ORIGINAL ARTICLE

Tool profiling for generation of discrete helical surfaces

Nicolae Oancea & Ionut Popa & Virgil Gabriel Teodor &

Victor G. Oancea

Received: 30 July 2009 /Accepted: 15 December 2009 /Published online: 21 January 2010# Springer-Verlag London Limited 2010

Abstract The envelope theory provides a solid mathemat-ical tool for accurate computations and design of millingtools that generate helical surfaces such as disk or end milltools. Given an analytical description of the helical surfaceto be generated, and considering that the machined surfaceand milling tools are reciprocally enwrapping during thegeneration motions, one can deduce equations for themilling tool profile. However, this usually involves tran-scendent equations that require significant computationalresources for their solution. More importantly, analyticalrepresentations of the surfaces to be generated are notalways available, as in the case of measured discretesurfaces. To alleviate the issue, polynomial approximationsof the entire helical surface or only of a spatial curve alongthe surface can be used as already published in previousworks. The present study focuses on the methodology offinding the simplest polynomials when only a small numberof points are known or measured from the surface to begenerated. The numerical simulation and real examplesshow that third degree Bezier polynomials are suitableapproximations for many practical situations. In such cases,not only computation complexity is highly reduced but alsothe measurement process is significantly simplified. Thepaper presents a methodology for the design of disk-cutting

tools reciprocally enwrapping cylindrical helical surfaces ofconstant pitch when the helical surface generatrix is knownin discrete form. The suggested method is using a smallnumber of points belonging to the helical surface to bemachined. Due to the use of a low-order Bezier poly-nomials representation of the helical surface generatrix, asmall approximation error is introduced as shown in ournumerical examples. Using as a reference the analyticalrepresentation of the generatrix, the exact axial section ofthe disk tool can be computed and compared to the axialsection obtained using the methodology described in thispaper. The approximation errors seem to be acceptable formany technical applications. Numerical simulations andexamples were performed using a Java-based implementa-tion of the suggested algorithm.

Keywords Discrete surfaces . Bezier polynomials .

Approximate profiling

1 Introduction

The design of cutting tools bounded by surfaces ofrevolution for the machining of helical surfaces flutesrepresent a problem which can be solved analytically usingthe enwrapping surfaces generation principles [4, 5]. Theanalytical methods based on the fundamental theorems ofenwrapping surfaces lead to rigorous determination of toolprofiles. Examples include disk or cylindrical tools.

In many practical applications, the surfaces generated byenwrapping are not associated with helical surfaces of teethused for torque transmission for which a very preciseprofile is needed. Consequently, in certain cases such ashelical drills or cylindrical milling cutters, absolute rigorousflute generation is not always necessary [1, 3].

N. Oancea : I. Popa :V. G. Teodor (*)“Dunărea de Jos” University of Galaţi,Domnească street, no. 111,Galatzi, Romaniae-mail: virgil.teodor@ugal.ro

V. G. OanceaSIMULIA,166 Valley St,Providence, RI, 02909, USA

Int J Adv Manuf Technol (2010) 50:37–46DOI 10.1007/s00170-009-2492-y

Small degree Bezier polynomials [2] are consideredsuitable analytical substitutes for the discrete measuredprofiles. Therefore, applying fundamental envelope theoremson those polynomial substitutions leads to simple but yetpowerful and precise mathematical models for tool profiling.

Instead of directly measuring and approximating thehelical 3D surface using specific methods (NURBS curvesor mesh subdivision), the surface is indirectly reconstructedby measuring and approximating a generatrix by a Bezierpolynomial and applying a helical movement to thisgeneratrix.

Several approximate methods exist. More recently, amethod for profiling tools for generating helical surfacesgiven in discrete form, which use a circular arc approxi-mation obtained from cubic splines was developed in [9].Methods for helical surfaces reconstruction which usepolynomials methods for profile’s cross-section approxima-tion based on sampled points using the instantaneousmotion method, in order to profile disk tools for thesesurfaces grinding are presented in [8].

