-circular periphery 84 (circular arc) centered about the right marking hole 74 and having a radius r equal to one-half the distance between the pair of outer holes 72, 74.
One corner 86 of the template 70 defines a circular arc of radius r, largely for aesthetic reasons.
Fig. LO shows how the template 70 is used to trisect an acute angle. Features associated with the angle and trisecting lines are labeled with reference numbers common to the angle shown in figs. 2a-2b. The user sights through the left hole 72 to position the left hole 72 over the left arm 38 of the angle, and simultaneously positions the lateral side edge 76 to intersect the vertex 42 of the angle, and the part-circular periphery 84 of the projection 82 in tangential relationship with the right arm 40. One trisecting line 32 can then be drawn immediately along the lateral side edge 76 or marked through the central hole 80 for drawing. The other trisecting line 32 is marleed through the right marking hole 74, and then drawn with a straight edge. This template 70 observes the same basic principle of operation described with respect to the template 10 of fig. 1.
Figs. 11 and 12 illustrate a particular application for the template 70. In this instance the angle 36 is drawn on a vertical wall surface 90. The object of trisecting the angle 36 is to delineate areas within the angle 36 that can be painted different colors. The wall surface 90 is located above a horizontal member 92 (in cross-section) with a flat upper horizontal surface 94. The vertex 42 associated with the angle 36 is positioned at a side edge 96 of the member 92, and the arm 40 of the angle 36 is substa~adally coi~acident with the upper surface 94. The template 70 is oriented ~lat agauast the wall surface 90 with the circular periphezy rested against the upper horizontal surface 94, and tl~e end of the side edge 76 distant from the projection 82 contacting the surface 94. The user slides the template 70 along the upper surface 94 in the direction of the arrow in fig. 11 until the sighting hole 72 aligns with the now upper angle arm 38 as shown i.~a fig. 12. The arrangement is particularly simple because the user is obliged only to ensure that the sighting hole 72 aligns with the a~agle arm 38. More specifically, the circular periphery 84 remains continually in tangential relationship with the upper arm 40 and the lateral side edge 76 intersects the angle's vertex 42 automatically upon reaching the side.
edge 96 of the horizontal member 92.
It will be appreciated that particular embodiments of the invexxtion have been illustrated and described, and that changes may be made thereto without departing from the scope of the appended claims.
This invention zelates to a devices and methods for trisecting an angle.
BACKGROUND OF THE INVENTIc~N
'Various devices have been proposed to enable trisectiuag of angle. Most are complex, cvznparatively expensive, difficult to use or suffer Borne significant shortcoming, as will be apparent from the prior art references below.
U.S. Patent No. 1,145,369 to Kaplan describes a device that relies on a pair of external liurtiting members pivotally connected to each other with complex links. The mechanism is cumbersome and relatively expensive.
U.S. Patent No. 1,764,5$1 to Shibuya describes angle trisector that uses a modified protractor. The device is potentially inexpensive to produce but the user rrtust select proper graduations and perform calculations, making use complex.
U.S. Patent No. 2,222,853 to Neurohr describes an angle trisector which uses a pair of side arms and a set of bars or arms pivotally connected to the side arms. The device is relatively expensive to produce and cumbersome tv use or transport.
German disclosure No. 1611808, published on February 13, 1968, describes a template for trisecting an angle. The tezrrplate has a central horizontal Iine and five holes edually spaced along the line by a distance r.
The center hole serves as the center of a circle of radius r. The periphery of the template is defined by two intersecting circles of radius 2r centered about the outermost holes. When the periphery is properly centered within angle, a pair of hypothetical lines tbxough the vertex and tangential to the circle effectively trisect the angle. Although potezrtially inexpensive, a major shortcoming is that the two trisecting lines cannot be drawn or readily mazked for drawing.
Russian Patent No. SU1735061-A1 to Memyrin describes a mechanism compzisi.ng czanl~s and slides linked to trisect an angle. "fhe mechanism is complex, expensive and cuzzrbezsorxre.
