(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: cg96.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Roman Courier Times-Bold Times-Italic Helvetica %%+ Symbol %%EndComments %DVIPSCommandLine: dvips -D600 -o cg96.ps cg96.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1996.02.15:1246 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict 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Fw(r)o(etur)o(n)19 b Fy(\()p Fz(x;)14 b(y)s Fy(\))2132 5197 y FG(The)28 b(follo)n(wing)f(algorithm,)h(which)g (multiplies)g(an)h(e)o(xpansion)2049 5290 y(by)20 b(a)g (\257oating-point)e(v)n(alue,)h(is)j(ne)n(w)-5 b(.)p eop %%Page: 5 5 5 4 bop -150 83 a Fw(Theor)o(em)20 b(6)41 b Fv(Let)21 b Fz(e)h Fy(=)551 21 y Fh(P)639 41 y Ft(m)639 108 y(i)p Fg(=)p FA(1)760 83 y Fz(e)799 95 y Ft(i)847 83 y Fv(be)e(an)f(or)m(der) m(ed)g(nono)o(verlapping)-150 176 y(e)n(xpansion)i(of)i Fz(m)g(p)p Fv(-bit)f(components,)f(and)g(let)i Fz(b)g Fv(be)f(a)h Fz(p)p Fv(-bit)f(value)-150 268 y(wher)m(e)k Fz(p)k FC(\025)g FG(4)p Fv(.)45 b(Then)25 b(the)h(following)f (algorithm)f(will)j(pr)l(oduce)d(a)-150 375 y(nono)o(verlapping)e(e)n (xpansion)j Fz(h)h Fv(suc)o(h)g(that)f Fz(h)31 b Fy(=)1345 312 y Fh(P)1432 333 y FA(2)p Ft(m)1432 399 y(i)p Fg(=)p FA(1)1554 375 y Fz(h)1602 387 y Ft(i)1660 375 y Fy(=)g Fz(be)p Fv(,)-150 467 y(wher)m(e)f Fz(h)g Fv(is)h(also)e(or)m(der)m (ed,)i(e)n(xcept)f(that)f(any)g(of)h(the)f 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5346.19 12124.43 l 5356.92 12047.92 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 4953 13053 m 3153 12003 l gs col0 s gr n 3288.61 12151.57 m 3153.00 12003.00 l 3349.08 12047.92 l 3360.81 12124.43 l 3288.61 12151.57 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 4953 13053 m 4053 12003 l gs col0 s gr n 4132.40 12187.82 m 4053.00 12003.00 l 4223.51 12109.73 l 4209.69 12185.72 l 4132.40 12187.82 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 7954 13054 m 7654 12004 l gs col0 s gr n 7649.06 12205.10 m 7654.00 12004.00 l 7764.44 12172.13 l 7720.43 12235.27 l 7649.06 12205.10 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 7954 13054 m 9454 12004 l gs col0 s gr n 9262.30 12064.95 m 9454.00 12004.00 l 9331.12 12163.26 l 9257.88 12142.13 l 9262.30 12064.95 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10354 13054 m 10654 12004 l gs col0 s gr n 10543.56 12172.13 m 10654.00 12004.00 l 10658.94 12205.10 l 10588.57 12235.27 l 10543.56 12172.13 l clp gs 0.00 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[133.3] 0 setdash n 7204 10804 m 11104 10804 l 11104 10204 l 7204 10204 l clp gs col0 s gr [] 0 setdash 15.000 slw % Polyline n 7807 15307 m 8107 14707 l gs col0 s gr n 7967.47 14851.90 m 8107.00 14707.00 l 8074.80 14905.56 l 8000.17 14922.16 l 7967.47 14851.90 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8707 15307 m 8407 14707 l gs col0 s gr n 8439.20 14905.56 m 8407.00 14707.00 l 8546.53 14851.90 l 8514.83 14922.16 l 8439.20 14905.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8106 14106 m 7956 13356 l gs col0 s gr n 7934.82 13556.04 m 7956.00 13356.00 l 8052.49 13532.50 l 8003.57 13591.84 l 7934.82 13556.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8406 14106 m 8556 13356 l gs col0 s gr n 8459.51 13532.50 m 8556.00 13356.00 l 8577.18 13556.04 l 8509.43 13591.84 l 8459.51 13532.50 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9607 15307 m 9907 14707 l gs col0 s gr n 9767.47 14851.90 m 9907.00 14707.00 l 9874.80 14905.56 l 9800.17 14922.16 l 9767.47 14851.90 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10507 15307 m 10207 14707 l gs col0 s gr n 10239.20 14905.56 m 10207.00 14707.00 l 10346.53 14851.90 l 10314.83 14922.16 l 10239.20 14905.