In this paper, we have developed specific algorithms forthe profiling of disk tools reciprocally enveloping withhelical flutes with constant pitch (as shown in Nikolaev [5],“minimum distance” method [6], “in-plane generatingtrajectory” [7] method) for the case when the surfacegeneratrix is not known analytically but only by coordinatesat a few points (as little as three or four points).Comparisons with numerical results obtained using rigor-ous analytical methods suggest that the error level isacceptable.

2 Disk-tool profiling methods

In this section, the classical method for computing the disk-tool profile, reciprocally enwrapping a cylindrical helicalsurface of constant pitch, based on the Nikolaev conditionis presented [5].

In many practical situations, one may know or measureonly a small number of points (as little as four points) alongthe planar or spatial generatrix of the surface to begenerated (workpiece surface). In these cases, the genera-trix can be substituted by a small order (two or three)Bezier polynomial Px(λ), PY(λ), and PZ(λ) where λ2[0,1].

In the helical movement of axis ~V and helical parameterp, the approximated helical surface of constant pitch can beexpressed as:

XYZ

������������ ¼

cosϕ � sinϕ 0sinϕ cosϕ 00 0 1

������������:

PX lð ÞPY lð ÞPZ lð Þ

������������þ

00pϕ

������������; ð1Þ

where λ and 8 are variables parameters.

The polynomial coefficients can be determined ingeneral using a fitting method such as least squaresfrom the reduced number of the actual points on thecurve for which the coordinates are known. In the casesbelow, only a very small number of points are con-sidered such that direct determination of the coefficientsis possible.

While an approximation, this representation can betreated as an analytical set of equations. Consequently, itis possible to use the fundamental theorems of surfacesenwrapping to determine the peripheral surface of the toolwhich would generate by enwrapping the desired helicalsurface. The helical surface can be expressed genericallyfrom 1, as:

Π :

ΠX l;ϕð Þ ¼ PX lð Þ � cosϕ� PY lð Þ � sinϕ;ΠY l;ϕð Þ ¼ PX lð Þ � sinϕþ PY lð Þ � cosϕ;ΠZ l;ϕð Þ ¼ PZ lð Þ þ p � ϕ:

�������ð2Þ

The normal to the approximated helical surface Π(λ, 8),can be written as

~NΠ ¼~i ~j ~kΠX

0ϕð Þ ΠY

0ϕð Þ p

ΠX0lð Þ ΠY

0lð Þ P

0Z lð Þ

������������ ;

ð3Þ

where ′ denotes derivative with respect to either of theindependent parameters.

The Nikolaev enwrapping condition [5] can be written as

N�!

; r1!; A

!� �¼ 0 ð4Þ

where

r1!¼ ΠX l;ϕð Þ � a½ � � i

!þΠY l;ϕð Þ � j!

þΠZ l;ϕð Þ � k!; ð5Þ

A!¼ � sin að Þ j!þ cos að Þ k! ð6Þand a and α are technological parameters as shown inFig. 1.

The characteristic curve is the curve of tangency pointsbetween the tool and the machined surface during themachining motions.

From Eqs. 3, 4, and 5, one can determine pointsbelonging to the characteristic curve CΠ, illustrated inFig. 2.

This would be accomplished for three or four values ofthe λ parameter, corresponding to the approximationpolynomial level for the helical surface generatrix inEq. 6. By expressing these coordinates in the tool’s

38 Int J Adv Manuf Technol (2010) 50:37–46

reference system illustrated in Fig. 1, one determines thesethree or four points on the disk tool S:

XiCΠYiCΠZiCΠ

������������ ¼

1 0 00 cos að Þ sin að Þ0 � sin að Þ cos að Þ

������������:

XCΠ

YCΠ

ZCΠ

�������������

a00

������������

24

35

ð7ÞThe disk tool’s characteristic curve is shown in Fig. 2

and can be written as a set of coordinates,

CΠ : XiCΠ; YiCΠ ; ZiCΠ

� �; i ¼ 1; 2; 3ð Þ: ð8Þ

To determine SA, the axial profile of the tool, oneobserves that:

SA

ZiCΠ

� �i¼ Hi;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX 2

iCΠ

þ Y 2iCΠ

q� i

¼ Ri;

��������i ¼ 1; 2; 3ð Þ ð9Þ

for a second-order polynomial, or i=1,2,3,4 for a third-order polynomial.

