THE AUTOMATIC PLOTTING OF THE ARCS OF CIRCLE IN THE CHAMFERED HEXAGONAL PRISM CASE REPRESENTATION

Headnote

Abstract: The paper presents a set of equations that stands to the basis of a computer program and to the software themselves, that offers to the user the possibility to automatic draw the arcs used to approximate the representation of a chamfer hexagonal prism. Also, the authors present the way to use software that generates the arcs that approximate hyperbolas resulting on a chamfered hexagonal prism.

Key words: Hexagonal Prism, AutoCAD, AutoLisp, Arcs, Hyperbolas.

1. INTRODUCTION

On mechanical parts, the hexagonal prism is a surface that frequently meets. When a mechanical part has a thread, this surface enables the wrench to rotate it in order to attach or remove it. Whatever kind of wrench is used, it must be attached to the part in an easy manner. For this reason, the hexagonal prism needs to be chamfered in all cases. This operation leads to a complicated situation from the graphical representations point of view.

The rotating movement of the prism along with the trajectory of the cutting tool creates the surface of a circular cone during the chamfering operation of a hexagonal prism. An arc of hyperbola will emerge from the intersection of the cone surface and the prism face, which imagines a plan surface [1]. But drawing the arc of hyperbola, in the orthogonal projection, is challenging and requires a lot of time. It is well known that, in these instances, the arcs of hyperbola are approximated with arcs of circle. These circular arcs do, however, create certain representation shortcomings, which must be kept under control during the drawing process by respecting certain rules of construction. So, when three faces of the prism are seen and the pen and the compass are used, there must be represented three arcs, one must have the radius Rl and two of them must have the radius R. In addition, line 1 '2' must be represented and its symmetry with respect to the axis, [2], [3], [4], Figure 1.

When two faces of prism are seen, the hyperbola arcs may easily be approximated with two arcs of circle as can be seen in Figure 1.

For a good approximation of the hyperbola arcs, the values for radii R, Rl and R2 are established in respect with the diameter of the bolt rod - d, by using some accepted mathematical relations. As long as the computer graphics are not used, everything looks good, but when this is used, some errors become visible and, in many cases, can be disturbing.

Thus, the arc with the radius R is not tangent either to the line given by points 3' and 5' or to the line given by the points 1' and 2'. More than that, this arc of circle doesn't end in point 1', Figure 2.

The errors are not big and, in some instances, can be covered by the thickness of the line. However, even in these situations, computer programs are always able to identify the specific points on an object, for example, the endpoints of this. When discussing AutoCAD specifically, it can be said that these ones not only affect the selection process of the specific points but, by way of consequence, will affect the use of the OTRACK and POLAR technologies too, leading to a number of additional chained errors.

Analysing the remaining arcs, namely those with the radii Rl and R2, it can be observed that they are not introducing any errors. Therefore, it may be said that the problems only arise when three faces of prism are visible.

Starting from those presented above, and taking into account that the chamfered hexagonal prism is present not only in case of the bolt but in many other situations, the authors decided to build a computer program capable to automatic represent these arcs of circle and to fix the previous described errors.

2. ANALYSIS AND SOLUTIONS

The authors began their investigation by examining the errors that the arc of circle, with a radius of R produces under all conditions. They observed that this arc of circle intersects twice the line given by points 2' and 5', once on the right side and once on the left side of the point 3' and that its end point, considering the fact that this starts from point 4', is situated below from point 1'. So, it creates three intersection points on each side, that do not exist in reality, and may produce errors in the drawing process, Figure 3.

The location where the arc ends, below point 1', has the highest chance of producing errors, mainly related with OTRACK or POLAR technologies. The reason for this is that the arc's end is located very close near to point 1'. Also, at high risk is the point situated on the left side of point 3'. It could be necessary to select point 2' and instead of this one to selected the intersection point. The intersection point situated on the right side of the point 3' may present a high risk only in cases in that the user works at a very large scale.

The authors attempted to use an arc that goes through points 1', 3', and 4' in the first scenario. But, this arc of circle intersects the line given by points 1' and 2' surpassing it to the outside of the projection, Figure 4.

This solution gives not only an intersection point between points 1' and 2' but in addition leads to a representation of prism that is unacceptable. There is a good chance that the previously indicated intersection point will be chosen incorrectly as the reference point.

The authors also considered the possibility of using two arcs of circle in instead of one. The first arc, which goes between points 4' and 3', and the second, which goes between points 3' and 1', Figure 5.

As can been seen in Figure 6, which is a detail taken near point 1', the arc of circle that runs through points 3' and 1', intersects the line given by points 1' and 2' near point 1'. This intersection point is a high-risk point because it is situated very close to point 1' and is very difficult to observe.

It is the same situation in the case of the arc of circle that goes through points 3' and 4'. This one intersects the line given by points 2' and 5' close to point 3', on his right side, Figure 7.

This intersection point is not at a high-risk, because of his position but, in some circumstances, it can cause issues in the drawing process.

Another solution analyses by authors were to use again two arcs of circle, the first one passing through points 4' and 3' as did the arc used in the case presented above, but having a different center placed on the same recall line. Furthermore, the second one, which passes via points 3' and 5', differs from the arc utilized in the preceding instance, Figure 8.