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9906 14106 m 9756 13356 l gs col0 s gr n 9734.82 13556.04 m 9756.00 13356.00 l 9852.49 13532.50 l 9803.57 13591.84 l 9734.82 13556.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10206 14106 m 10356 13356 l gs col0 s gr n 10259.51 13532.50 m 10356.00 13356.00 l 10377.18 13556.04 l 10309.43 13591.84 l 10259.51 13532.50 l clp gs 0.00 setgray ef gr gs col0 s gr 30.000 slw % Polyline [133.3] 0 setdash n 7504 13504 m 9004 13504 l 9004 12904 l 7504 12904 l clp gs col0 s gr [] 0 setdash % Polyline [133.3] 0 setdash n 9304 13504 m 10804 13504 l 10804 12904 l 9304 12904 l clp gs col0 s gr [] 0 setdash 15.000 slw % Polyline n 3000 10350 m 4350 9000 l gs col0 s gr n 4171.81 9093.34 m 4350.00 9000.00 l 4256.66 9178.19 l 4180.79 9170.21 l 4171.81 9093.34 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 7800 10350 m 4650 9000 l gs col0 s gr n 4802.84 9130.78 m 4650.00 9000.00 l 4850.11 9020.48 l 4871.09 9095.04 l 4802.84 9130.78 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 3900 10350 m 5850 9000 l gs col0 s gr n 5657.99 9059.96 m 5850.00 9000.00 l 5726.29 9158.62 l 5653.17 9137.11 l 5657.99 9059.96 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8700 10350 m 6150 9000 l gs col0 s gr n 6291.61 9142.86 m 6150.00 9000.00 l 6347.76 9036.81 l 6362.61 9112.79 l 6291.61 9142.86 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 4800 10350 m 7350 9000 l gs col0 s gr n 7152.24 9036.81 m 7350.00 9000.00 l 7208.39 9142.86 l 7138.39 9112.79 l 7152.24 9036.81 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9600 10350 m 7650 9000 l gs col0 s gr n 7773.71 9158.62 m 7650.00 9000.00 l 7842.01 9059.96 l 7847.83 9137.11 l 7773.71 9158.62 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 5700 10350 m 8850 9000 l gs col0 s gr n 8649.89 9020.48 m 8850.00 9000.00 l 8697.16 9130.78 l 8629.91 9095.04 l 8649.89 9020.48 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10500 10350 m 9150 9000 l gs col0 s gr n 9243.34 9178.19 m 9150.00 9000.00 l 9328.19 9093.34 l 9320.21 9170.21 l 9243.34 9178.19 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 6000 7350 m 5400 6600 l gs col0 s gr n 5473.09 6787.41 m 5400.00 6600.00 l 5566.79 6712.45 l 5550.43 6787.91 l 5473.09 6787.41 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 4500 7350 m 5100 6600 l gs col0 s gr n 4933.21 6712.45 m 5100.00 6600.00 l 5026.91 6787.41 l 4950.57 6787.91 l 4933.21 6712.45 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 7500 7350 m 8100 6600 l gs col0 s gr n 7933.21 6712.45 m 8100.00 6600.00 l 8026.91 6787.41 l 7950.57 6787.91 l 7933.21 6712.45 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9000 7350 m 8400 6600 l gs col0 s gr n 8473.09 6787.41 m 8400.00 6600.00 l 8566.79 6712.45 l 8550.43 6787.91 l 8473.09 6787.41 l clp gs 0.00 setgray ef gr gs col0 s gr 30.000 slw % Polyline [133.3] 0 setdash n 2403 10803 m 6303 10803 l 6303 10203 l 2403 10203 l clp gs col0 s gr [] 0 setdash 15.000 slw % Polyline n 750 9975 m 2550 9975 l gs col0 s gr % Polyline n 3150 9150 m 3150 9000 l gs col0 s gr n 3090.00 9192.00 m 3150.00 9000.00 l 3210.00 9192.00 l 3150.50 9240.50 l 3090.00 9192.