For a second-order Bezier polynomial, the approximateaxial section thus determined can be used to determine thepolynomial coefficients for the axial section using,

Q :QH lð Þ ¼ l2DH þ 2l 1� lð ÞEH þ 1� lð Þ2FH ;

QR lð Þ ¼ l2DR þ 2l 1� lð ÞER þ 1� lð Þ2FR;

����� ð10Þ

where

Qð0Þ ¼ SA1; Q1

2

� ¼ SA2; Qð1Þ ¼ SA3: ð11Þ

Once the coefficients DH, DR, EH, ER, FH, FR aredetermined, the approximate coordinates of any point onthe tool’s axial section can be determined.

The process is thus completed: starting only from threeof four points on the planar generatrix of the helical surfaceone can obtain an approximate profile of the required tool.This requires fewer computations than in the case ofclassical methods. Moreover, it allows for the determinationof the tool profile even when the desired profile to beobtained is known only at very reduced number of points,such as obtained from measuring devices (coordinatemeasuring machine).

Note that enwrapping condition 4 and axial profile of thetool 9 should not be evaluated for dense values of λ, butonly for the d+1 values: lk ¼ k�1

d , 1 � k � d þ 1; where dis the polynomial degree of P. Another Bezier polynomial,Q, is used to interpolate the axial profile of the tool.

2.1 The “minimum distance method”

The method (Fig. 3) is based on the following observation(theorem) described in [6]: the contact curve between ahelical cylindrical surface with constant pitch and arevolving surface is the geometrical place of points belongsto helical surface, for which, in planes perpendicularly onthe revolving surface axis, the distance to this axis isminimum.

Fig. 2 Disk tool’s peripheral surface axial section

Fig. 1 The Nikolaev enwrapping condition

Fig. 3 Disk tool—the minimum distance method [6]

Int J Adv Manuf Technol (2010) 50:37–46 39

Transforming the Π surface equations to the disk toolreference system from condition

�ΠY l;ϕð Þ sin a þΠZ l;ϕð Þ cos a ¼ H ; ð12Þ

with H variable is determined the dependency

ϕ ¼ ϕ l;Hð Þ ð13Þ

which, associated with Eq. 2, determine the intersectioncurve between Π with plane Z1=H, in principle, the (CΠ)Hcurves family:

CΠð ÞHX1 ¼ ΠX l;ϕð Þ � a;

Y1 ¼ ΠY l;ϕð Þ cos a þΠZ l;ϕð Þ sin a:

����� ð14Þ

The minimum distance of current point from CΠ curve to~A axis is

dmin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX1

2 l;Hð Þ þ Y12 l;Hð Þq�

min

: ð15Þ

The minimum of this distance is obtained for

X1 l;Hð ÞX 01lþ Y1 l;Hð ÞY 0

1l ¼ 0; ð16Þ

equation which represent the enwrapping condition, con-formal to the presented theorem. Thus, for the two surfacesS and Π—the condition to determine the characteristiccurve can be obtained.

The assembly of Eqs. 2 and 16 represent the character-istic curve of the two enveloping surfaces, in principle, withequations:

CΠ

X1 ¼ X1ðHÞ;Y1 ¼ Y1ðHÞ;Z1 ¼ H ;

�������ð17Þ

similarly to 8. By revolving the CΠ curve around the disktool’s axis, the S surface is generated—the primaryperipheral surface of the disk tool, reciprocally enwrappingwith the Π surface, of which axial section is obtained isshown in Eq. 9.

