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Research Article A Comparison of Standard One-Step DDA CircularInterpolators with a New Cheap Two-Step Algorithm

Leonid Moroz1 Jan L CieVliNski2 Marta Stakhiv1 and Volodymyr Maksymovych1

983089 Lviv Polytechnic National University S Bandery Street 983089983090 Lviv 983095983097983088983089983091 Ukraine983090 Uniwersytet w Białymstoku Wydział Fizyki ul Lipowa 983092983089 983089983093-983092983090983092 Białystok Poland

Correspondence should be addressed to Jan L Cieslinski janekalphauwbedupl

Received 983090983096 June 983090983088983089983091 Accepted 983090983089 October 983090983088983089983091 Published 983090983088 January 983090983088983089983092

Academic Editor Jing-song Hong

Copyright copy 983090983088983089983092 Leonid Moroz et al Tis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present and study existing digital differential analyzer (DDA) algorithms or circle generation including an improved two-stepDDA algorithm which can be implemented solely in terms o elementary shifs addition and subtraction

1 Introduction

Digital interpolation algorithms are widely used in machinetools with numerical control graphics displays and plottersand manipulation robots Circles and circular arcs requently appear in computer graphics computer-controlled printingand automated control see [983089] One o most popular methodsor generation o circles and arcs is known as digital differen-tial analyzer (DDA) Digital circular interpolators (or angularsweep generators) based on DDA approach are widely used[983090ndash983094] Tere are many papers where characteristics researchresults o such interpolators are systematized [983095 983096]

In this paper a new improved two-step algorithm orDDA circle generation is presented In paper [983097] the ideao applying a Nystrom two-step scheme to circle generationappeared or the 1047297rst time Our next paper [983089983088] developedtheoretical and geometric aspects o this method In thepresent paper we ocus on practical and experimental issuesMoreover herewe consider circles o arbitrary radius whilein [983097 983089983088] we assumed = 1

Te accuracy o this method is higher than the accuracy o other known algorithms Because o its simplicity (it usesonly elementary shifadditionand subtraction) this methodcan also be used in numerical control planning mechanismsand so orth

2 DDA Algorithms

Te general class o DDA algorithms or circles generation isbased on obvious trigonometric transormations describingrotation o vector in the coordinate plane 1038389-1103925 (Figure 983089)

1038389+1 = sin (907317 + Δ907317) = sin907317 cosΔ907317 + cos907317 sinΔ907317= 1038389 cosΔ907317 + 1103925 sinΔ907317

1103925+1 = cos (907317 + Δ907317) = cos907317 cosΔ907317 + sin907317 sinΔ907317

= 1103925 cos

Δ907317 minus 1038389 sin

Δ907317

(983089)

Advantages o DDA algorithms include simplicity and highspeed in generating circle point coordinate 1038389+1 1103925+1

Substituting Δ907317 = = 2minus1038389 where is an integer (usually ge 3) we rewrite (983089) as

1038389+1 = 1038389 cos + 1103925 sin 1103925+1 = 1103925 cos minus 1038389 sin (983090)

which obviously can be expressed by a rotation matrix

= 983131cos minus sin sin cos 983133 (983091)

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2014 Article ID 916539 6 pageshttpdxdoiorg1011552014916539

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983090 Modelling and Simulation in Engineering

y

x

Δ

R

xn yn

xn+1 yn+1

F983145983143983157983154983141 983089 Rotation o vector in the coordinate plane 1038389-1103925

o avoid expensive computation o trigonometric unc-tions DDA algorithms use simpler (cheaper) expressionsinstead o cos and sin For example by replacing trigono-metric unctions by truncated aylor series one gets so-called simultaneous DDA algorithms [983091] Te determinant o the rotation matrix is 983089 Matrices o DDA algorithms havedifferent determinants and det asymp 1 only approximatelyTe closer it equals 983089 the more accurate is the correspondingcircular interpolator [983090] Approximating cos asymp 1 and sin asymp we obtain the simplest (and least accurate) DDA algorithm[983090 983089983089]

1038389+1 = 1038389 + 1103925 1103925+1 = 1103925 minus 1038389 det = 1 + 2

(983092)

Much more accurate algorithm is obtained using cos asymp 1 minus(22) and sin asymp [983095 983089983090 983089983091]

1038389+1 = 8520081 minus 2

2 8520091038389 + 1103925 1103925+1 = 8520081 minus 2

2 8520091103925 minus 1038389

det = 1 + 44 (983093)

Approximating cos and sin with 1047297rst two terms o aylorseries we obtain the ollowing algorithm [983089983092]

1038389+1 = 8520081 minus 2

2 8520091038389 + 852008 minus 3

6 8520091103925

1103925+1 = 8520081 minus 2

2 8520091103925 minus 852008 minus 3

6 8520091038389

det = 1 minus 4

4 + 6

36

(983094)

Another second order scheme different orm truncated ay-lor series is proposed in [983090]

1038389+1 = 8520081 minus 2

2 852009 1038389 + 852008 minus 3

4 8520091103925

1103925+1 = 8520081 minus 2

2 8520091103925 minus 852008 minus 3

4 852009 1038389det = 1 minus 44 +

616

(983095)

All the above DDA algorithms have systematic radial error ateach step which is de1047297ned as

Δ = radic10383892 + 11039252

minus (983096)

In [983095] given the recurrence relations or algorithms that donot have radial errors

1038389+1 = 4 minus 2

4 + 21038389 + 44 + 211039251103925+1 = 4 minus

2

4 + 21103925 minus 44 + 21038389

(983097)

However implementation o these algorithms requires 1047298oat-ing point operations [983095] Replacing the coefficients o (983097) withthe 1047297rst two terms o aylor series [983090] we obtain

4 minus 24 + 2 asymp 1 minus

22

44 + 2 asymp minus

34 (983089983088)

As a result we obtain obviously algorithm given by (983095)A lower level o radial error with much simpler hardware

implementation can be achieved using the ollowing algo-rithm [983091 983089983091]

1038389+1 = 8520081 minus 2

2 852009 1038389 + 852008 minus 3

8 8520091103925

1103925+1 = 8520081 minus 2

2 8520091103925 minus 852008 minus 3

8 852009 1038389

det = 1 + 6

64

(983089983089)

Another scheme worthwhile to be mentioned is the so-calledldquomagic circlerdquo interpolator [983093 983095 983089983093]

1038389+1 = 1038389 + 1103925 1103925+1 = 9830801 minus 2983081 1103925 minus 1038389det = 1 (983089983090)

Tis interpolator is more accurate than the simplest scheme(983092) and requires the use o only two multipliers However inpractice it is not used or large because then the generatedldquocirclerdquo becomes an ellipse [983090 983089983093]

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Modelling and Simulation in Engineering 983091

983137983138983148983141 983089 Maximum values o the absolute radial error ( = 21038389)

Number o algorithms = 3 = 5 = 6 = 8 = 10 = 12(983092) 3972 3332 3234 3159 3145 3142(983093) 0012 0771 minus 3 0192 minus 3 0120 minus 4 0749 minus 6 0468 minus 7(983094) minus041 minus 2 minus0257 minus 3 minus0641 minus 4 minus0400 minus 5 minus025 minus 6 minus0156 minus 7(983095) minus0012 minus0771 minus 3 minus0192 minus 3 minus0120 minus 4 minus074 minus 6 minus0468 minus 7(983089983089) 012 minus 4 047 minus 7 029 minus 8 011 minus 10 045 minus 13 017 minus 15(983089983090) 0262 0253 0251 0250 0250 0250(983089983091) 0262 0252 0251 0250 0250 minus0250(983089983092) minus0063 minus0015 minus0781 minus 2 minus0195 minus 2 minus048 minus 3 minus0122 minus 3(983089983093) minus2677 minus3020 minus3079 minus3124 minus3136 minus3140(983089983094) 018 minus 7 031 minus 6 012 minus 5 019 minus 4 030 minus 3 049 minus 2

Tere is also a group o sequential DDA algorithms [983089983090]Here are some o them

1038389+1

= 1038389

+ 1103925

1103925+1

= 1103925

minus 1038389+1

(983089983091)

1038389+1 = 1038389 + 1103925 1103925+1 = 1103925 minus 1038389+11103925+2 = 1103925+1 minus 1038389+1 1038389+2 = 1038389+1 + 1103925+2 (983089983092)

1038389+1 = 8520081 minus 2

2 8520091038389 + 1103925 1103925+1 = 8520081 minus 2

2 8520091103925 minus 1038389+1(983089983093)

wo-step algorithm

1038389+2 = 1038389+1 cos minus 1038389 1103925+2 = 1103925+1 cos minus 1103925 (983089983094)

with initial conditions

10383890

=

11039250

= 0

10383891

= cos

and

11039251 = sin is known as a ldquodirectrdquo or biquad orm o thecircular interpolator [983089983092] Note that its implementation needsonly two multipliers

In the process o circle points generation radial errorsaccumulate and the law o accumulation is speci1047297c to a givenalgorithm Note that in most cases the circle radius is chosen

to satisy condition 21038389minus1 le le 21038389 [983090 983089983090 983089983093] Obviouslythe greatest error will occur when = 21038389 In able 983089 we givemaximum values o the absolute radial systematic errors or = 21038389 and = 2(1) = [221038389] that is or the ull circlegeneration Initial conditions are chosen to be10383890 = 0 11039250 =

Apart rom radial error an important role in such devices(especially in computer numerical control (CNC) interpola-

tors) is played by the chord error ch [983095] which should notexceed one basic length unit (BLU) Unlike radial error thechord error does not accumulate In [983095] it is shown that oraster circle point generation or most accurate methods (983093)ndash

(983095) (983097) Δ907317 can be chosen rom condition Δ907317 = 991770 8 atch = 1 (we recall that Δ907317 equiv ) that is

le 221038389+3ch or le 221038389+3 (983089983095)

Since the CNC systems radius can reach value max

= 108BLU the CNC interpolators angles range Δ907317 = = 2minus1038389

where = 312 overlapping all possible radius values range(the though adopted algorithms can work and at

gt 12)

In able 983090 we give maximum values o absolute radialsystematic error or most accurate algorithms in case =221038389+3

From these results we see that algorithm (983089983089) is the most

accurate Tereore in next sections we choose algorithm (983089983089)in order to compare it with two new circular interpolatorsintroduced in [983089983088]

3 New DDA Algorithms

In the proposed two-step DDA algorithm the values1038389+2 and1103925+2 are determined by the ollowing ormula

1038389+2 = 1038389 minus 2 sdot sin sdot 1103925+1 1103925+2 = 1103925 + 2 sdot sin sdot 1038389+1(983089983096)

Tis is an exact algorithm which can be derived as ollows (or

another approach see [983089983088]) We consider the rotation romthe point 1038389 1103925 to 1038389+2 1103925+2 see Figure 983090 Hence we obtainthe ollowing equations

1038389+1 = 1038389 cos minus 1103925 sin 1103925+1 = 1103925 cos + 1038389 sin (983089983097)

1038389+2 = 1038389+1 cos minus 1103925+1 sin 1103925+2 = 1103925+1 cos + 1038389+1 sin (983090983088)

From (983090983088) we obtain values 1038389+1 and minus1103925+1

1038389+1 = 1103925+2 minus 1103925+1 cos sin

minus1103925+1 = 1038389+2 minus 1038389+1 cos sin (983090983089)

Substituting values given by (983090983089) into (983090983088) we get

1038389+2 minus 21038389+1 cos + 1038389 = 01103925+2 minus 21103925+1 cos + 1103925 = 0 (983090983090)

Here we recognize the exact discretization o the classicalharmonic oscillator equation [983089983094 983089983095]From(983090983090) it ollows that

1038389+1 = 1038389+2 + 1038389

2cos

1103925+1 = 1103925+2 + 1103925

2cos

(983090983091)