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2850 9225 m 2850 9000 l gs col0 s gr n 2790.00 9192.00 m 2850.00 9000.00 l 2910.00 9192.00 l 2850.50 9240.50 l 2790.00 9192.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1950 9375 m 1950 9000 l gs col0 s gr n 1890.00 9192.00 m 1950.00 9000.00 l 2010.00 9192.00 l 1950.50 9240.50 l 1890.00 9192.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1650 9450 m 1650 9000 l gs col0 s gr n 1590.00 9192.00 m 1650.00 9000.00 l 1710.00 9192.00 l 1650.50 9240.50 l 1590.00 9192.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 750 9600 m 750 9000 l gs col0 s gr n 690.00 9192.00 m 750.00 9000.00 l 810.00 9192.00 l 750.50 9240.50 l 690.00 9192.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 450 9675 m 450 9000 l gs col0 s gr n 390.00 9192.00 m 450.00 9000.00 l 510.00 9192.00 l 450.50 9240.50 l 390.00 9192.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1050 9900 m 7350 9900 l gs col0 s gr % Polyline n 1950 9750 m 3450 9750 l gs col0 s gr % Polyline n 2250 9675 m 8250 9675 l gs col0 s gr % Polyline n 3150 9525 m 4350 9525 l gs col0 s gr % Polyline n 3450 9450 m 9150 9450 l gs col0 s gr % Polyline n 2850 10275 m 2850 10500 l gs col0 s gr % Polyline n 3750 10050 m 3750 10500 l gs col0 s gr % Polyline n 4650 9825 m 4650 10500 l gs col0 s gr % Polyline n 7650 10200 m 7650 10500 l gs col0 s gr % Polyline n 8550 9975 m 8550 10500 l gs col0 s gr % Polyline n 9450 9750 m 9450 10500 l gs col0 s gr % Polyline n 1650 7350 m 2100 6900 l gs col0 s gr n 1921.81 6993.34 m 2100.00 6900.00 l 2006.66 7078.19 l 1930.79 7070.21 l 1921.81 6993.34 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2850 7350 m 2400 6900 l gs col0 s gr n 2493.34 7078.19 m 2400.00 6900.00 l 2578.19 6993.34 l 2570.21 7070.21 l 2493.34 7078.19 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 6753 3603 m 6753 2853 l gs col0 s gr n 6693.00 3045.00 m 6753.00 2853.00 l 6813.00 3045.00 l 6753.50 3093.50 l 6693.00 3045.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2100 5250 m 2400 4800 l gs col0 s gr n 2243.57 4926.47 m 2400.00 4800.00 l 2343.42 4993.04 l 2267.37 5000.19 l 2243.57 4926.47 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 5250 4950 m 6600 4200 l gs col0 s gr n 6403.02 4240.79 m 6600.00 4200.00 l 6461.30 4345.69 l 6390.70 4317.05 l 6403.02 4240.79 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8250 4950 m 6900 4200 l gs col0 s gr n 7038.70 4345.69 m 6900.00 4200.00 l 7096.98 4240.79 l 7110.30 4317.05 l 7038.70 4345.69 l clp gs 0.00 setgray ef gr gs col0 s gr 30.000 slw % Polyline 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9453 m 3903 9453 l gs col0 s gr 60.000 slw % Polyline n 9307 18607 m 9307 18757 l 9607 18757 l 9607 18607 l gs col0 s gr % Polyline n 8707 18607 m 8707 18757 l 9007 18757 l 9007 18607 l gs col0 s gr % Polyline n 8407 18607 m 8407 18757 l 8707 18757 l 8707 18607 l gs col0 s gr % Ellipse n 10512 19812 300 300 0 360 DrawEllipse gs col0 s gr % Polyline n 10512 19811 m 10725 19599 l gs col0 s gr % Polyline n 10512 19811 m 10725 20023 l gs col0 s gr % Polyline n 10511 19812 m 10298 19600 l gs col0 s gr % Polyline n 10511 19812 m 10298 20024 l gs col0 s gr % Ellipse n 9612 19812 300 300 0 360 DrawEllipse gs col0 s gr % Polyline n 9612 19811 m 9825 19599 l gs col0 s gr % Polyline n 9612 19811 m 9825 20023 l gs col0 s gr % Polyline n 9611 19812 m 9398 19600 l gs col0 s gr % Polyline n 9611 19812 m 