2.2 “In-plane trajectories generating” method

In addition, the algorithm is also based on the in-planetrajectories generation method as illustrated in [7] (seeFig. 4). In a nutshell, this method shows that the revolutionsurface, reciprocally enwrapping with a cylindrical surface

Fig. 4 Disk tool—in-plane trajectories generating methods [7]

Fig. 5 Measured points, P–, P+ polynomials

Fig. 6 Circular profile in frontal plane (disk-mill tool with helicalteeth)

40 Int J Adv Manuf Technol (2010) 50:37–46

with constant pitch, is formed by the parallels circles familyof the surfaces representing the enveloping of the in-planetrajectories of the helical surface’s profile, from the crossingplanes of the revolution surfaces axis, in the revolvingmotion around this axis.

In the X,Y,Z reference system, the parametrical equationsof the Π helical surface with λ and 8 variable parametersare determined.

The crossing plane of ~A axis of disk tool, plane parallelwith X1Y1, intersects the Π surface after the ΠT curve.

The equation for the crossing plane—the T plane can bewritten as

Z1 ¼ H H variableð Þ: ð18ÞThe parametrical equations of the helical surface Π

which are reported in the reference system associated withthe disk tool can be written as:

X1 ¼ ΠX ϕ; lð Þ � a;

Y1 ¼ ΠY ϕ; lð Þ cos a �ΠZ ϕ; lð Þ sin a;Z1 ¼ ΠY ϕ; lð Þ sina þΠZ ϕ; lð Þ cos a;

ð19Þ

which allow to write the Eq. 18 in the form

�ΠY ϕ; lð Þ sin a þΠZ ϕ; lð Þ cos a ¼ H ð20ÞThe assembly of Eqs. 19 and 20 determines, in X1Y1Z1

reference system, the crossing section, the ΣT curve withequations:

ΠTX1 ¼ X1 lð Þ;Y1 ¼ Y1 lð Þ:

����� ð21Þ

In the revolving motion around Z1 axis ΠT, Eq. 21becomes the generating curves family:

ΠTð ÞvX1 ¼ X1 lð Þ cos v� Y1 lð Þ sin v;Y1 ¼ X1 lð Þ sin vþ Y1 lð Þ cos v;Z1 ¼ H :

������ ð22Þ

The enwrapping of these generating curves family, inconformity with the in-plane trajectories generation meth-od, is the parallel circle of the S surface from the T plane.

The enveloping is obtained associating with the Eq. 22the enveloping condition

X01l

X01v

¼ Y01l

Y01v

: ð23Þ

The assembly of Eqs. 19, 20, and 23 for H variable,represent, on the Π surface, the characteristic curve CΠ,S inthe form:

CΠ;S

X1 ¼ X1 l;ϕð Þ;Y1 ¼ Y1 l;ϕð Þ;Z1 ¼ Z1 l;ϕð Þ;Z1 l;ϕð Þ ¼ H ; H variableð Þ;X

01l� Y 0

1v¼ Y

01l� X 0

1v;

����������ð24Þ

equivalent with the form in 8 and 17.

3 Applications

3.1 Polynomial approximation of measured points

Before presenting experimental results, a methodology foreffective construction of the polynomial approximation ofthe measured points should be introduced.

Consider M ¼ m1; . . . ;mnf g 2 R3 and ordered set ofmeasured points in the 3D space.

One should find a polynomial P of degree d, where d<nis as small as possible, such that: dist P;mið Þ � "; 8i; 1 �i � n, where ε is an arbitrary positive real (maximumadmitted error).