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983092 Modelling and Simulation in Engineering

983137983138983148983141 983090 Maximum values o the absolute radial error ( = 221038389+3)

Number o algorithms = 3 = 5 = 6 = 8 = 10 = 12(983093) 07975 01973 0984 minus 1 0246 minus 1 0614 minus 2 0153 minus 2(983094) minus02642 minus0657 minus 1 minus0328 minus 1 minus0818 minus 2 minus0205 minus 2 minus0511 minus 3(983095) minus07932 minus01972 minus0984 minus 1 minus0246 minus 1 0614 minus 2 0153 minus 2(983089983089) 078 minus 3 012 minus 4 015 minus 5 023 minus 7 037 minus 9 057 minus 11

Substituting obtained values in (983090983089) we get

1038389+1 = 1103925+2 minus 104861610486161103925+2 + 11039251048617 2 cos 1048617 cos sin

minus1103925+1 = 1038389+2 minus 104861610486161038389+2 + 10383891048617 2 cos 1048617 cos sin

(983090983092)

which can easily be reduced to

1038389+1 = 1103925+2

minus 1103925

2 sin minus1103925+1 = 1038389+2

minus 1038389

2 sin (983090983093)

that is to (983089983096) Te algorithm works with initial conditions10383890 = 11039250 = 0 10383891 = cos and 11039251 = sin where = 21038389( is an integer) Approximating sin asymp one can transorm(983089983096) into another two-step algorithm

1038389+2 = 1038389 minus 2 sdot sdot 1103925+1 1103925+2 = 1103925 + 2 sdot sdot 1038389+1 (983090983094)

with 10383890 = 11039250 = 0 10383891 = 1057306 1 minus 2 11039251 = Tis algorithmwas 1047297rst proposed in [983097] see also [983089983088]

Functional diagram o interpolator which reproduces(983090983094) is shown in Figure 983091 Interpolator contains 983094 registers

(Register 983089 Register 983090 Register 983091 Register 983092 Outputmdash1038389+2and Outputmdash1103925+2) 983090 multipliers (Right Shif Register 983089 RightShif Register 983090) subtracter and adder

In this scheme at the beginning (beore the interpolatorstarts) values o 10383890 and 11039250 are recorded in Register 983090 andRegister 983092 respectively while values o 10383891 and11039251 are recordedin Register 983089 and Register 983091 Ten values o 1038389+1 and 1038389 arestored in Register 983089 and Register 983090 and values o 1103925+1 and 1103925are stored in Register 983091 and Register 983092 Right Shif Register983089 and Right Shif Register 983090 perorm multiplication o values1103925+1 and 1038389+1 by 2 Tus at outputs o subtractor and adderparallel binary codes are ormed in accordance with (983090983094)Rounded values arerecorded in Output1038389+2 and Output1103925+2

able 983091 contains maximum values o absolute radialsystematic error o proposed algorithm given by (983090983094) aferull circle simulation in the case

= 24 = 32 = 2minus1038389 = 21038389 = 3 10

= 21 (983090983095)

where is structure elements datapath (in bits) and isnumber o iterations able 983092 contains the same data or case = 221038389+3 Note that results given in ables 983089 and 983090 do notinclude round-off errors hence they do not depend on

y

x

10383891038389

xn

yn

xn+1

yn+1

xn+2

yn+2

xn yn

xn+1 yn+1

xn+2 yn+2

F983145983143983157983154983141 983090 Generation o 983091 subsequent points along the circle

Register 1

Register 1

Register 2

Register 2

Register 3

Register 4

Output Output

Right Shif Right Shif

Round Round

xn yn

xn+1 yn+1

xn+2 yn+2

++

++ minusminus

F983145983143983157983154983141 983091 Functional diagram or the circular interpolator (983090983094)

Analogical data or algorithm given by (983089983089) are given inable 983093 (we consider as beore case (983090983095)) able 983094 containsthe same data or = 221038389+3

Comparing errors in ables 983091 and 983093 and in ables 983092 and983094 we see that the absolute radial errors o our algorithm (983090983094)(ables 983091 and 983092) are smaller than errors o well-known algo-rithm (983089983089) (see ables 983093 and 983094) For example we can compare

values o errors at = 21038389 ( = 10) and = 24 given inables 983091 and 983093 Consequently we will get that an absolute

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Modelling and Simulation in Engineering 983093

1314

151617181920212223

3 4 5 6 7 8 9 10

(m)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

Proposed (26)

Well-known (11)

F983145983143983157983154983141 983092 Number o correct bits o result as a unction o at =21038389 and = 24

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

9

11

13

15

17

19

21

23

C o r r e c

t b i t s

N u m

b e r o

f r e s u

l t s

F983145983143983157983154983141 983093 Number o correct bits o result as a unction o at =221038389+3 and = 24

983137983138983148983141 983091 Maximum values o the absolute radial error (

= 21038389) or

proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 7712 minus 8 minus2567 minus 7 4377 minus 10 minus9137 minus 10983092 1215 minus 7 minus6462 minus 7 1284 minus 9 minus2666 minus 9983093 2458 minus 7 minus1267 minus 6 9422 minus 10 minus5362 minus 9983094 4643 minus 7 minus2645 minus 6 2223 minus 9 minus8292 minus 9983095 7045 minus 7 minus4942 minus 6 3614 minus 9 minus1856 minus 8983096 1655 minus 6 minus9678 minus 6 6936 minus 9 minus3794 minus 8983097 3258 minus 6 minus1898 minus 5 1245 minus 8 minus7470 minus 8983089983088 6758 minus 6 minus3761 minus 5 2510 minus 8 minus1456 minus 7

radial error is smaller by (max(able 5))(max(able 3)) =(1267 minus 5)(6792 minus 6) = 1876 timesWe also see that the smaller is the greater is the

difference between the corresponding errors o both algo-rithms For example or ( = 3) and = 32 we obtainthat an absolute radial error is smaller by (max(able 6))(max(able 4)) = (7781minus4)(4533minus10) asymp 1716523 times

Finally Figures 983092 983093 983094 and 983095 present numbers o correct

bits as a unction o or = 21038389 and = 221038389+3 and or

= 24 and

= 32 respectively

983137983138983148983141 983092 Maximum values o the absolute radial error ( = 221038389+3)or proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 6965 minus 8 minus3866 minus 7 4533 minus 10 minus1671 minus 9983092 1768 minus 7 minus8276 minus 7 4654 minus 10 minus3550 minus 9983093 3010 minus 7 minus1176 minus 6 1260 minus 9 minus5494 minus 9983094 4245 minus 7 minus2292 minus 6 1923 minus 9 minus9138 minus 9983095 8948 minus 7 minus5040 minus 6 3707 minus 9 minus1927 minus 8983096 1454 minus 6 minus9834 minus 6 8918 minus 9 minus3704 minus 8983097 3424 minus 6 minus1936 minus 5 1307 minus 8 minus7389 minus 8983089983088 6792 minus 6 minus3648 minus 5 2792 minus 8 minus1437 minus 7

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

16

18

2022

24

26

28

30

C o r r

e c t b i t s

N u m

b e

r o

f r e s u

l t s

F983145983143983157983154983141 983094 Number o correct bits o result as a unction o at =21038389 and = 32

983137983138983148983141 983093 Maximum values o the absolute radial error ( = 21038389) orAlgorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min

983091 1214 minus 5 0 1215 minus 5 0983092 9641 minus 7 minus8277 minus 7 7525 minus 7 0983093 5496 minus 7 minus2386 minus 6 4552 minus 8 0983094 5508 minus 7 minus4660 minus 6 5781 minus 9 minus1682 minus 8983095 1235 minus 6 minus9674 minus 6 6069 minus 9 minus3579 minus 8983096 4009 minus 6 minus1858 minus 5 1524 minus 8 minus7158 minus 8983097 6478 minus 6 minus3651 minus 5 2034 minus 8 minus1484 minus 7983089983088

1268 minus 5 minus7285 minus 5 5564 minus 8 minus2853 minus 7Presented 1047297gures clearly show that the proposed algo-

rithm (983090983094) yields greater number o correct bits in compar-ison to algorithm (983089983089)

4 Conclusion

A new circle generation DDA algorithm has been proposedIn terms o hardware implementation and precision o circlegeneration the proposed DDA algorithm or circle interpola-toris more efficient as compared with existing algorithms Wehope that the proposed algorithm will 1047297nd some applications

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983094 Modelling and Simulation in Engineering

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

9

14

19

24

29

F983145983143983157983154983141 983095 Number o correct bits o result as a unction o at =221038389+3 and = 32

983137983138983148983141 983094 Maximum values o the absolute radial error ( = 221038389+3)or Algorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min983091 7780 minus 4 0 7781 minus 4 0983092 9650 minus 5 0 9632 minus 5 0983093 1165 minus 5 0 1204 minus 5 0983094 2469 minus 6 minus3933 minus 6 1501 minus 6 0983095 1731 minus 6 minus9497 minus 6 1878 minus 7 0983096 2359 minus 6 minus1942 minus 5 3598 minus 8 minus6302 minus 8983097 5371 minus 6 minus3779 minus 5 2655 minus 8 minus1421 minus 7983089983088 1057 minus 5 minus7478 minus 5 5028 minus 8 minus2914 minus 7

in computer graphics (display screens) computer-controlledprinting automated control and so orth

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

Acknowledgment

Te second author (Jan L Cieslinski) is partly support-ed by the National Science Centre (NCN) Grant no 983090983088983089983089983088983089BS983089983088983093983089983091983095

References[983089] W Wang and C Wang ldquoDifference method or generation o

circular arcs and ellipsesrdquo Computer-Aided Design vol983090983089 no 983089pp 983091983091ndash983091983095 983089983097983096983097

[983090] N Matsushiro ldquoA new digital differential analyzer or circl gen-erationrdquo IEICE Transactions on Information and Systems volE983096983089-D no 983090 pp 983090983091983097ndash983090983092983090 983089983097983097983096

[983091] I Chami ldquoA high precision digital differential analyzer orcircular interpolationrdquo Tishreen University Journal vol 983090983097 no983089 983090983088983088983095

[983092] H Hama and Okumoto ldquoFast transorm into polar coordi-nate by generalized DDArdquo Journal of the Institute of TelevisionEngineers of Japan vol 983092983095 no 983089 pp 983097983095ndash983089983088983088 983089983097983097983091

[983093] X Y Fan Y H Guo and S C Li ldquoNew digital differential ana-lyzer interpolation algorithmrdquo Advanced Science Letters vol 983094no 983089 pp 983094983097983090ndash983094983097983093 983090983088983089983090

[983094] X L Yan W L Wang and N Xue ldquoResearch and simulation o an efficient circular interpolation algorithmrdquo Advanced Materi-als Research vol 983093983092983090-983093983092983091 pp 983089983090983088983092ndash983089983090983088983096 983090983088983089983090

[983095] O Masory and Y Koren ldquoReerence-word circular interpolatorsor CNC systemsrdquo Journal of Engineering for Industry vol 983089983088983092no 983092 pp 983092983088983088ndash983092983088983093 983089983097983096983090

[983096] Y Katayama H Harasaki and M Yano ldquoParallel algorithm o˜three dimensional digital differential analyzerrdquo in Proceedings of the Annual Convention ITE Japan pp 983089983096983093ndash983089983096983094 983089983097983097983091

[983097] L V Moroz ldquoA method or studying characteristics o digitalsine-cosine generatorsrdquo Information Extraction and Processing vol 983091983094 no 983089983089983090 pp 983096983092ndash983097983088 983090983088983089983090 (Ukrainian)

[983089983088] J L Cieslinski and L V Moroz ldquoFast exact digital differentialanalyzer or circle generationrdquo httparxivorgabs983089983091983088983092983092983097983095983092