U.S. Patent No. 5,210,951 to Chen describes an instrument for trisecting an angle that has two circular plates and foot pointers. Two pointers are intended to define either an acute or obtuse angle. Two other pointezs ate intended to divide the angle into three equal angles when oriented pezpendicular to one other. The device is relatively expensive but awkward to use.
BRIEF SUMMARY OF TIxE INVENTION
In general, the invention provides a template that in effect identifies three congruent triangles between an angle to be tzisected. The terrrplate zelies on aligning and marking points (prefezably defined by sir~aple clearance holes) and a circular arc, all observing specific geometric relationships. This arzangerzxent allows the template to be positiozred to r~aazl~
trisecting lines, effectively along certain sides of the hypothetical triangles. This operating principle need not be understood to use the template but will be described in greater detail below with reference to the dzawings.
In one aspect, the invention provides a template fvz trisecting an angle displayed on a flat surface. The template comprises a first aligning point adapted foz visual location over one arm of the angle, a second point where a marking hole is located, a line intended to be placed into intersecta~ag relationship with the vertex of the angle, and a circular arc intezrded tv be aligned in tangential relationship with the other arm of the angle. The lure is perpendicular to a hypothetical line between the first and second points and alig~aed axially with the midpoint of the hypothetical litre. The cizcular arc adapted for visual location in tangential relationship with the othez azzo,.
The czzcular arc is centered about the second point and has a radius equal to one half of the distance between the fiarst and second points.
Xn one e~oabodiznent,.the invention provides a transparent angle-xrisecting template that compzises a pain of clearance holes, and a pair of circles of equal radius, each centered about a different clearance holes. The circles have a radius equal to one-half of the distance between the clearance holes, and consequently side-by-side forzniz~g a point contact. A central clearance hole is located at the midpoint of a hypothetical line between the pair of clearance holes, substantially at the point of contact between the two cizcles.
A line extends perpendicular to the hypothetical line in aligned with the midpoint of the hypothetical line. This embodiment is symmetric and gives the user the option of using eithez circle to orient the template against one arm of the angle.
The cenkral clearance hole of the ether cizcle can be located over the other arm.
In this embodiment, the peripheral shape of the template is immaterial.
In another embodiment, the invention provides a template whose periphery is instrumental to trisecting an angle. The template coznprises a pair of clearance holes, a lateral side edge perpendicular to a hypothetical line between tl~e clearance holes and aligned with the midpoint of the line, and a projection that extends laterally relative to the side edge. The projection has a part--circular periphery centered about one of the clearance holes and has a radius equal to one-half the distance between the clearance holes. The template is positioned zelative to an angle in substantially the same manner as discussed above. However, o~ace oriented, the latezal side edge can be used to immediately draw one trisecting line. The other trisecting line is marked through the clearance hole centered in the part-circular peziphery, and can be drawn with a straight edge extended thzough the vertex of the angle to the zxaazl~.
Other aspects 4f the invention will be apparent from a description below of preferred embodiments and will be nnore speci:fzcally defined in the appended claims.
BRIEF DESCRIPTION Q,~I; '~'~~~~AWINGS
The invention will be better understood with respect to drawings ~, which:
Fig. 1 is a plan view of a template for trisecting angles;
Fig. 2a shows how the tezx~,plate is positioned for marking the position of trisecting lines in an acute angle;
Fig. 2b shows how trisecting lanes are drawn after marking;
Fig. 2c illustrates the priztciples underlying the invention;
Fig. Zd illustrates how the trisecting lines can be verified with the template;
Fig. 3 shows how the template can be used to trisect a right angle;
Fig. .4 shows how the template can be used to trisect an obtuse angle;
Fig. 5 shows how the template can be used to trisect a 180 degree angle;
Fig. 6 shows how the tezr~plate can be used to trisect a reflective angle;
Fig. 7 shows how the template can be used to bisect an angle;
Fig. 8 shows a method of using the temple to trisect a very small angle;
Fig. 9 is a plant view of an alternative template;
Fig. 10 shOwS how the template of fig. 9 is positioned fox marking of trisecting lines izt an acute angle;
Fig. 11 and atad 12 are elevations! views showing how a particular application of the template, specifically trisecting an angle on a vertical surface adjacent a flat horizontal member that can support the template for sliding movement.