λ λ1=0 l2� ¼ 0.315 l2þ ¼0.376 l3� ¼0.621 l3þ ¼0.678 λ4=1

X 35.991 35.727 35.616 34.982 34.792 33.439

Y 0.001 2.390 2.843 4.597 4.989 7.034

Z 0.000 0.003 0.004 0.003 0.003 0.002

Table 1 Measured points

Fig. 7 Axial profiles of disk tool

Int J Adv Manuf Technol (2010) 50:37–46 41

For a fixed value of d, choose: j�1; . . . ; j�dþ1 and

jþ1; . . . ; jþdþ1 calculate l�1 ; . . . ; l

�dþ1

and lþ1 ; . . . ; lþdþ1

, respec-tively, such that, for each 1 � k � d þ 1;

l�k¼

Pj�ii¼2

dist mi�1 � mið ÞPni¼2

dist mi�1 � mið Þ; is as close as possible to

k � 1

dbut slightly smaller;

ð25Þ

lþk¼

Pjþii¼2

dist mi�1 � mið ÞPni¼2

dist mi�1 � mið Þ; is as close as possible to

k � 1

dbut slightly bigger;

ð26Þ

Then one can solve the equations below to determinepolynomial coefficients (see Fig. 5).

P� l�1

� �¼ mj�1

� � �P� l�

k

� �¼ mj�1

� � �P� l�

dþ1

� �¼ mj�dþ1

8>>>>>>>>>><>>>>>>>>>>:

ð27Þ

λ Approximated profile Theoretical profile Error [mm]

RA [mm] HA [mm] RT [mm] HT [mm]

0.000 144.0000 0.0000 143.9998 −0.0039 0.0039

0.150 144.1638 −1.5793 144.1603 −1.5807 0.0037

0.345 144.8366 −3.5289 144.8394 −3.5285 0.0029

0.500 145.6842 −4.9347 145.6865 −4.9326 0.0031

0.649 146.7101 −6.1483 146.7062 −6.1512 0.0049

0.800 147.8907 −7.2127 147.8809 −7.2235 0.0146

1.000 149.6397 −8.4091 149.6359 −8.4094 0.0038

Table 2 Comparative results:approximated and theoreticaltool profile

Maximum error is 0.0146,obtained for λ=0.800

Fig. 8 Errors diagram for presented profile (k=20)

AB measured circle arc

λ λ1=0 l2� ¼0.287 l2þ ¼0.430 l3� ¼0.572 l3þ ¼0.715 λ4=1

X [mm] 34.05 32.78 32.24 31.81 31.49 31.21

Y [mm] 4.63 5.12 5.54 6.06 6.66 7.98

B′C measured straight line

λ λ1=0 l2� ¼0.180 l2þ ¼0.453 l3� ¼0.638 l3þ ¼0.819 λ4=1

X [mm] 38.88 38.05 36.8 35.95 35.12 34.29

Y [mm] 4.19 4.26 4.37 4.46 4.53 4.6

Table 3 Measured profiles

42 Int J Adv Manuf Technol (2010) 50:37–46

and

Pþ lþ1

� �¼ mjþ1

� � �Pþ lþ

k

� �¼ mjþ1

� � �Pþ lþ

dþ1

� �¼ mjþdþ1

8>>>>>>>>>><>>>>>>>>>>:

ð28Þ

Let P be a polynomial with coefficients obtained as theaverage of corresponding coefficients in P– and P+. Then onecan evaluate the approximation error as: maxni¼1 dist P;mið Þð Þ.

If the error obtained as above is not satisfactory, thepolynomial degree d can be increased and the algorithmabove can be repeated. In the worst-case scenario, thealgorithm stops, eventually, when d=n, and then errors turnto 0.

Coordinates may be measured using 3D coordinatesmeasuring machine. Internal machine software is able toapply a probe radius compensation algorithm can provideaccurate measured points. Therefore, our algorithm willignore the problem of probe radius.

3.2 Numerical examples and physical experiments

In this section, there are presented several numericalexamples and physical tests. All the experiments weredriven as follows. An analytical representation of a curvewas considered as a generatrix for the theoretical helicalsurface. The profile of the disk tool generating this surface isderived by theoretical methods. Secondly, a real measureddiscrete curve is considered, as a generatrix of the real helicalsurfaces. The approximated profile of the disk tool generat-ing the real surface is computed by the methodologypresented above. The axial profiles of the both determineddisk tools are compared and the propagation of measurementerror is analyzed. We can conclude that the methodology

presented here is introducing an acceptable approximationerror along the intrinsic measurement error.