[983089983089] W M Newman and R F Sproull Principles of InteractiveComputer Graphics McGraw-Hill 983089983097983095983097

[983089983090] F S Lim Y S Wong and M Rahman ldquoCircular interpolators

or numerical control a comparison o the modi1047297ed DDAtechniques and an LSI interpolatorrdquo Computers in Industry vol983089983096 no 983089 pp 983092983089ndash983093983090 983089983097983097983090

[983089983091] R F Eschenbach and B M Oliver ldquoAn efficient coordinate rota-tion algorithmrdquo IEEE Transactions on Computers vol 983090983095 no 983089983090pp 983089983089983095983096ndash983089983089983096983088 983089983097983095983096

[983089983092] H Hasegawa and Y Okada ldquoApparatus and method or con-trolling a moving vehicle utilizing a digital differential analysiscircuitrdquo US Patent 983093983091983095983097983091983093983091 983089983097983097983093

[983089983093] C Pokorny Computer Graphics An Object Oriented Approachtothe Artand Science FranklinBeedle amp Associates Publishing983089983097983097983092

[983089983094] J L Cieslinski and B Ratkiewicz ldquoOn simulations o the

classical harmonic oscillator equation by difference equationsrdquo Advances in Difference Equations vol 983090983088983088983094 Article ID 983092983088983089983095983089983090983088983088983094

[983089983095] J L Cieslinski ldquoOn the exact discretization o the classicalharmonic oscillator equationrdquo Journal of Difference Equationsand Applications vol 983089983095 no 983089983089 pp 983089983094983095983091ndash983089983094983097983092 983090983088983089983089

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 77

Submit your manuscripts at

httpwwwhindawicom

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 27

983090 Modelling and Simulation in Engineering

y

x

Δ

R

xn yn

xn+1 yn+1

F983145983143983157983154983141 983089 Rotation o vector in the coordinate plane 1038389-1103925

o avoid expensive computation o trigonometric unc-tions DDA algorithms use simpler (cheaper) expressionsinstead o cos and sin For example by replacing trigono-metric unctions by truncated aylor series one gets so-called simultaneous DDA algorithms [983091] Te determinant o the rotation matrix is 983089 Matrices o DDA algorithms havedifferent determinants and det asymp 1 only approximatelyTe closer it equals 983089 the more accurate is the correspondingcircular interpolator [983090] Approximating cos asymp 1 and sin asymp we obtain the simplest (and least accurate) DDA algorithm[983090 983089983089]

1038389+1 = 1038389 + 1103925 1103925+1 = 1103925 minus 1038389 det = 1 + 2

(983092)

Much more accurate algorithm is obtained using cos asymp 1 minus(22) and sin asymp [983095 983089983090 983089983091]

1038389+1 = 8520081 minus 2

2 8520091038389 + 1103925 1103925+1 = 8520081 minus 2

2 8520091103925 minus 1038389

det = 1 + 44 (983093)

Approximating cos and sin with 1047297rst two terms o aylorseries we obtain the ollowing algorithm [983089983092]

1038389+1 = 8520081 minus 2

2 8520091038389 + 852008 minus 3

6 8520091103925

1103925+1 = 8520081 minus 2

2 8520091103925 minus 852008 minus 3

6 8520091038389

det = 1 minus 4

4 + 6

36

(983094)

Another second order scheme different orm truncated ay-lor series is proposed in [983090]

1038389+1 = 8520081 minus 2

2 852009 1038389 + 852008 minus 3

4 8520091103925

1103925+1 = 8520081 minus 2

2 8520091103925 minus 852008 minus 3

4 852009 1038389det = 1 minus 44 +

616

(983095)

All the above DDA algorithms have systematic radial error ateach step which is de1047297ned as

Δ = radic10383892 + 11039252

minus (983096)

In [983095] given the recurrence relations or algorithms that donot have radial errors

1038389+1 = 4 minus 2

4 + 21038389 + 44 + 211039251103925+1 = 4 minus

2

4 + 21103925 minus 44 + 21038389

(983097)

However implementation o these algorithms requires 1047298oat-ing point operations [983095] Replacing the coefficients o (983097) withthe 1047297rst two terms o aylor series [983090] we obtain

4 minus 24 + 2 asymp 1 minus

22

44 + 2 asymp minus

34 (983089983088)

As a result we obtain obviously algorithm given by (983095)A lower level o radial error with much simpler hardware

implementation can be achieved using the ollowing algo-rithm [983091 983089983091]

1038389+1 = 8520081 minus 2

2 852009 1038389 + 852008 minus 3

8 8520091103925

1103925+1 = 8520081 minus 2

2 8520091103925 minus 852008 minus 3

8 852009 1038389

det = 1 + 6

64

(983089983089)

Another scheme worthwhile to be mentioned is the so-calledldquomagic circlerdquo interpolator [983093 983095 983089983093]

1038389+1 = 1038389 + 1103925 1103925+1 = 9830801 minus 2983081 1103925 minus 1038389det = 1 (983089983090)

Tis interpolator is more accurate than the simplest scheme(983092) and requires the use o only two multipliers However inpractice it is not used or large because then the generatedldquocirclerdquo becomes an ellipse [983090 983089983093]

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 37

Modelling and Simulation in Engineering 983091

983137983138983148983141 983089 Maximum values o the absolute radial error ( = 21038389)

Number o algorithms = 3 = 5 = 6 = 8 = 10 = 12(983092) 3972 3332 3234 3159 3145 3142(983093) 0012 0771 minus 3 0192 minus 3 0120 minus 4 0749 minus 6 0468 minus 7(983094) minus041 minus 2 minus0257 minus 3 minus0641 minus 4 minus0400 minus 5 minus025 minus 6 minus0156 minus 7(983095) minus0012 minus0771 minus 3 minus0192 minus 3 minus0120 minus 4 minus074 minus 6 minus0468 minus 7(983089983089) 012 minus 4 047 minus 7 029 minus 8 011 minus 10 045 minus 13 017 minus 15(983089983090) 0262 0253 0251 0250 0250 0250(983089983091) 0262 0252 0251 0250 0250 minus0250(983089983092) minus0063 minus0015 minus0781 minus 2 minus0195 minus 2 minus048 minus 3 minus0122 minus 3(983089983093) minus2677 minus3020 minus3079 minus3124 minus3136 minus3140(983089983094) 018 minus 7 031 minus 6 012 minus 5 019 minus 4 030 minus 3 049 minus 2

Tere is also a group o sequential DDA algorithms [983089983090]Here are some o them

1038389+1

= 1038389

+ 1103925

1103925+1

= 1103925

minus 1038389+1

(983089983091)

1038389+1 = 1038389 + 1103925 1103925+1 = 1103925 minus 1038389+11103925+2 = 1103925+1 minus 1038389+1 1038389+2 = 1038389+1 + 1103925+2 (983089983092)

1038389+1 = 8520081 minus 2

2 8520091038389 + 1103925 1103925+1 = 8520081 minus 2

2 8520091103925 minus 1038389+1(983089983093)

wo-step algorithm

1038389+2 = 1038389+1 cos minus 1038389 1103925+2 = 1103925+1 cos minus 1103925 (983089983094)

with initial conditions

10383890

=

11039250

= 0

10383891

= cos

and

11039251 = sin is known as a ldquodirectrdquo or biquad orm o thecircular interpolator [983089983092] Note that its implementation needsonly two multipliers

In the process o circle points generation radial errorsaccumulate and the law o accumulation is speci1047297c to a givenalgorithm Note that in most cases the circle radius is chosen

to satisy condition 21038389minus1 le le 21038389 [983090 983089983090 983089983093] Obviouslythe greatest error will occur when = 21038389 In able 983089 we givemaximum values o the absolute radial systematic errors or = 21038389 and = 2(1) = [221038389] that is or the ull circlegeneration Initial conditions are chosen to be10383890 = 0 11039250 =

Apart rom radial error an important role in such devices(especially in computer numerical control (CNC) interpola-

tors) is played by the chord error ch [983095] which should notexceed one basic length unit (BLU) Unlike radial error thechord error does not accumulate In [983095] it is shown that oraster circle point generation or most accurate methods (983093)ndash

(983095) (983097) Δ907317 can be chosen rom condition Δ907317 = 991770 8 atch = 1 (we recall that Δ907317 equiv ) that is

le 221038389+3ch or le 221038389+3 (983089983095)

Since the CNC systems radius can reach value max

= 108BLU the CNC interpolators angles range Δ907317 = = 2minus1038389

where = 312 overlapping all possible radius values range(the though adopted algorithms can work and at

gt 12)

In able 983090 we give maximum values o absolute radialsystematic error or most accurate algorithms in case =221038389+3

From these results we see that algorithm (983089983089) is the most

accurate Tereore in next sections we choose algorithm (983089983089)in order to compare it with two new circular interpolatorsintroduced in [983089983088]

3 New DDA Algorithms

In the proposed two-step DDA algorithm the values1038389+2 and1103925+2 are determined by the ollowing ormula

1038389+2 = 1038389 minus 2 sdot sin sdot 1103925+1 1103925+2 = 1103925 + 2 sdot sin sdot 1038389+1(983089983096)

Tis is an exact algorithm which can be derived as ollows (or

another approach see [983089983088]) We consider the rotation romthe point 1038389 1103925 to 1038389+2 1103925+2 see Figure 983090 Hence we obtainthe ollowing equations

1038389+1 = 1038389 cos minus 1103925 sin 1103925+1 = 1103925 cos + 1038389 sin (983089983097)

1038389+2 = 1038389+1 cos minus 1103925+1 sin 1103925+2 = 1103925+1 cos + 1038389+1 sin (983090983088)

From (983090983088) we obtain values 1038389+1 and minus1103925+1

1038389+1 = 1103925+2 minus 1103925+1 cos sin

minus1103925+1 = 1038389+2 minus 1038389+1 cos sin (983090983089)

Substituting values given by (983090983089) into (983090983088) we get

1038389+2 minus 21038389+1 cos + 1038389 = 01103925+2 minus 21103925+1 cos + 1103925 = 0 (983090983090)

Here we recognize the exact discretization o the classicalharmonic oscillator equation [983089983094 983089983095]From(983090983090) it ollows that

1038389+1 = 1038389+2 + 1038389

2cos

1103925+1 = 1103925+2 + 1103925

2cos

(983090983091)

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 47

983092 Modelling and Simulation in Engineering

983137983138983148983141 983090 Maximum values o the absolute radial error ( = 221038389+3)

Number o algorithms = 3 = 5 = 6 = 8 = 10 = 12(983093) 07975 01973 0984 minus 1 0246 minus 1 0614 minus 2 0153 minus 2(983094) minus02642 minus0657 minus 1 minus0328 minus 1 minus0818 minus 2 minus0205 minus 2 minus0511 minus 3(983095) minus07932 minus01972 minus0984 minus 1 minus0246 minus 1 0614 minus 2 0153 minus 2(983089983089) 078 minus 3 012 minus 4 015 minus 5 023 minus 7 037 minus 9 057 minus 11

Substituting obtained values in (983090983089) we get

1038389+1 = 1103925+2 minus 104861610486161103925+2 + 11039251048617 2 cos 1048617 cos sin

minus1103925+1 = 1038389+2 minus 104861610486161038389+2 + 10383891048617 2 cos 1048617 cos sin

(983090983092)

which can easily be reduced to

1038389+1 = 1103925+2

minus 1103925

2 sin minus1103925+1 = 1038389+2

minus 1038389

2 sin (983090983093)

that is to (983089983096) Te algorithm works with initial conditions10383890 = 11039250 = 0 10383891 = cos and 11039251 = sin where = 21038389( is an integer) Approximating sin asymp one can transorm(983089983096) into another two-step algorithm