Fig. 1 shows a template 10, a flat menczber foamed of transparent plastic. The ter~aplate 10 lass a zectangular periphezy 12 and a thickness between one-sixteenth and one-eighth inches. The exact dimensions, shape and thickness of this template 10 are not czitical. The template 10 need only be shaped to seat against a surface on which an angle has been drawn, such as a sheet of drawing paper.
Viewed in the orientation of fig. l, the template 10 cozzrpzi.ses identical left and right ci.zcles 14, 16 with a conazazorx zadius r. The cizcles 14, 16 may be defined on the upper dace 18 of the template 10 by scoring oz with conventional marking material. The left circle 14 is centered about a left marking hole 20 (effectively defining a left marking or sighting point), and the tight circle 16, about another right marking hole 22 (defining a right marking or sighting point). In fzg. l, a hypothetical line 24 has been shown in phantozza outline, extending between the marking holes 20, 22. The holes 20, 22 ate spaced apazt by a distance equal to twice the radius r, and the circles 14, 16 are consequently side-by-side forming a point contact at the midpoint (not specifically indicated) of the hypothetical line 24. A central marking hole 26 is fozrmed in the template 10 midway between the outer holes 20, 22, at the midpoint of the hypothetical line 24. A Line 30 is formed on the upper face 1$, perpendicular to the hypothetical line 24 and aligned with the midpoint of the hypothetical line 24.
Fig 2a and 2b 5hOW5 how the template 10 can be used to mark trisecting lines 32, 34 on an acute angle 36. The angle 36 is defined by a pair of arms 3$, 40 that intersect to define a vertex 42, and zaaay have been drawn previously on a drawing sheet (not illustrated). The template 10 is displaced by hand until the following conditions are simultaneously met: the hole 20 is located over the angle arm 38; the circle 16 is located in tartgezttial relationship with the angle arrn 40; and the line 30 intersects the vertex 42 of the angle 36, au as shown in fig. 2a. A pencil or other mazkiztg instntment may then be used to place marks 44, 46 on the drawing sheet through the central marking hole 26 and xnarking holes 22. These marks 44, 46 are shown as filled circles in fi.g.
2a.
The template 10 is then removed, and trisecting lines 32, 34 as shown in fig. Zb are drawn. Each of the trisecting lines 32, 34 is formed by aligzzing a ruler or other straight edge edge with the vertex 42 and one of the marks 44, 46. The general principle of operation inherent in the invention will be apparent in fig. 2b. The geometry of the tepnplate 10 effectively identifies three triangles 48, 50, 52 between the angle arms 38, 40 of the angle 36, when the template IO is positioned as required relative to the angle 36. Elementary geometry indicates that the three triangles 48, 50, 52 are congruent, and that their angles (not numbered) proximate to the vertex 42 are identical. The central marking hale 26 and the marking hole 22 are consequently oriented to fall on trisecting lines 32, 34. This template 10 is symmetric and consequently the hole 22 can be used as a sighting point to be positioned on the angle arm 40, and the hole 20 may then be used to mark a trisecting line.
Fig. 2d ialustrates another aspect of the template 10, naz~oely, detecting human erzox and verifying that trisec'g lines 32, 34 do in fact trisect the angle 36. Fig. 2d shows the same angle 36 and the same trisecting lines 32, 34 produced in by the steps apparent in figs. 2a-Zc_ zn this instance, the template 10 has been rotated so that the hole 22 can be overlayed with the lower angle arm 40 and the upper circle 14 can be placed in tangential relationship with the upper angle arm 38. 'this could serve as an alternative arrangement in which the trisecting lines are marked for drawing through upper hole 20 and the central hole 2b. In this instance, the user simply sights through the upper hole 20 and the central hole 26 to confirm that the previously drawn lines 32, 34 pass through the holes 20, 32, confitxning that they do trisect the angle 36.