& Disk-tool to generate a worm with circular generatrixin the cross plane

In Fig. 6 is presented the mill tool with helical teeth.Consider that the following coordinates are known:

– the center OC[XOC,YOC], [25;0];– angle θ=0 to 0.696 rad;– radius of circle arc, R=11 mm;– coordinates of the circular arc ends, A [XA, YA] and

B [XB, YB].

The analytical equations of the circular profile are:

X ¼ XOC þ R cos q;

Y ¼ YOC � R sin q;ð29Þ

In Table 1 are presented the coordinates measured onsurface’s generatrix.

In Fig. 7 and Table 2 are presented the numericalcoordinates of the axial profile of disk tool reciprocallyenveloping with helical surfaces with circular frontalgeneratrix, determined by the two methods with the erroroutlined in the last column (a=180 mm, p=87.5 mm, seeFig. 1).

The value of absolute error is calculated as the minimumof the function:

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRAi � RTj

�2 þ HAi � HTj

�2q� min

; i ¼ 1 . . . n; j ¼ 1 . . .mð Þ;

ð30Þpresented in Table 2. In Fig. 8 is presented the evolution ofthese errors regarding the theoretically profile. The toolwas regarded as a grinding wheel with Rex≈150 mm

Fig. 9 Measured transversal section of the helical surface: AB circlearc; B′C straight line segment

Fig. 10 Measuring of profile on 3D Micro Hite measuring machine

Int J Adv Manuf Technol (2010) 50:37–46 43

λ Approximated profile Theoretical profile Error [mm]

RA [mm] HA [mm] RT [mm] HT [mm]

0.0000 43.2389 −1.7441 43.2479 −1.7414 0.0094

0.1500 43.7527 −2.0270 43.7601 −2.0195 0.0106

0.3000 44.2037 −2.4023 44.2090 −2.3995 0.0060

0.3150 44.2447 −2.4444 44.2494 −2.4419 0.0054

0.3330 44.2928 −2.4959 44.2969 −2.4940 0.0046

0.5000 44.6782 −3.0229 44.6780 −3.0239 0.0010

0.7000 44.9743 −3.7456 44.9757 −3.7444 0.0019

0.7250 44.9973 −3.8408 44.9994 −3.8389 0.0028

0.7500 45.0171 −3.9367 45.0206 −3.9380 0.0037

0.9500 45.0563 −4.7167 45.0599 −4.7183 0.0039

1.0000 45.0331 −4.9111 45.0359 −4.8976 0.0138

Table 4 Tool profile—circle arcgeneratrix

Maximum error is 0.0138 whenparameter λ=1

λ Approximated profile Theoretical profile Error [mm]

RA [mm] HA [mm] RT [mm] HT [mm]

0.000 37.92 −0.561 37.919 −0.572 0.011

0.050 38.1721 −0.6056 38.1699 −0.6176 0.012

0.150 38.6764 −0.6978 38.6756 −0.7114 0.013

0.300 39.4314 −0.8442 39.4306 −0.8572 0.013

0.358 39.6830 −0.8950 39.6800 −0.9070 0.012

0.400 39.9337 −0.9468 39.9298 −0.9575 0.011

0.500 40.4353 −1.0533 40.4334 −−1.0620 0.008

0.600 40.9360 −1.1634 40.9364 −1.1695 0.006

0.643 41.1860 −1.2200 41.1850 −1.2240 0.004

0.700 41.4359 −1.2772 41.4336 −1.2790 0.002

0.800 41.9350 −1.3944 41.9352 −1.3929 0.001

0.900 42.4333 −1.5150 42.4361 −1.5099 0.005

1.00 42.9310 −1.6390 42.9310 −1.6290 0.010

Table 5 Tool profile—straightline generatrix

Maximum error is 0.013 whenparameter λ=0.3

Fig. 11 Errors diagram for straight line (B′C segment) and circle arc(AB arc) (k=10) Fig. 12 Disk tool profile (Rex≈45 mm, a=77.5 mm)