1038389+2 = 1038389 minus 2 sdot sdot 1103925+1 1103925+2 = 1103925 + 2 sdot sdot 1038389+1 (983090983094)

with 10383890 = 11039250 = 0 10383891 = 1057306 1 minus 2 11039251 = Tis algorithmwas 1047297rst proposed in [983097] see also [983089983088]

Functional diagram o interpolator which reproduces(983090983094) is shown in Figure 983091 Interpolator contains 983094 registers

(Register 983089 Register 983090 Register 983091 Register 983092 Outputmdash1038389+2and Outputmdash1103925+2) 983090 multipliers (Right Shif Register 983089 RightShif Register 983090) subtracter and adder

In this scheme at the beginning (beore the interpolatorstarts) values o 10383890 and 11039250 are recorded in Register 983090 andRegister 983092 respectively while values o 10383891 and11039251 are recordedin Register 983089 and Register 983091 Ten values o 1038389+1 and 1038389 arestored in Register 983089 and Register 983090 and values o 1103925+1 and 1103925are stored in Register 983091 and Register 983092 Right Shif Register983089 and Right Shif Register 983090 perorm multiplication o values1103925+1 and 1038389+1 by 2 Tus at outputs o subtractor and adderparallel binary codes are ormed in accordance with (983090983094)Rounded values arerecorded in Output1038389+2 and Output1103925+2

able 983091 contains maximum values o absolute radialsystematic error o proposed algorithm given by (983090983094) aferull circle simulation in the case

= 24 = 32 = 2minus1038389 = 21038389 = 3 10

= 21 (983090983095)

where is structure elements datapath (in bits) and isnumber o iterations able 983092 contains the same data or case = 221038389+3 Note that results given in ables 983089 and 983090 do notinclude round-off errors hence they do not depend on

y

x

10383891038389

xn

yn

xn+1

yn+1

xn+2

yn+2

xn yn

xn+1 yn+1

xn+2 yn+2

F983145983143983157983154983141 983090 Generation o 983091 subsequent points along the circle

Register 1

Register 1

Register 2

Register 2

Register 3

Register 4

Output Output

Right Shif Right Shif

Round Round

xn yn

xn+1 yn+1

xn+2 yn+2

++

++ minusminus

F983145983143983157983154983141 983091 Functional diagram or the circular interpolator (983090983094)

Analogical data or algorithm given by (983089983089) are given inable 983093 (we consider as beore case (983090983095)) able 983094 containsthe same data or = 221038389+3

Comparing errors in ables 983091 and 983093 and in ables 983092 and983094 we see that the absolute radial errors o our algorithm (983090983094)(ables 983091 and 983092) are smaller than errors o well-known algo-rithm (983089983089) (see ables 983093 and 983094) For example we can compare

values o errors at = 21038389 ( = 10) and = 24 given inables 983091 and 983093 Consequently we will get that an absolute

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 57

Modelling and Simulation in Engineering 983093

1314

151617181920212223

3 4 5 6 7 8 9 10

(m)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

Proposed (26)

Well-known (11)

F983145983143983157983154983141 983092 Number o correct bits o result as a unction o at =21038389 and = 24

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

9

11

13

15

17

19

21

23

C o r r e c

t b i t s

N u m

b e r o

f r e s u

l t s

F983145983143983157983154983141 983093 Number o correct bits o result as a unction o at =221038389+3 and = 24

983137983138983148983141 983091 Maximum values o the absolute radial error (

= 21038389) or

proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 7712 minus 8 minus2567 minus 7 4377 minus 10 minus9137 minus 10983092 1215 minus 7 minus6462 minus 7 1284 minus 9 minus2666 minus 9983093 2458 minus 7 minus1267 minus 6 9422 minus 10 minus5362 minus 9983094 4643 minus 7 minus2645 minus 6 2223 minus 9 minus8292 minus 9983095 7045 minus 7 minus4942 minus 6 3614 minus 9 minus1856 minus 8983096 1655 minus 6 minus9678 minus 6 6936 minus 9 minus3794 minus 8983097 3258 minus 6 minus1898 minus 5 1245 minus 8 minus7470 minus 8983089983088 6758 minus 6 minus3761 minus 5 2510 minus 8 minus1456 minus 7

radial error is smaller by (max(able 5))(max(able 3)) =(1267 minus 5)(6792 minus 6) = 1876 timesWe also see that the smaller is the greater is the

difference between the corresponding errors o both algo-rithms For example or ( = 3) and = 32 we obtainthat an absolute radial error is smaller by (max(able 6))(max(able 4)) = (7781minus4)(4533minus10) asymp 1716523 times

Finally Figures 983092 983093 983094 and 983095 present numbers o correct

bits as a unction o or = 21038389 and = 221038389+3 and or

= 24 and

= 32 respectively

983137983138983148983141 983092 Maximum values o the absolute radial error ( = 221038389+3)or proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 6965 minus 8 minus3866 minus 7 4533 minus 10 minus1671 minus 9983092 1768 minus 7 minus8276 minus 7 4654 minus 10 minus3550 minus 9983093 3010 minus 7 minus1176 minus 6 1260 minus 9 minus5494 minus 9983094 4245 minus 7 minus2292 minus 6 1923 minus 9 minus9138 minus 9983095 8948 minus 7 minus5040 minus 6 3707 minus 9 minus1927 minus 8983096 1454 minus 6 minus9834 minus 6 8918 minus 9 minus3704 minus 8983097 3424 minus 6 minus1936 minus 5 1307 minus 8 minus7389 minus 8983089983088 6792 minus 6 minus3648 minus 5 2792 minus 8 minus1437 minus 7

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

16

18

2022

24

26

28

30

C o r r

e c t b i t s

N u m

b e

r o

f r e s u

l t s

F983145983143983157983154983141 983094 Number o correct bits o result as a unction o at =21038389 and = 32

983137983138983148983141 983093 Maximum values o the absolute radial error ( = 21038389) orAlgorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min

983091 1214 minus 5 0 1215 minus 5 0983092 9641 minus 7 minus8277 minus 7 7525 minus 7 0983093 5496 minus 7 minus2386 minus 6 4552 minus 8 0983094 5508 minus 7 minus4660 minus 6 5781 minus 9 minus1682 minus 8983095 1235 minus 6 minus9674 minus 6 6069 minus 9 minus3579 minus 8983096 4009 minus 6 minus1858 minus 5 1524 minus 8 minus7158 minus 8983097 6478 minus 6 minus3651 minus 5 2034 minus 8 minus1484 minus 7983089983088

1268 minus 5 minus7285 minus 5 5564 minus 8 minus2853 minus 7Presented 1047297gures clearly show that the proposed algo-

rithm (983090983094) yields greater number o correct bits in compar-ison to algorithm (983089983089)

4 Conclusion

A new circle generation DDA algorithm has been proposedIn terms o hardware implementation and precision o circlegeneration the proposed DDA algorithm or circle interpola-toris more efficient as compared with existing algorithms Wehope that the proposed algorithm will 1047297nd some applications

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 67

983094 Modelling and Simulation in Engineering

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

9

14

19

24

29

F983145983143983157983154983141 983095 Number o correct bits o result as a unction o at =221038389+3 and = 32

983137983138983148983141 983094 Maximum values o the absolute radial error ( = 221038389+3)or Algorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min983091 7780 minus 4 0 7781 minus 4 0983092 9650 minus 5 0 9632 minus 5 0983093 1165 minus 5 0 1204 minus 5 0983094 2469 minus 6 minus3933 minus 6 1501 minus 6 0983095 1731 minus 6 minus9497 minus 6 1878 minus 7 0983096 2359 minus 6 minus1942 minus 5 3598 minus 8 minus6302 minus 8983097 5371 minus 6 minus3779 minus 5 2655 minus 8 minus1421 minus 7983089983088 1057 minus 5 minus7478 minus 5 5028 minus 8 minus2914 minus 7

in computer graphics (display screens) computer-controlledprinting automated control and so orth

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

Acknowledgment

Te second author (Jan L Cieslinski) is partly support-ed by the National Science Centre (NCN) Grant no 983090983088983089983089983088983089BS983089983088983093983089983091983095

References[983089] W Wang and C Wang ldquoDifference method or generation o

circular arcs and ellipsesrdquo Computer-Aided Design vol983090983089 no 983089pp 983091983091ndash983091983095 983089983097983096983097

[983090] N Matsushiro ldquoA new digital differential analyzer or circl gen-erationrdquo IEICE Transactions on Information and Systems volE983096983089-D no 983090 pp 983090983091983097ndash983090983092983090 983089983097983097983096

[983091] I Chami ldquoA high precision digital differential analyzer orcircular interpolationrdquo Tishreen University Journal vol 983090983097 no983089 983090983088983088983095

[983092] H Hama and Okumoto ldquoFast transorm into polar coordi-nate by generalized DDArdquo Journal of the Institute of TelevisionEngineers of Japan vol 983092983095 no 983089 pp 983097983095ndash983089983088983088 983089983097983097983091

[983093] X Y Fan Y H Guo and S C Li ldquoNew digital differential ana-lyzer interpolation algorithmrdquo Advanced Science Letters vol 983094no 983089 pp 983094983097983090ndash983094983097983093 983090983088983089983090

[983094] X L Yan W L Wang and N Xue ldquoResearch and simulation o an efficient circular interpolation algorithmrdquo Advanced Materi-als Research vol 983093983092983090-983093983092983091 pp 983089983090983088983092ndash983089983090983088983096 983090983088983089983090

[983095] O Masory and Y Koren ldquoReerence-word circular interpolatorsor CNC systemsrdquo Journal of Engineering for Industry vol 983089983088983092no 983092 pp 983092983088983088ndash983092983088983093 983089983097983096983090

[983096] Y Katayama H Harasaki and M Yano ldquoParallel algorithm o˜three dimensional digital differential analyzerrdquo in Proceedings of the Annual Convention ITE Japan pp 983089983096983093ndash983089983096983094 983089983097983097983091

[983097] L V Moroz ldquoA method or studying characteristics o digitalsine-cosine generatorsrdquo Information Extraction and Processing vol 983091983094 no 983089983089983090 pp 983096983092ndash983097983088 983090983088983089983090 (Ukrainian)

[983089983088] J L Cieslinski and L V Moroz ldquoFast exact digital differentialanalyzer or circle generationrdquo httparxivorgabs983089983091983088983092983092983097983095983092

[983089983089] W M Newman and R F Sproull Principles of InteractiveComputer Graphics McGraw-Hill 983089983097983095983097

[983089983090] F S Lim Y S Wong and M Rahman ldquoCircular interpolators

or numerical control a comparison o the modi1047297ed DDAtechniques and an LSI interpolatorrdquo Computers in Industry vol983089983096 no 983089 pp 983092983089ndash983093983090 983089983097983097983090

[983089983091] R F Eschenbach and B M Oliver ldquoAn efficient coordinate rota-tion algorithmrdquo IEEE Transactions on Computers vol 983090983095 no 983089983090pp 983089983089983095983096ndash983089983089983096983088 983089983097983095983096

[983089983092] H Hasegawa and Y Okada ldquoApparatus and method or con-trolling a moving vehicle utilizing a digital differential analysiscircuitrdquo US Patent 983093983091983095983097983091983093983091 983089983097983097983093

[983089983093] C Pokorny Computer Graphics An Object Oriented Approachtothe Artand Science FranklinBeedle amp Associates Publishing983089983097983097983092

[983089983094] J L Cieslinski and B Ratkiewicz ldquoOn simulations o the

classical harmonic oscillator equation by difference equationsrdquo Advances in Difference Equations vol 983090983088983088983094 Article ID 983092983088983089983095983089983090983088983088983094