The versatility of the trisecting template 10 will be zzzoze apparent from figs. 3-6 which illustrate how the template can be used respectively to trisect a angle, an obtuse angle, a 180 degree angle, and a reflective angle.
Common reference numbers have been used to identify angle arms, vertices, and trisecting ?tines in figs. 3-6, the same reference numbers used in figs.
2a--c.
The same general method for positioning the template 10 relative to a displayed angle, marking the drawing sheet to identify points on trisecting lines 32, 34, and actually drawing trisecting lines 32, 34, is used in each instance and may be understood with reference to the description of figs. 2a and 2b. The template 10 would normally be removed befvze drawing trisecting lines 32, 34, but has been illustrated together with the resulting trisecting lines 32, 34 as this arrangement better illustrates the geometric relationship between features on the tez~aplate 10, the angle in issue, and the trisecting lines ultimately draw~a.
Reference is made tv fig. 7 which illustrates how the te~onplate 10 can also be used tv bisect an angle 36. The template 10 is displaced zelative to the angle 36 until each of the holes 20, 22 are located over the az~xts 38, 40 of the angle 36. Tlxe line 30 of the template is simultaneously positioned to intersect the vertex 42 of the angle 36. It will be recalled that the line 30 is perpendicular to the hypothetical line 84 (not shown in fig. 7) extending between the centers of the pair of holes 20, 22, and essentially bisects the line 74, thus being equidistant from the holes 20, 22. Accordingly, when placed in the suggested orientation, the bisector 53 can be znazl~ed through the central hole 26, and then drawn through the mark and the vertex 42. The steps of znarlsSng and drawing the bisector 53 have been omitted to avoid needless reproduction _7_ of drawings, and the template 10 is effectively shown repositioned over the angle 36 and the bisecting line 53 after drawing.
How the template 10 may be used to trisect a very small angle will explained with reference to fig. 8. The angle requiring trisecting is identified as s2 is in fig. 8, preferably ~orrrxed on a transparency, and for purposes o~ exemplification may be 3 degrees. The object is not to apply the tezx~plate 10 directly to the small angle SZ, which is not feasible, but to produce another trae,sparency (assumed to be the plane of the drawing page) that cac~
be used afterward to mark and draw trisecting lines 32, 34. A line 56 is drawn ~rozn the vertex 42 of angle s2 to define with the angle arm 40 a larger angle of 60 degrees (arbitrarily selected). The template 10 is then used to trisect the 60--degree angle to produce trisecting lip,es 58, 60. The template 10 zs then used to trisect the larger combined 63-degree angle to produce trisecting lines 62, 64. As apparent ~rozzr fig. $, the Lines 58, 62 form an angle of one-third SZ
or 1 degree, and the lines 60, 64 ~orzz~. an angle of two-thirds S2 or 2 degrees.
The transparency so produced can be overlaid by the transparency bearing the angle S2, and rotated to position the trisecting lines 58, 62 relative to the angle Sa iuato alignment with the upper angle arm 38 to mark or draw the trisecting line 32. 'The Litaes 60, 64 can then be rotated into alignrne~ut with the upper arm to z~aark or draw the trisecting line 34. Since the working triseetors 58-64 represent one-third S2 and twv-thirds SZ only a single pair is required to draw both trisectors 32, 34.
Several points should be noted regarding trisection of the angle SZ. First, angle values have been indicated in fig. 8 solely for exemplification. .
Second, since the trisector 10 does not rely on mathematical calculations or use of protractors, the value o~ S~ and the value of the large angle used to facilitate trisection o~ angle ~2 are immaterial and need not be known.
Reference is ~oaade to fig. 9 which illustrates another template 70 _g_ for trisecting angles. This template 70 is a thin planar nrAember formed of opaque plastic. In the orientation of fig. 9, the template 70 is seen to comprise left and right circular holes 72, 74, the left hole 72 serving as a sighting hole and the right hole 74 as a marking hole. The template 70 has a lateral side edge 76 oriented perpendicular to a hypothetical line 78 between the left and right holes 72, 74 and aligned with the midpoint of the hypothetical line 78. The side edge 76 effectively defines a line corresponding in essence to the line 30 associated with the terzlplate 10 of fig. 1, and performing the same function.