44 Int J Adv Manuf Technol (2010) 50:37–46

(149.64 mm). The error is at level 1.10−2mm which is goodenough for this tool’s type, calculated from the Table 2, inthe form:

HAe ¼ HT þ HT � HAð Þ � k;RAe ¼ RT þ RT � RAð Þ � k; ð31Þ

where HAe and RAe are the calculated coordinates for scalederrors and k multiplier factor.

& Disk tool for generation of composite profiles

Consider another example of two measured profiles,tangent and adjacent, measured on the same helical surface(see Table 3 and Fig. 9).

The profile was measured on a 3D coordinate measuringmachine of type 3D Micro Hite as shown in Fig. 10.

Along the measured profiles, the following theoreticalprofiles were considered:

– straight line: [X,Y,Z]=[38.86; 4.18] and [X,Y,Z]=[34.30; 4.61]

– circle arc having center in [X,Y,Z]=[34.63; 8], radius:3.41 mm;

– distance between tool’s and blank axis, a=77.5 mm;– helical parameter of piece, p=87.5 mm.

In Tables 4 and 5 are presented comparative results ofapproximated and theoretical axial sections of the disk tool.Approximation error, along the measurement error, isoutlined in the last column.

In Fig. 11, errors are presented for the straight line andfor circle arc, calculated with Eq. 31.

The value of the absolute error is calculated as in theprevious example. The errors evolution along the composite

profile (shown in Fig. 11) show that the profile of disk tool(mill tool) with external radius Rex≈45 mm (45.0359 mm)have a maximum deviation on level 1.10−2 mm, small enoughto be acceptable for the profile of the gap of a mill tool.

Also, the final numerical examples reveal the tangencycontinuity is very well preserved in the case of compositeapproximated tool profile (Fig. 12).

4 Software considerations

A software application, developed in the Java programminglanguage, was specially designed for this purpose. InFig. 13, we present a screenshot of the Java applet, wherewe can define:

– coordinates of an arbitrary number of the points,measured on generatrix curve (numerical values canbe imported from CSV files);

– outer diameter of the helical surface(D);– the tool type (disk tool in our case).

To estimate the error for the method proposed in thispaper, and for comparative purpose, we can also use in ourapplication a series of analytically defined generatrix curves(circle arc or straight line). Numerical results can beexported to comma-separated-values files.

5 Conclusions

In this paper, we have presented a novel method toapproximate via Bezier polynomials of small order a

Fig. 13 Applet screenshot(disk tool)

Int J Adv Manuf Technol (2010) 50:37–46 45

cylindrical helical surface with constant pitch and in-planegeneratrix known in discrete form by a reduced number ofpoints (3 or 4). Specific algorithms were developed for thedetermination of the axial profile of the associated disktool, reciprocally enveloping with the helical surface,known in discreet form and expressed by Bezier replacingpolynomials.

Several examples illustrated the accuracy of the method.Besides the computational advantage (faster execution), themain appeal of the method is that the profile to be generatedcan be represented by the coordinates of a small number ofpoints. These points can be eventually obtained fromphysical measurements on the generatrix.

The presented algorithm is characterized by a satisfac-tory precision for typical industrial practice, especially forsurfaces which are not associated with large load torquesuch as gears.

The proposed algorithm is based on the Nikolaevtheorem but may be applied to other method known in thereciprocally enveloping surface studies.

Finally, the results above suggest that the intrinsicapproximation of the method is quite acceptable assuggested by the comparisons with the results obtainedusing classical methods.

Acknowledgement The authors gratefully acknowledge the finan-cial support of the Romanian Ministry of Education, Research, andInnovation through grant PN_II_ID_656/2007.

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