[983089983095] J L Cieslinski ldquoOn the exact discretization o the classicalharmonic oscillator equationrdquo Journal of Difference Equationsand Applications vol 983089983095 no 983089983089 pp 983089983094983095983091ndash983089983094983097983092 983090983088983089983089

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 77

Submit your manuscripts at

httpwwwhindawicom

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 37

Modelling and Simulation in Engineering 983091

983137983138983148983141 983089 Maximum values o the absolute radial error ( = 21038389)

Number o algorithms = 3 = 5 = 6 = 8 = 10 = 12(983092) 3972 3332 3234 3159 3145 3142(983093) 0012 0771 minus 3 0192 minus 3 0120 minus 4 0749 minus 6 0468 minus 7(983094) minus041 minus 2 minus0257 minus 3 minus0641 minus 4 minus0400 minus 5 minus025 minus 6 minus0156 minus 7(983095) minus0012 minus0771 minus 3 minus0192 minus 3 minus0120 minus 4 minus074 minus 6 minus0468 minus 7(983089983089) 012 minus 4 047 minus 7 029 minus 8 011 minus 10 045 minus 13 017 minus 15(983089983090) 0262 0253 0251 0250 0250 0250(983089983091) 0262 0252 0251 0250 0250 minus0250(983089983092) minus0063 minus0015 minus0781 minus 2 minus0195 minus 2 minus048 minus 3 minus0122 minus 3(983089983093) minus2677 minus3020 minus3079 minus3124 minus3136 minus3140(983089983094) 018 minus 7 031 minus 6 012 minus 5 019 minus 4 030 minus 3 049 minus 2

Tere is also a group o sequential DDA algorithms [983089983090]Here are some o them

1038389+1

= 1038389

+ 1103925

1103925+1

= 1103925

minus 1038389+1

(983089983091)

1038389+1 = 1038389 + 1103925 1103925+1 = 1103925 minus 1038389+11103925+2 = 1103925+1 minus 1038389+1 1038389+2 = 1038389+1 + 1103925+2 (983089983092)

1038389+1 = 8520081 minus 2

2 8520091038389 + 1103925 1103925+1 = 8520081 minus 2

2 8520091103925 minus 1038389+1(983089983093)

wo-step algorithm

1038389+2 = 1038389+1 cos minus 1038389 1103925+2 = 1103925+1 cos minus 1103925 (983089983094)

with initial conditions

10383890

=

11039250

= 0

10383891

= cos

and

11039251 = sin is known as a ldquodirectrdquo or biquad orm o thecircular interpolator [983089983092] Note that its implementation needsonly two multipliers

In the process o circle points generation radial errorsaccumulate and the law o accumulation is speci1047297c to a givenalgorithm Note that in most cases the circle radius is chosen

to satisy condition 21038389minus1 le le 21038389 [983090 983089983090 983089983093] Obviouslythe greatest error will occur when = 21038389 In able 983089 we givemaximum values o the absolute radial systematic errors or = 21038389 and = 2(1) = [221038389] that is or the ull circlegeneration Initial conditions are chosen to be10383890 = 0 11039250 =

Apart rom radial error an important role in such devices(especially in computer numerical control (CNC) interpola-

tors) is played by the chord error ch [983095] which should notexceed one basic length unit (BLU) Unlike radial error thechord error does not accumulate In [983095] it is shown that oraster circle point generation or most accurate methods (983093)ndash

(983095) (983097) Δ907317 can be chosen rom condition Δ907317 = 991770 8 atch = 1 (we recall that Δ907317 equiv ) that is

le 221038389+3ch or le 221038389+3 (983089983095)

Since the CNC systems radius can reach value max

= 108BLU the CNC interpolators angles range Δ907317 = = 2minus1038389

where = 312 overlapping all possible radius values range(the though adopted algorithms can work and at

gt 12)

In able 983090 we give maximum values o absolute radialsystematic error or most accurate algorithms in case =221038389+3

From these results we see that algorithm (983089983089) is the most

accurate Tereore in next sections we choose algorithm (983089983089)in order to compare it with two new circular interpolatorsintroduced in [983089983088]

3 New DDA Algorithms

In the proposed two-step DDA algorithm the values1038389+2 and1103925+2 are determined by the ollowing ormula

1038389+2 = 1038389 minus 2 sdot sin sdot 1103925+1 1103925+2 = 1103925 + 2 sdot sin sdot 1038389+1(983089983096)

Tis is an exact algorithm which can be derived as ollows (or

another approach see [983089983088]) We consider the rotation romthe point 1038389 1103925 to 1038389+2 1103925+2 see Figure 983090 Hence we obtainthe ollowing equations

1038389+1 = 1038389 cos minus 1103925 sin 1103925+1 = 1103925 cos + 1038389 sin (983089983097)

1038389+2 = 1038389+1 cos minus 1103925+1 sin 1103925+2 = 1103925+1 cos + 1038389+1 sin (983090983088)

From (983090983088) we obtain values 1038389+1 and minus1103925+1

1038389+1 = 1103925+2 minus 1103925+1 cos sin

minus1103925+1 = 1038389+2 minus 1038389+1 cos sin (983090983089)

Substituting values given by (983090983089) into (983090983088) we get

1038389+2 minus 21038389+1 cos + 1038389 = 01103925+2 minus 21103925+1 cos + 1103925 = 0 (983090983090)

Here we recognize the exact discretization o the classicalharmonic oscillator equation [983089983094 983089983095]From(983090983090) it ollows that

1038389+1 = 1038389+2 + 1038389

2cos

1103925+1 = 1103925+2 + 1103925

2cos

(983090983091)

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 47

983092 Modelling and Simulation in Engineering

983137983138983148983141 983090 Maximum values o the absolute radial error ( = 221038389+3)

Number o algorithms = 3 = 5 = 6 = 8 = 10 = 12(983093) 07975 01973 0984 minus 1 0246 minus 1 0614 minus 2 0153 minus 2(983094) minus02642 minus0657 minus 1 minus0328 minus 1 minus0818 minus 2 minus0205 minus 2 minus0511 minus 3(983095) minus07932 minus01972 minus0984 minus 1 minus0246 minus 1 0614 minus 2 0153 minus 2(983089983089) 078 minus 3 012 minus 4 015 minus 5 023 minus 7 037 minus 9 057 minus 11

Substituting obtained values in (983090983089) we get

1038389+1 = 1103925+2 minus 104861610486161103925+2 + 11039251048617 2 cos 1048617 cos sin

minus1103925+1 = 1038389+2 minus 104861610486161038389+2 + 10383891048617 2 cos 1048617 cos sin

(983090983092)

which can easily be reduced to

1038389+1 = 1103925+2

minus 1103925

2 sin minus1103925+1 = 1038389+2

minus 1038389

2 sin (983090983093)

that is to (983089983096) Te algorithm works with initial conditions10383890 = 11039250 = 0 10383891 = cos and 11039251 = sin where = 21038389( is an integer) Approximating sin asymp one can transorm(983089983096) into another two-step algorithm

1038389+2 = 1038389 minus 2 sdot sdot 1103925+1 1103925+2 = 1103925 + 2 sdot sdot 1038389+1 (983090983094)

with 10383890 = 11039250 = 0 10383891 = 1057306 1 minus 2 11039251 = Tis algorithmwas 1047297rst proposed in [983097] see also [983089983088]

Functional diagram o interpolator which reproduces(983090983094) is shown in Figure 983091 Interpolator contains 983094 registers

(Register 983089 Register 983090 Register 983091 Register 983092 Outputmdash1038389+2and Outputmdash1103925+2) 983090 multipliers (Right Shif Register 983089 RightShif Register 983090) subtracter and adder

In this scheme at the beginning (beore the interpolatorstarts) values o 10383890 and 11039250 are recorded in Register 983090 andRegister 983092 respectively while values o 10383891 and11039251 are recordedin Register 983089 and Register 983091 Ten values o 1038389+1 and 1038389 arestored in Register 983089 and Register 983090 and values o 1103925+1 and 1103925are stored in Register 983091 and Register 983092 Right Shif Register983089 and Right Shif Register 983090 perorm multiplication o values1103925+1 and 1038389+1 by 2 Tus at outputs o subtractor and adderparallel binary codes are ormed in accordance with (983090983094)Rounded values arerecorded in Output1038389+2 and Output1103925+2

able 983091 contains maximum values o absolute radialsystematic error o proposed algorithm given by (983090983094) aferull circle simulation in the case

= 24 = 32 = 2minus1038389 = 21038389 = 3 10

= 21 (983090983095)

where is structure elements datapath (in bits) and isnumber o iterations able 983092 contains the same data or case = 221038389+3 Note that results given in ables 983089 and 983090 do notinclude round-off errors hence they do not depend on

y

x

10383891038389

xn

yn

xn+1

yn+1

xn+2

yn+2

xn yn

xn+1 yn+1

xn+2 yn+2

F983145983143983157983154983141 983090 Generation o 983091 subsequent points along the circle

Register 1

Register 1

Register 2

Register 2

Register 3

Register 4

Output Output

Right Shif Right Shif

Round Round

xn yn

xn+1 yn+1

xn+2 yn+2

++

++ minusminus

F983145983143983157983154983141 983091 Functional diagram or the circular interpolator (983090983094)

Analogical data or algorithm given by (983089983089) are given inable 983093 (we consider as beore case (983090983095)) able 983094 containsthe same data or = 221038389+3

Comparing errors in ables 983091 and 983093 and in ables 983092 and983094 we see that the absolute radial errors o our algorithm (983090983094)(ables 983091 and 983092) are smaller than errors o well-known algo-rithm (983089983089) (see ables 983093 and 983094) For example we can compare

values o errors at = 21038389 ( = 10) and = 24 given inables 983091 and 983093 Consequently we will get that an absolute

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 57

Modelling and Simulation in Engineering 983093

1314

151617181920212223

3 4 5 6 7 8 9 10

(m)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

Proposed (26)

Well-known (11)

F983145983143983157983154983141 983092 Number o correct bits o result as a unction o at =21038389 and = 24

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

9

11

13

15

17

19

21

23

C o r r e c

t b i t s

N u m

b e r o

f r e s u

l t s

F983145983143983157983154983141 983093 Number o correct bits o result as a unction o at =221038389+3 and = 24

983137983138983148983141 983091 Maximum values o the absolute radial error (

= 21038389) or

proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 7712 minus 8 minus2567 minus 7 4377 minus 10 minus9137 minus 10983092 1215 minus 7 minus6462 minus 7 1284 minus 9 minus2666 minus 9983093 2458 minus 7 minus1267 minus 6 9422 minus 10 minus5362 minus 9983094 4643 minus 7 minus2645 minus 6 2223 minus 9 minus8292 minus 9983095 7045 minus 7 minus4942 minus 6 3614 minus 9 minus1856 minus 8983096 1655 minus 6 minus9678 minus 6 6936 minus 9 minus3794 minus 8983097 3258 minus 6 minus1898 minus 5 1245 minus 8 minus7470 minus 8983089983088 6758 minus 6 minus3761 minus 5 2510 minus 8 minus1456 minus 7

radial error is smaller by (max(able 5))(max(able 3)) =(1267 minus 5)(6792 minus 6) = 1876 timesWe also see that the smaller is the greater is the

difference between the corresponding errors o both algo-rithms For example or ( = 3) and = 32 we obtainthat an absolute radial error is smaller by (max(able 6))(max(able 4)) = (7781minus4)(4533minus10) asymp 1716523 times