A
central marking hole 80 is located midway between the left and right holes 72, 74 at the rnidpoint of the hypothetical line 78. The template 70 has a projection 82 that extends laterally relative to the side edge 76 and has a part-circular periphery 84 (circular arc) centered about the right marking hole 74 and having a radius r equal to one-half the distance between the pair of outer holes 72, 74.
One corner 86 of the template 70 defines a circular arc of radius r, largely for aesthetic reasons.
Fig. LO shows how the template 70 is used to trisect an acute angle. Features associated with the angle and trisecting lines are labeled with reference numbers common to the angle shown in figs. 2a-2b. The user sights through the left hole 72 to position the left hole 72 over the left arm 38 of the angle, and simultaneously positions the lateral side edge 76 to intersect the vertex 42 of the angle, and the part-circular periphery 84 of the projection 82 in tangential relationship with the right arm 40. One trisecting line 32 can then be drawn immediately along the lateral side edge 76 or marked through the central hole 80 for drawing. The other trisecting line 32 is marleed through the right marking hole 74, and then drawn with a straight edge. This template 70 observes the same basic principle of operation described with respect to the template 10 of fig. 1.
Figs. 11 and 12 illustrate a particular application for the template 70. In this instance the angle 36 is drawn on a vertical wall surface 90. The object of trisecting the angle 36 is to delineate areas within the angle 36 that can be painted different colors. The wall surface 90 is located above a horizontal member 92 (in cross-section) with a flat upper horizontal surface 94. The vertex 42 associated with the angle 36 is positioned at a side edge 96 of the member 92, and the arm 40 of the angle 36 is substa~adally coi~acident with the upper surface 94. The template 70 is oriented ~lat agauast the wall surface 90 with the circular periphezy rested against the upper horizontal surface 94, and tl~e end of the side edge 76 distant from the projection 82 contacting the surface 94. The user slides the template 70 along the upper surface 94 in the direction of the arrow in fig. 11 until the sighting hole 72 aligns with the now upper angle arm 38 as shown i.~a fig. 12. The arrangement is particularly simple because the user is obliged only to ensure that the sighting hole 72 aligns with the a~agle arm 38. More specifically, the circular periphery 84 remains continually in tangential relationship with the upper arm 40 and the lateral side edge 76 intersects the angle's vertex 42 automatically upon reaching the side.
edge 96 of the horizontal member 92.
It will be appreciated that particular embodiments of the invexxtion have been illustrated and described, and that changes may be made thereto without departing from the scope of the appended claims.
Claims (7)
1. A template for trisecting an angle defined between a pair of straight arms that intersect at a vertex, the template comprising:
means defining on the template a first point adapted for visual location ever one of the arms;
a second point surrounded by a marking hole;
a trisector line adapted for visual location in intersecting relationship with the vertex, the trisector line oriented perpendicular to a hypothetical line between the first and second points and aligned axially with the midpoint of the hypothetical line; and, a circular arc adapted for visual location in tangential relationship with the other arm, the circular arc centered about the second point and having a radius equal to one half of the distance between the first and second points.
means defining on the template a first point adapted for visual location ever one of the arms;
a second point surrounded by a marking hole;
a trisector line adapted for visual location in intersecting relationship with the vertex, the trisector line oriented perpendicular to a hypothetical line between the first and second points and aligned axially with the midpoint of the hypothetical line; and, a circular arc adapted for visual location in tangential relationship with the other arm, the circular arc centered about the second point and having a radius equal to one half of the distance between the first and second points.
2. The template of claim 1 in which the template comprises:
a marking hole surrounding the first point;
a marking hole at the midpoint of the hypothetical line.
a marking hole surrounding the first point;
a marking hole at the midpoint of the hypothetical line.