As can be seen in Figure 8, the arc that runs through points 4' and 3' has the center in point 10' and the arc that runs through points 3' and 5' has the center in point 11'. The arc that goes through points 4' and 3' is tangent to the line given by points 2' and 3' and the arc that goes through point 3' and 5' is also tangent to this line. An important detail is that the point 3' is the point where an arc ends and the other starts. Thus, it can observe that in this case there are only two risk points, namely the points 3' and 5'. The software may accidentally treat these, when object snap mode is active, as intersections or endpoints and it could be a disadvantage in some instances. But the risk of happening is not high and that is because point 5' is far from points 1' and 2' and also the point 3' is far from point 2'.

On the other hand, the authors have noted that the approximation of the hyperbola arcs from the perspective of the graphical representation is adequate.

The authors decided to use this method. Consequently, they established that, in order to use this method of approximation, it is necessary to calculate the coordinates of these points using a convenient system of coordinates. More than that it is very important that all the coordinates to be calculating according to the thread dimension from the case of bolt. This dimension is usually noted, in specialized literature, with the letter d. The diameter of the circle that is circumscribed to the hexagon and which defines the hexagonal prism can be obtained if the thread dimension is multiplied by two. This dimension is usually noted with the letter D.

The authors understood that the use of the d and D dimensions in the method proposed by them ensure a comprehensive relationship with the classical theory that resolve the studied problem.

In the case in that the hexagonal prism is projected in such way to be seen 3 faces, the authors established the origin of the axes in the top and the middle of the hexagonal prism, Figure 8. The authors' software will use this location as the insertion point for the circles arcs that approximate the hyperbola arcs in the future. Consequently, the OY axis is mistaken for the bolt's axis and its position is helpful for deducing the calculus relations for the coordinates of the points used in the arcs' representation, in addition helps the building of to the software targeted by the authors.

The values for the point's coordinates 1 are given by the equations (1) and (2), Figure 8.

The value for the X coordinate of point 2 is given by the equation (3). The Y coordinate is equal to 0, Figure 8.

The authors calculated that the value for the X coordinate of point 3 is given by the equation (4), and the Y coordinate of point 3 is equal to zero, Figure 8.

The calculated values of the X and Y coordinates of point 4 are given by the equation (5) and (6), Figure 8.

The values for the point's coordinates 6 are given by the equations (7) and (8), Figure 8.

And, finally, the values for the point's coordinates 9 are given by the equations (9) and (10), Figure 8.

To determine the position of points 5, 10 and 11 in the coordinates system adopted by the authors, there is no need to calculate the values for the X or Y coordinates. The positions of these points may be established using the facilities offered by the AutoLISP programming language. For example, point 8' may be obtained using angle and distance AutoLISP functions and further once point 8' been known the point 10' can be obtained using de inters AutoLISP function [5], [6].

In the case in that the hexagonal prism is projected in such way to be seen 2 faces, the authors established the origin of the axes in the top and the middle of the hexagonal prism, Figure 9.

In this case, it needs to be represented just two arcs of circle, much like in the classical solution when the pencil and the compass are used. To represent one of these arcs it is necessary to know the coordinates of the points 1", 4" and 12", Figure 9.

The value for the Y coordinate of point 1" is given by the equation (11). The X coordinate of point 1" is equal to zero, Figure 9.

The values for the point's coordinates 4" are given by the equations (13) and (11), Figure 9.

Finally, the values for the point's coordinates 12" are given by the equations (14) and (15), Figure 9.

3. THE COMPUTER PROGRAM

The authors have developed software, named RepHex, using the AutoLISP programming language, that allows the user to automatically generate a representation of a chamfered hexagonal prism when it is projected to be seen as three faces, two faces, or when only is seen the hexagon that lies at the base of the prism. This is based on the solutions that were previously presented.

The user is invited to specify the diameter of circle circumscribed about the hexagon that lies on the prism base, Figure 10. This diameter of circle is equal to the diameter d multiplied by two. And, in addition the user must specify the height of the prism.

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Then, the user is invited to specify which is the projection that must be represented, Figure 11.

The last thing that must be done by the user is to specify in the graphical space where is located the point that represents the top middle center of the projection, if it is the case of two or three faces representation of the prism, or the center of the hexagon if it case of the top projection.

4. CONCLUSIONS

The authors propose a way to approximate the arcs of hyperbola with arcs of circle, for the case of chamfered hexagonal prism, more suitable than the classical solution, for its use in the building process of a software designated to represents these arcs in orthogonal projection.

This method is based on points whose coordinates can be determined with equations deduced by the authors. These equations make this way of approximation easy to implement in many other software.

The software built by the authors helps users to quickly represent, in orthogonal projection, a chamfered prism, when three or two faces are projected and when the prism is projected from the top as well.

References

REFERENCES

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[2] Gaba, Victor. (1979). Bazele desenului tehnic, Rotaprint, Institul Politehnic Iasi.

[3] Popa, Constantin, Strobel Gheorghe., Anghel, Alina., Pruna, Liviu., Onofrei, Ligia., Danaila, Vanda., Antonescu, Ion. (1996). Desen tehnic, Editura "Gh. Asachi", Iasi.

[4] Giesecke, Frederick., Alva, Mitchell., Spencer, Henry, Cecil., Hill, Ivan, Leroy., Dygdon, John, Thomas., Novak, James., Lockhart, Shavna. (2008). Technical Drawing (Thirteenth Edition), Pearson Prentice Hall, New York.

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[6] Head, O., George. (1992). AutoLISP in Plain English. A Practical Guide for Non-Programmers, Ventana Press, Chapel Hill