Finally Figures 983092 983093 983094 and 983095 present numbers o correct

bits as a unction o or = 21038389 and = 221038389+3 and or

= 24 and

= 32 respectively

983137983138983148983141 983092 Maximum values o the absolute radial error ( = 221038389+3)or proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 6965 minus 8 minus3866 minus 7 4533 minus 10 minus1671 minus 9983092 1768 minus 7 minus8276 minus 7 4654 minus 10 minus3550 minus 9983093 3010 minus 7 minus1176 minus 6 1260 minus 9 minus5494 minus 9983094 4245 minus 7 minus2292 minus 6 1923 minus 9 minus9138 minus 9983095 8948 minus 7 minus5040 minus 6 3707 minus 9 minus1927 minus 8983096 1454 minus 6 minus9834 minus 6 8918 minus 9 minus3704 minus 8983097 3424 minus 6 minus1936 minus 5 1307 minus 8 minus7389 minus 8983089983088 6792 minus 6 minus3648 minus 5 2792 minus 8 minus1437 minus 7

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

16

18

2022

24

26

28

30

C o r r

e c t b i t s

N u m

b e

r o

f r e s u

l t s

F983145983143983157983154983141 983094 Number o correct bits o result as a unction o at =21038389 and = 32

983137983138983148983141 983093 Maximum values o the absolute radial error ( = 21038389) orAlgorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min

983091 1214 minus 5 0 1215 minus 5 0983092 9641 minus 7 minus8277 minus 7 7525 minus 7 0983093 5496 minus 7 minus2386 minus 6 4552 minus 8 0983094 5508 minus 7 minus4660 minus 6 5781 minus 9 minus1682 minus 8983095 1235 minus 6 minus9674 minus 6 6069 minus 9 minus3579 minus 8983096 4009 minus 6 minus1858 minus 5 1524 minus 8 minus7158 minus 8983097 6478 minus 6 minus3651 minus 5 2034 minus 8 minus1484 minus 7983089983088

1268 minus 5 minus7285 minus 5 5564 minus 8 minus2853 minus 7Presented 1047297gures clearly show that the proposed algo-

rithm (983090983094) yields greater number o correct bits in compar-ison to algorithm (983089983089)

4 Conclusion

A new circle generation DDA algorithm has been proposedIn terms o hardware implementation and precision o circlegeneration the proposed DDA algorithm or circle interpola-toris more efficient as compared with existing algorithms Wehope that the proposed algorithm will 1047297nd some applications

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 67

983094 Modelling and Simulation in Engineering

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

9

14

19

24

29

F983145983143983157983154983141 983095 Number o correct bits o result as a unction o at =221038389+3 and = 32

983137983138983148983141 983094 Maximum values o the absolute radial error ( = 221038389+3)or Algorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min983091 7780 minus 4 0 7781 minus 4 0983092 9650 minus 5 0 9632 minus 5 0983093 1165 minus 5 0 1204 minus 5 0983094 2469 minus 6 minus3933 minus 6 1501 minus 6 0983095 1731 minus 6 minus9497 minus 6 1878 minus 7 0983096 2359 minus 6 minus1942 minus 5 3598 minus 8 minus6302 minus 8983097 5371 minus 6 minus3779 minus 5 2655 minus 8 minus1421 minus 7983089983088 1057 minus 5 minus7478 minus 5 5028 minus 8 minus2914 minus 7

in computer graphics (display screens) computer-controlledprinting automated control and so orth

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

Acknowledgment

Te second author (Jan L Cieslinski) is partly support-ed by the National Science Centre (NCN) Grant no 983090983088983089983089983088983089BS983089983088983093983089983091983095

References[983089] W Wang and C Wang ldquoDifference method or generation o

circular arcs and ellipsesrdquo Computer-Aided Design vol983090983089 no 983089pp 983091983091ndash983091983095 983089983097983096983097

[983090] N Matsushiro ldquoA new digital differential analyzer or circl gen-erationrdquo IEICE Transactions on Information and Systems volE983096983089-D no 983090 pp 983090983091983097ndash983090983092983090 983089983097983097983096

[983091] I Chami ldquoA high precision digital differential analyzer orcircular interpolationrdquo Tishreen University Journal vol 983090983097 no983089 983090983088983088983095

[983092] H Hama and Okumoto ldquoFast transorm into polar coordi-nate by generalized DDArdquo Journal of the Institute of TelevisionEngineers of Japan vol 983092983095 no 983089 pp 983097983095ndash983089983088983088 983089983097983097983091

[983093] X Y Fan Y H Guo and S C Li ldquoNew digital differential ana-lyzer interpolation algorithmrdquo Advanced Science Letters vol 983094no 983089 pp 983094983097983090ndash983094983097983093 983090983088983089983090

[983094] X L Yan W L Wang and N Xue ldquoResearch and simulation o an efficient circular interpolation algorithmrdquo Advanced Materi-als Research vol 983093983092983090-983093983092983091 pp 983089983090983088983092ndash983089983090983088983096 983090983088983089983090

[983095] O Masory and Y Koren ldquoReerence-word circular interpolatorsor CNC systemsrdquo Journal of Engineering for Industry vol 983089983088983092no 983092 pp 983092983088983088ndash983092983088983093 983089983097983096983090

[983096] Y Katayama H Harasaki and M Yano ldquoParallel algorithm o˜three dimensional digital differential analyzerrdquo in Proceedings of the Annual Convention ITE Japan pp 983089983096983093ndash983089983096983094 983089983097983097983091

[983097] L V Moroz ldquoA method or studying characteristics o digitalsine-cosine generatorsrdquo Information Extraction and Processing vol 983091983094 no 983089983089983090 pp 983096983092ndash983097983088 983090983088983089983090 (Ukrainian)

[983089983088] J L Cieslinski and L V Moroz ldquoFast exact digital differentialanalyzer or circle generationrdquo httparxivorgabs983089983091983088983092983092983097983095983092

[983089983089] W M Newman and R F Sproull Principles of InteractiveComputer Graphics McGraw-Hill 983089983097983095983097

[983089983090] F S Lim Y S Wong and M Rahman ldquoCircular interpolators

or numerical control a comparison o the modi1047297ed DDAtechniques and an LSI interpolatorrdquo Computers in Industry vol983089983096 no 983089 pp 983092983089ndash983093983090 983089983097983097983090

[983089983091] R F Eschenbach and B M Oliver ldquoAn efficient coordinate rota-tion algorithmrdquo IEEE Transactions on Computers vol 983090983095 no 983089983090pp 983089983089983095983096ndash983089983089983096983088 983089983097983095983096

[983089983092] H Hasegawa and Y Okada ldquoApparatus and method or con-trolling a moving vehicle utilizing a digital differential analysiscircuitrdquo US Patent 983093983091983095983097983091983093983091 983089983097983097983093

[983089983093] C Pokorny Computer Graphics An Object Oriented Approachtothe Artand Science FranklinBeedle amp Associates Publishing983089983097983097983092

[983089983094] J L Cieslinski and B Ratkiewicz ldquoOn simulations o the

classical harmonic oscillator equation by difference equationsrdquo Advances in Difference Equations vol 983090983088983088983094 Article ID 983092983088983089983095983089983090983088983088983094

[983089983095] J L Cieslinski ldquoOn the exact discretization o the classicalharmonic oscillator equationrdquo Journal of Difference Equationsand Applications vol 983089983095 no 983089983089 pp 983089983094983095983091ndash983089983094983097983092 983090983088983089983089

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 77

Submit your manuscripts at

httpwwwhindawicom

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 47

983092 Modelling and Simulation in Engineering

983137983138983148983141 983090 Maximum values o the absolute radial error ( = 221038389+3)

Number o algorithms = 3 = 5 = 6 = 8 = 10 = 12(983093) 07975 01973 0984 minus 1 0246 minus 1 0614 minus 2 0153 minus 2(983094) minus02642 minus0657 minus 1 minus0328 minus 1 minus0818 minus 2 minus0205 minus 2 minus0511 minus 3(983095) minus07932 minus01972 minus0984 minus 1 minus0246 minus 1 0614 minus 2 0153 minus 2(983089983089) 078 minus 3 012 minus 4 015 minus 5 023 minus 7 037 minus 9 057 minus 11

Substituting obtained values in (983090983089) we get

1038389+1 = 1103925+2 minus 104861610486161103925+2 + 11039251048617 2 cos 1048617 cos sin

minus1103925+1 = 1038389+2 minus 104861610486161038389+2 + 10383891048617 2 cos 1048617 cos sin

(983090983092)

which can easily be reduced to

1038389+1 = 1103925+2

minus 1103925

2 sin minus1103925+1 = 1038389+2

minus 1038389

2 sin (983090983093)

that is to (983089983096) Te algorithm works with initial conditions10383890 = 11039250 = 0 10383891 = cos and 11039251 = sin where = 21038389( is an integer) Approximating sin asymp one can transorm(983089983096) into another two-step algorithm

1038389+2 = 1038389 minus 2 sdot sdot 1103925+1 1103925+2 = 1103925 + 2 sdot sdot 1038389+1 (983090983094)

with 10383890 = 11039250 = 0 10383891 = 1057306 1 minus 2 11039251 = Tis algorithmwas 1047297rst proposed in [983097] see also [983089983088]

Functional diagram o interpolator which reproduces(983090983094) is shown in Figure 983091 Interpolator contains 983094 registers

(Register 983089 Register 983090 Register 983091 Register 983092 Outputmdash1038389+2and Outputmdash1103925+2) 983090 multipliers (Right Shif Register 983089 RightShif Register 983090) subtracter and adder

In this scheme at the beginning (beore the interpolatorstarts) values o 10383890 and 11039250 are recorded in Register 983090 andRegister 983092 respectively while values o 10383891 and11039251 are recordedin Register 983089 and Register 983091 Ten values o 1038389+1 and 1038389 arestored in Register 983089 and Register 983090 and values o 1103925+1 and 1103925are stored in Register 983091 and Register 983092 Right Shif Register983089 and Right Shif Register 983090 perorm multiplication o values1103925+1 and 1038389+1 by 2 Tus at outputs o subtractor and adderparallel binary codes are ormed in accordance with (983090983094)Rounded values arerecorded in Output1038389+2 and Output1103925+2

able 983091 contains maximum values o absolute radialsystematic error o proposed algorithm given by (983090983094) aferull circle simulation in the case

= 24 = 32 = 2minus1038389 = 21038389 = 3 10

= 21 (983090983095)

where is structure elements datapath (in bits) and isnumber o iterations able 983092 contains the same data or case = 221038389+3 Note that results given in ables 983089 and 983090 do notinclude round-off errors hence they do not depend on

y

x

10383891038389

xn

yn

xn+1

yn+1

xn+2

yn+2

xn yn

xn+1 yn+1

xn+2 yn+2

F983145983143983157983154983141 983090 Generation o 983091 subsequent points along the circle

Register 1

Register 1

Register 2

Register 2

Register 3

Register 4

Output Output

Right Shif Right Shif

Round Round

xn yn

xn+1 yn+1

xn+2 yn+2

++

++ minusminus

F983145983143983157983154983141 983091 Functional diagram or the circular interpolator (983090983094)

Analogical data or algorithm given by (983089983089) are given inable 983093 (we consider as beore case (983090983095)) able 983094 containsthe same data or = 221038389+3

Comparing errors in ables 983091 and 983093 and in ables 983092 and983094 we see that the absolute radial errors o our algorithm (983090983094)(ables 983091 and 983092) are smaller than errors o well-known algo-rithm (983089983089) (see ables 983093 and 983094) For example we can compare

values o errors at = 21038389 ( = 10) and = 24 given inables 983091 and 983093 Consequently we will get that an absolute

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 57

Modelling and Simulation in Engineering 983093

1314

151617181920212223

3 4 5 6 7 8 9 10

(m)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

Proposed (26)

Well-known (11)

F983145983143983157983154983141 983092 Number o correct bits o result as a unction o at =21038389 and = 24

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

9

11

13

15

17

19

21

23

C o r r e c

t b i t s

N u m

b e r o

f r e s u

l t s

F983145983143983157983154983141 983093 Number o correct bits o result as a unction o at =221038389+3 and = 24