3. The template of claim 1 comprising:
a lateral side edge defining the trisector line;
a projection extending laterally outward relative to the side edge and defining the circular arc.
a lateral side edge defining the trisector line;
a projection extending laterally outward relative to the side edge and defining the circular arc.
4. The template of claim 3 in which the template comprises:
a marking hole surrounding the first point;
a marking hole at the midpoint of the hypothetical line.
a marking hole surrounding the first point;
a marking hole at the midpoint of the hypothetical line.
5. A transparent template for trisecting an angle, the template comprising:
a pair of circles of equal radius;
a pair of clearance holes, each of the clearance holes located at the center of a different one of the circles, the pair of clearance holes spaced by a distance equal to twice the radius of the circles whereby the circles are side by side;
a central clearance hole located at the midpoint of a hypothetical line between the pair of clearance holes;
a line oriented perpendicular to the hypothetical line and aligned with the midpoint of the hypothetical line.
a pair of circles of equal radius;
a pair of clearance holes, each of the clearance holes located at the center of a different one of the circles, the pair of clearance holes spaced by a distance equal to twice the radius of the circles whereby the circles are side by side;
a central clearance hole located at the midpoint of a hypothetical line between the pair of clearance holes;
a line oriented perpendicular to the hypothetical line and aligned with the midpoint of the hypothetical line.
6. A generally planar template for trisecting an angle, the template comprising:
a pair of clearance holes;
a lateral side edge oriented perpendicular to a hypothetical line between the clearance holes and aligned with the midpoint of the hypothetical line; and, a projection extending laterally relative to the side edge, the projection comprising a part-circular periphery centered about one of the clearance holes and having a radius equal to one-half the distance between the pair of clearance holes.
a pair of clearance holes;
a lateral side edge oriented perpendicular to a hypothetical line between the clearance holes and aligned with the midpoint of the hypothetical line; and, a projection extending laterally relative to the side edge, the projection comprising a part-circular periphery centered about one of the clearance holes and having a radius equal to one-half the distance between the pair of clearance holes.
7. The template of claim 6 further comprising a central clearance hole located at the midpoint of the hypothetical line.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA 2441543 CA2441543A1 (en) | 2003-09-18 | 2003-09-18 | Template and method for trisecting an angle |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA 2441543 CA2441543A1 (en) | 2003-09-18 | 2003-09-18 | Template and method for trisecting an angle |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA2441543A1 true CA2441543A1 (en) | 2005-03-18 |
Family
ID=34318753
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA 2441543 Abandoned CA2441543A1 (en) | 2003-09-18 | 2003-09-18 | Template and method for trisecting an angle |
Country Status (1)
| Country | Link |
|---|---|
| CA (1) | CA2441543A1 (en) |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US10994569B2 (en) | 2018-02-06 | 2021-05-04 | Ronald Harvey Rosenfield | Angle trisector, as validated to perform accurately over a wide range of device settings by a novel geometric forming process; also capable of portraying finite lengths that only could be approximated by means of otherwise applying a compass and straightedge to a given length of unity; that furthermore functions as a level whose inherent geometry could be adapted for many other uses such as being incorporated into the design of a hydraulic car lift |
| CN114604024A (en) * | 2022-03-25 | 2022-06-10 | 王兆民 | Triangular equi-divider |
-
2003
- 2003-09-18 CA CA 2441543 patent/CA2441543A1/en not_active Abandoned
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US10994569B2 (en) | 2018-02-06 | 2021-05-04 | Ronald Harvey Rosenfield | Angle trisector, as validated to perform accurately over a wide range of device settings by a novel geometric forming process; also capable of portraying finite lengths that only could be approximated by means of otherwise applying a compass and straightedge to a given length of unity; that furthermore functions as a level whose inherent geometry could be adapted for many other uses such as being incorporated into the design of a hydraulic car lift |
| CN114604024A (en) * | 2022-03-25 | 2022-06-10 | 王兆民 | Triangular equi-divider |
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| Date | Code | Title | Description |
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| EEER | Examination request | ||
| FZDE | Dead |