983137983138983148983141 983091 Maximum values o the absolute radial error (

= 21038389) or

proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 7712 minus 8 minus2567 minus 7 4377 minus 10 minus9137 minus 10983092 1215 minus 7 minus6462 minus 7 1284 minus 9 minus2666 minus 9983093 2458 minus 7 minus1267 minus 6 9422 minus 10 minus5362 minus 9983094 4643 minus 7 minus2645 minus 6 2223 minus 9 minus8292 minus 9983095 7045 minus 7 minus4942 minus 6 3614 minus 9 minus1856 minus 8983096 1655 minus 6 minus9678 minus 6 6936 minus 9 minus3794 minus 8983097 3258 minus 6 minus1898 minus 5 1245 minus 8 minus7470 minus 8983089983088 6758 minus 6 minus3761 minus 5 2510 minus 8 minus1456 minus 7

radial error is smaller by (max(able 5))(max(able 3)) =(1267 minus 5)(6792 minus 6) = 1876 timesWe also see that the smaller is the greater is the

difference between the corresponding errors o both algo-rithms For example or ( = 3) and = 32 we obtainthat an absolute radial error is smaller by (max(able 6))(max(able 4)) = (7781minus4)(4533minus10) asymp 1716523 times

Finally Figures 983092 983093 983094 and 983095 present numbers o correct

bits as a unction o or = 21038389 and = 221038389+3 and or

= 24 and

= 32 respectively

983137983138983148983141 983092 Maximum values o the absolute radial error ( = 221038389+3)or proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 6965 minus 8 minus3866 minus 7 4533 minus 10 minus1671 minus 9983092 1768 minus 7 minus8276 minus 7 4654 minus 10 minus3550 minus 9983093 3010 minus 7 minus1176 minus 6 1260 minus 9 minus5494 minus 9983094 4245 minus 7 minus2292 minus 6 1923 minus 9 minus9138 minus 9983095 8948 minus 7 minus5040 minus 6 3707 minus 9 minus1927 minus 8983096 1454 minus 6 minus9834 minus 6 8918 minus 9 minus3704 minus 8983097 3424 minus 6 minus1936 minus 5 1307 minus 8 minus7389 minus 8983089983088 6792 minus 6 minus3648 minus 5 2792 minus 8 minus1437 minus 7

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

16

18

2022

24

26

28

30

C o r r

e c t b i t s

N u m

b e

r o

f r e s u

l t s

F983145983143983157983154983141 983094 Number o correct bits o result as a unction o at =21038389 and = 32

983137983138983148983141 983093 Maximum values o the absolute radial error ( = 21038389) orAlgorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min

983091 1214 minus 5 0 1215 minus 5 0983092 9641 minus 7 minus8277 minus 7 7525 minus 7 0983093 5496 minus 7 minus2386 minus 6 4552 minus 8 0983094 5508 minus 7 minus4660 minus 6 5781 minus 9 minus1682 minus 8983095 1235 minus 6 minus9674 minus 6 6069 minus 9 minus3579 minus 8983096 4009 minus 6 minus1858 minus 5 1524 minus 8 minus7158 minus 8983097 6478 minus 6 minus3651 minus 5 2034 minus 8 minus1484 minus 7983089983088

1268 minus 5 minus7285 minus 5 5564 minus 8 minus2853 minus 7Presented 1047297gures clearly show that the proposed algo-

rithm (983090983094) yields greater number o correct bits in compar-ison to algorithm (983089983089)

4 Conclusion

A new circle generation DDA algorithm has been proposedIn terms o hardware implementation and precision o circlegeneration the proposed DDA algorithm or circle interpola-toris more efficient as compared with existing algorithms Wehope that the proposed algorithm will 1047297nd some applications

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 67

983094 Modelling and Simulation in Engineering

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

9

14

19

24

29

F983145983143983157983154983141 983095 Number o correct bits o result as a unction o at =221038389+3 and = 32

983137983138983148983141 983094 Maximum values o the absolute radial error ( = 221038389+3)or Algorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min983091 7780 minus 4 0 7781 minus 4 0983092 9650 minus 5 0 9632 minus 5 0983093 1165 minus 5 0 1204 minus 5 0983094 2469 minus 6 minus3933 minus 6 1501 minus 6 0983095 1731 minus 6 minus9497 minus 6 1878 minus 7 0983096 2359 minus 6 minus1942 minus 5 3598 minus 8 minus6302 minus 8983097 5371 minus 6 minus3779 minus 5 2655 minus 8 minus1421 minus 7983089983088 1057 minus 5 minus7478 minus 5 5028 minus 8 minus2914 minus 7

in computer graphics (display screens) computer-controlledprinting automated control and so orth

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

Acknowledgment

Te second author (Jan L Cieslinski) is partly support-ed by the National Science Centre (NCN) Grant no 983090983088983089983089983088983089BS983089983088983093983089983091983095

References[983089] W Wang and C Wang ldquoDifference method or generation o

circular arcs and ellipsesrdquo Computer-Aided Design vol983090983089 no 983089pp 983091983091ndash983091983095 983089983097983096983097

[983090] N Matsushiro ldquoA new digital differential analyzer or circl gen-erationrdquo IEICE Transactions on Information and Systems volE983096983089-D no 983090 pp 983090983091983097ndash983090983092983090 983089983097983097983096

[983091] I Chami ldquoA high precision digital differential analyzer orcircular interpolationrdquo Tishreen University Journal vol 983090983097 no983089 983090983088983088983095

[983092] H Hama and Okumoto ldquoFast transorm into polar coordi-nate by generalized DDArdquo Journal of the Institute of TelevisionEngineers of Japan vol 983092983095 no 983089 pp 983097983095ndash983089983088983088 983089983097983097983091

[983093] X Y Fan Y H Guo and S C Li ldquoNew digital differential ana-lyzer interpolation algorithmrdquo Advanced Science Letters vol 983094no 983089 pp 983094983097983090ndash983094983097983093 983090983088983089983090

[983094] X L Yan W L Wang and N Xue ldquoResearch and simulation o an efficient circular interpolation algorithmrdquo Advanced Materi-als Research vol 983093983092983090-983093983092983091 pp 983089983090983088983092ndash983089983090983088983096 983090983088983089983090

[983095] O Masory and Y Koren ldquoReerence-word circular interpolatorsor CNC systemsrdquo Journal of Engineering for Industry vol 983089983088983092no 983092 pp 983092983088983088ndash983092983088983093 983089983097983096983090

[983096] Y Katayama H Harasaki and M Yano ldquoParallel algorithm o˜three dimensional digital differential analyzerrdquo in Proceedings of the Annual Convention ITE Japan pp 983089983096983093ndash983089983096983094 983089983097983097983091

[983097] L V Moroz ldquoA method or studying characteristics o digitalsine-cosine generatorsrdquo Information Extraction and Processing vol 983091983094 no 983089983089983090 pp 983096983092ndash983097983088 983090983088983089983090 (Ukrainian)

[983089983088] J L Cieslinski and L V Moroz ldquoFast exact digital differentialanalyzer or circle generationrdquo httparxivorgabs983089983091983088983092983092983097983095983092

[983089983089] W M Newman and R F Sproull Principles of InteractiveComputer Graphics McGraw-Hill 983089983097983095983097

[983089983090] F S Lim Y S Wong and M Rahman ldquoCircular interpolators

or numerical control a comparison o the modi1047297ed DDAtechniques and an LSI interpolatorrdquo Computers in Industry vol983089983096 no 983089 pp 983092983089ndash983093983090 983089983097983097983090

[983089983091] R F Eschenbach and B M Oliver ldquoAn efficient coordinate rota-tion algorithmrdquo IEEE Transactions on Computers vol 983090983095 no 983089983090pp 983089983089983095983096ndash983089983089983096983088 983089983097983095983096

[983089983092] H Hasegawa and Y Okada ldquoApparatus and method or con-trolling a moving vehicle utilizing a digital differential analysiscircuitrdquo US Patent 983093983091983095983097983091983093983091 983089983097983097983093

[983089983093] C Pokorny Computer Graphics An Object Oriented Approachtothe Artand Science FranklinBeedle amp Associates Publishing983089983097983097983092

[983089983094] J L Cieslinski and B Ratkiewicz ldquoOn simulations o the

classical harmonic oscillator equation by difference equationsrdquo Advances in Difference Equations vol 983090983088983088983094 Article ID 983092983088983089983095983089983090983088983088983094

[983089983095] J L Cieslinski ldquoOn the exact discretization o the classicalharmonic oscillator equationrdquo Journal of Difference Equationsand Applications vol 983089983095 no 983089983089 pp 983089983094983095983091ndash983089983094983097983092 983090983088983089983089

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 77

Submit your manuscripts at

httpwwwhindawicom

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 57

Modelling and Simulation in Engineering 983093

1314

151617181920212223

3 4 5 6 7 8 9 10

(m)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

Proposed (26)

Well-known (11)

F983145983143983157983154983141 983092 Number o correct bits o result as a unction o at =21038389 and = 24

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

9

11

13

15

17

19

21

23

C o r r e c

t b i t s

N u m

b e r o

f r e s u

l t s

F983145983143983157983154983141 983093 Number o correct bits o result as a unction o at =221038389+3 and = 24

983137983138983148983141 983091 Maximum values o the absolute radial error (

= 21038389) or

proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 7712 minus 8 minus2567 minus 7 4377 minus 10 minus9137 minus 10983092 1215 minus 7 minus6462 minus 7 1284 minus 9 minus2666 minus 9983093 2458 minus 7 minus1267 minus 6 9422 minus 10 minus5362 minus 9983094 4643 minus 7 minus2645 minus 6 2223 minus 9 minus8292 minus 9983095 7045 minus 7 minus4942 minus 6 3614 minus 9 minus1856 minus 8983096 1655 minus 6 minus9678 minus 6 6936 minus 9 minus3794 minus 8983097 3258 minus 6 minus1898 minus 5 1245 minus 8 minus7470 minus 8983089983088 6758 minus 6 minus3761 minus 5 2510 minus 8 minus1456 minus 7

radial error is smaller by (max(able 5))(max(able 3)) =(1267 minus 5)(6792 minus 6) = 1876 timesWe also see that the smaller is the greater is the

difference between the corresponding errors o both algo-rithms For example or ( = 3) and = 32 we obtainthat an absolute radial error is smaller by (max(able 6))(max(able 4)) = (7781minus4)(4533minus10) asymp 1716523 times

Finally Figures 983092 983093 983094 and 983095 present numbers o correct

bits as a unction o or = 21038389 and = 221038389+3 and or

= 24 and

= 32 respectively

983137983138983148983141 983092 Maximum values o the absolute radial error ( = 221038389+3)or proposed Algorithm (983090983094)

Absolute radial error

= 24 = 32Max Min Max Min

983091 6965 minus 8 minus3866 minus 7 4533 minus 10 minus1671 minus 9983092 1768 minus 7 minus8276 minus 7 4654 minus 10 minus3550 minus 9983093 3010 minus 7 minus1176 minus 6 1260 minus 9 minus5494 minus 9983094 4245 minus 7 minus2292 minus 6 1923 minus 9 minus9138 minus 9983095 8948 minus 7 minus5040 minus 6 3707 minus 9 minus1927 minus 8983096 1454 minus 6 minus9834 minus 6 8918 minus 9 minus3704 minus 8983097 3424 minus 6 minus1936 minus 5 1307 minus 8 minus7389 minus 8983089983088 6792 minus 6 minus3648 minus 5 2792 minus 8 minus1437 minus 7

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

16

18

2022

24

26

28

30

C o r r

e c t b i t s

N u m

b e

r o

f r e s u

l t s

F983145983143983157983154983141 983094 Number o correct bits o result as a unction o at =21038389 and = 32

983137983138983148983141 983093 Maximum values o the absolute radial error ( = 21038389) orAlgorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min

983091 1214 minus 5 0 1215 minus 5 0983092 9641 minus 7 minus8277 minus 7 7525 minus 7 0983093 5496 minus 7 minus2386 minus 6 4552 minus 8 0983094 5508 minus 7 minus4660 minus 6 5781 minus 9 minus1682 minus 8983095 1235 minus 6 minus9674 minus 6 6069 minus 9 minus3579 minus 8983096 4009 minus 6 minus1858 minus 5 1524 minus 8 minus7158 minus 8983097 6478 minus 6 minus3651 minus 5 2034 minus 8 minus1484 minus 7983089983088

1268 minus 5 minus7285 minus 5 5564 minus 8 minus2853 minus 7Presented 1047297gures clearly show that the proposed algo-

rithm (983090983094) yields greater number o correct bits in compar-ison to algorithm (983089983089)

4 Conclusion

A new circle generation DDA algorithm has been proposedIn terms o hardware implementation and precision o circlegeneration the proposed DDA algorithm or circle interpola-toris more efficient as compared with existing algorithms Wehope that the proposed algorithm will 1047297nd some applications

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 67

983094 Modelling and Simulation in Engineering

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

9

14

19

24

29

F983145983143983157983154983141 983095 Number o correct bits o result as a unction o at =221038389+3 and = 32

983137983138983148983141 983094 Maximum values o the absolute radial error ( = 221038389+3)or Algorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min983091 7780 minus 4 0 7781 minus 4 0983092 9650 minus 5 0 9632 minus 5 0983093 1165 minus 5 0 1204 minus 5 0983094 2469 minus 6 minus3933 minus 6 1501 minus 6 0983095 1731 minus 6 minus9497 minus 6 1878 minus 7 0983096 2359 minus 6 minus1942 minus 5 3598 minus 8 minus6302 minus 8983097 5371 minus 6 minus3779 minus 5 2655 minus 8 minus1421 minus 7983089983088 1057 minus 5 minus7478 minus 5 5028 minus 8 minus2914 minus 7

in computer graphics (display screens) computer-controlledprinting automated control and so orth

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

Acknowledgment

Te second author (Jan L Cieslinski) is partly support-ed by the National Science Centre (NCN) Grant no 983090983088983089983089983088983089BS983089983088983093983089983091983095

References[983089] W Wang and C Wang ldquoDifference method or generation o

circular arcs and ellipsesrdquo Computer-Aided Design vol983090983089 no 983089pp 983091983091ndash983091983095 983089983097983096983097

[983090] N Matsushiro ldquoA new digital differential analyzer or circl gen-erationrdquo IEICE Transactions on Information and Systems volE983096983089-D no 983090 pp 983090983091983097ndash983090983092983090 983089983097983097983096

[983091] I Chami ldquoA high precision digital differential analyzer orcircular interpolationrdquo Tishreen University Journal vol 983090983097 no983089 983090983088983088983095

[983092] H Hama and Okumoto ldquoFast transorm into polar coordi-nate by generalized DDArdquo Journal of the Institute of TelevisionEngineers of Japan vol 983092983095 no 983089 pp 983097983095ndash983089983088983088 983089983097983097983091

[983093] X Y Fan Y H Guo and S C Li ldquoNew digital differential ana-lyzer interpolation algorithmrdquo Advanced Science Letters vol 983094no 983089 pp 983094983097983090ndash983094983097983093 983090983088983089983090

[983094] X L Yan W L Wang and N Xue ldquoResearch and simulation o an efficient circular interpolation algorithmrdquo Advanced Materi-als Research vol 983093983092983090-983093983092983091 pp 983089983090983088983092ndash983089983090983088983096 983090983088983089983090

[983095] O Masory and Y Koren ldquoReerence-word circular interpolatorsor CNC systemsrdquo Journal of Engineering for Industry vol 983089983088983092no 983092 pp 983092983088983088ndash983092983088983093 983089983097983096983090

[983096] Y Katayama H Harasaki and M Yano ldquoParallel algorithm o˜three dimensional digital differential analyzerrdquo in Proceedings of the Annual Convention ITE Japan pp 983089983096983093ndash983089983096983094 983089983097983097983091

[983097] L V Moroz ldquoA method or studying characteristics o digitalsine-cosine generatorsrdquo Information Extraction and Processing vol 983091983094 no 983089983089983090 pp 983096983092ndash983097983088 983090983088983089983090 (Ukrainian)

[983089983088] J L Cieslinski and L V Moroz ldquoFast exact digital differentialanalyzer or circle generationrdquo httparxivorgabs983089983091983088983092983092983097983095983092

[983089983089] W M Newman and R F Sproull Principles of InteractiveComputer Graphics McGraw-Hill 983089983097983095983097

[983089983090] F S Lim Y S Wong and M Rahman ldquoCircular interpolators

or numerical control a comparison o the modi1047297ed DDAtechniques and an LSI interpolatorrdquo Computers in Industry vol983089983096 no 983089 pp 983092983089ndash983093983090 983089983097983097983090

[983089983091] R F Eschenbach and B M Oliver ldquoAn efficient coordinate rota-tion algorithmrdquo IEEE Transactions on Computers vol 983090983095 no 983089983090pp 983089983089983095983096ndash983089983089983096983088 983089983097983095983096

[983089983092] H Hasegawa and Y Okada ldquoApparatus and method or con-trolling a moving vehicle utilizing a digital differential analysiscircuitrdquo US Patent 983093983091983095983097983091983093983091 983089983097983097983093

[983089983093] C Pokorny Computer Graphics An Object Oriented Approachtothe Artand Science FranklinBeedle amp Associates Publishing983089983097983097983092

[983089983094] J L Cieslinski and B Ratkiewicz ldquoOn simulations o the

classical harmonic oscillator equation by difference equationsrdquo Advances in Difference Equations vol 983090983088983088983094 Article ID 983092983088983089983095983089983090983088983088983094

[983089983095] J L Cieslinski ldquoOn the exact discretization o the classicalharmonic oscillator equationrdquo Journal of Difference Equationsand Applications vol 983089983095 no 983089983089 pp 983089983094983095983091ndash983089983094983097983092 983090983088983089983089

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 77

Submit your manuscripts at

httpwwwhindawicom

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 67

983094 Modelling and Simulation in Engineering

3 4 5 6 7 8 9 10

(m)

Proposed (26)

Well-known (11)

C o r r e c t b i t s

N u m

b e r o

f r e s u

l t s

9

14

19

24

29

F983145983143983157983154983141 983095 Number o correct bits o result as a unction o at =221038389+3 and = 32

983137983138983148983141 983094 Maximum values o the absolute radial error ( = 221038389+3)or Algorithm (983089983089)

Absolute radial error

= 24 = 32Max Min Max Min983091 7780 minus 4 0 7781 minus 4 0983092 9650 minus 5 0 9632 minus 5 0983093 1165 minus 5 0 1204 minus 5 0983094 2469 minus 6 minus3933 minus 6 1501 minus 6 0983095 1731 minus 6 minus9497 minus 6 1878 minus 7 0983096 2359 minus 6 minus1942 minus 5 3598 minus 8 minus6302 minus 8983097 5371 minus 6 minus3779 minus 5 2655 minus 8 minus1421 minus 7983089983088 1057 minus 5 minus7478 minus 5 5028 minus 8 minus2914 minus 7

in computer graphics (display screens) computer-controlledprinting automated control and so orth

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

Acknowledgment

Te second author (Jan L Cieslinski) is partly support-ed by the National Science Centre (NCN) Grant no 983090983088983089983089983088983089BS983089983088983093983089983091983095

References[983089] W Wang and C Wang ldquoDifference method or generation o

circular arcs and ellipsesrdquo Computer-Aided Design vol983090983089 no 983089pp 983091983091ndash983091983095 983089983097983096983097

[983090] N Matsushiro ldquoA new digital differential analyzer or circl gen-erationrdquo IEICE Transactions on Information and Systems volE983096983089-D no 983090 pp 983090983091983097ndash983090983092983090 983089983097983097983096

[983091] I Chami ldquoA high precision digital differential analyzer orcircular interpolationrdquo Tishreen University Journal vol 983090983097 no983089 983090983088983088983095

[983092] H Hama and Okumoto ldquoFast transorm into polar coordi-nate by generalized DDArdquo Journal of the Institute of TelevisionEngineers of Japan vol 983092983095 no 983089 pp 983097983095ndash983089983088983088 983089983097983097983091

[983093] X Y Fan Y H Guo and S C Li ldquoNew digital differential ana-lyzer interpolation algorithmrdquo Advanced Science Letters vol 983094no 983089 pp 983094983097983090ndash983094983097983093 983090983088983089983090

[983094] X L Yan W L Wang and N Xue ldquoResearch and simulation o an efficient circular interpolation algorithmrdquo Advanced Materi-als Research vol 983093983092983090-983093983092983091 pp 983089983090983088983092ndash983089983090983088983096 983090983088983089983090

[983095] O Masory and Y Koren ldquoReerence-word circular interpolatorsor CNC systemsrdquo Journal of Engineering for Industry vol 983089983088983092no 983092 pp 983092983088983088ndash983092983088983093 983089983097983096983090

[983096] Y Katayama H Harasaki and M Yano ldquoParallel algorithm o˜three dimensional digital differential analyzerrdquo in Proceedings of the Annual Convention ITE Japan pp 983089983096983093ndash983089983096983094 983089983097983097983091

[983097] L V Moroz ldquoA method or studying characteristics o digitalsine-cosine generatorsrdquo Information Extraction and Processing vol 983091983094 no 983089983089983090 pp 983096983092ndash983097983088 983090983088983089983090 (Ukrainian)

[983089983088] J L Cieslinski and L V Moroz ldquoFast exact digital differentialanalyzer or circle generationrdquo httparxivorgabs983089983091983088983092983092983097983095983092

[983089983089] W M Newman and R F Sproull Principles of InteractiveComputer Graphics McGraw-Hill 983089983097983095983097

[983089983090] F S Lim Y S Wong and M Rahman ldquoCircular interpolators

or numerical control a comparison o the modi1047297ed DDAtechniques and an LSI interpolatorrdquo Computers in Industry vol983089983096 no 983089 pp 983092983089ndash983093983090 983089983097983097983090

[983089983091] R F Eschenbach and B M Oliver ldquoAn efficient coordinate rota-tion algorithmrdquo IEEE Transactions on Computers vol 983090983095 no 983089983090pp 983089983089983095983096ndash983089983089983096983088 983089983097983095983096

[983089983092] H Hasegawa and Y Okada ldquoApparatus and method or con-trolling a moving vehicle utilizing a digital differential analysiscircuitrdquo US Patent 983093983091983095983097983091983093983091 983089983097983097983093

[983089983093] C Pokorny Computer Graphics An Object Oriented Approachtothe Artand Science FranklinBeedle amp Associates Publishing983089983097983097983092

[983089983094] J L Cieslinski and B Ratkiewicz ldquoOn simulations o the

classical harmonic oscillator equation by difference equationsrdquo Advances in Difference Equations vol 983090983088983088983094 Article ID 983092983088983089983095983089983090983088983088983094

[983089983095] J L Cieslinski ldquoOn the exact discretization o the classicalharmonic oscillator equationrdquo Journal of Difference Equationsand Applications vol 983089983095 no 983089983089 pp 983089983094983095983091ndash983089983094983097983092 983090983088983089983089

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 77

Submit your manuscripts at

httpwwwhindawicom

8202019 916539pdf

httpslidepdfcomreaderfull916539pdf 77

Submit your manuscripts at

httpwwwhindawicom