9398 20024 l gs col0 s gr % Ellipse n 8712 19812 300 300 0 360 DrawEllipse gs col0 s gr % Polyline n 8712 19811 m 8925 19599 l gs col0 s gr % Polyline n 8712 19811 m 8925 20023 l gs col0 s gr % Polyline n 8711 19812 m 8498 19600 l gs col0 s gr % Polyline n 8711 19812 m 8498 20024 l gs col0 s gr % Ellipse n 7812 19812 300 300 0 360 DrawEllipse gs col0 s gr % Polyline n 7812 19811 m 8025 19599 l gs col0 s gr % Polyline n 7812 19811 m 8025 20023 l gs col0 s gr % Polyline n 7811 19812 m 7598 19600 l gs col0 s gr % Polyline n 7811 19812 m 7598 20024 l gs col0 s gr % Polyline n 7807 18607 m 7807 18757 l 8107 18757 l 8107 18607 l gs col0 s gr 15.000 slw % Polyline n 5251 7501 m 7276 7501 l gs col0 s gr % Polyline n 5401 7351 m 5401 7651 l gs col0 s gr 60.000 slw % Polyline n 12157 5857 m 12157 6457 l gs col0 s gr % Polyline n 11857 6457 m 12457 6457 l 12457 5857 l 11857 5857 l clp gs col0 s gr % Polyline n 11857 6157 m 12457 6157 l gs col0 s gr 15.000 slw % Polyline [100.0] 0 setdash n 6002 10652 m 6002 10802 l gs col0 s gr [] 0 setdash % Polyline [100.0] 0 setdash n 6302 10652 m 6302 10802 l gs col0 s gr [] 0 setdash % Polyline [100.0] 0 setdash n 6602 10652 m 6602 10802 l gs col0 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gs col0 s gr [] 0 setdash 15.000 slw % Polyline n 12610 23410 m 12910 22810 l gs col0 s gr n 12770.47 22954.90 m 12910.00 22810.00 l 12877.80 23008.56 l 12803.17 23025.16 l 12770.47 22954.90 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13510 23410 m 13210 22810 l gs col0 s gr n 13242.20 23008.56 m 13210.00 22810.00 l 13349.53 22954.90 l 13317.83 23025.16 l 13242.20 23008.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 12909 22209 m 12759 21459 l gs col0 s gr n 12737.82 21659.04 m 12759.00 21459.00 l 12855.49 21635.50 l 12806.57 21694.84 l 12737.82 21659.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13209 22209 m 13359 21459 l gs col0 s gr n 13262.51 21635.50 m 13359.00 21459.00 l 13380.18 21659.04 l 13312.43 21694.84 l 13262.51 21635.50 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14410 23410 m 14710 22810 l gs col0 s gr n 14570.47 22954.90 m 14710.00 22810.00 l 14677.80 23008.56 l 14603.17 23025.16 l 14570.47 22954.90 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 15310 23410 m 15010 22810 l gs col0 s gr n 15042.20 23008.56 m 15010.00 22810.00 l 15149.53 22954.90 l 15117.83 23025.16 l 15042.20 23008.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14709 22209 m 14559 21459 l gs col0 s gr n 14537.82 21659.04 m 14559.00 21459.00 l 14655.49 21635.50 l 14606.57 21694.84 l 14537.82 21659.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 15009 22209 m 15159 21459 l gs col0 s gr n 15062.51 21635.50 m 15159.00 21459.00 l 15180.18 21659.04 l 15112.43 21694.84 l 15062.51 21635.50 l clp gs 0.00 setgray ef gr gs col0 s gr 30.000 slw % Polyline [133.3] 0 setdash n 12307 21607 m 13807 21607 l 13807 21007 l 12307 21007 l clp gs col0 s gr [] 0 setdash % Polyline [133.3] 0 setdash n 14107 21607 m 15607 21607 l 15607 21007 l 14107 21007 l clp gs col0 s gr [] 0 setdash % Polyline [133.3] 0 setdash n 7206 18906 m 11106 18906 l 11106 18306 l 7206 18306 l clp gs col0 s gr [] 0 setdash 15.000 slw % Polyline n 7807 19507 m 7802 18752 l gs col0 s gr n 7743.27 18944.39 m 7802.00 18752.00 l 7863.27 18943.60 l 7804.09 18992.49 l 7743.27 18944.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8707 19507 m 8702 18752 l gs col0 s gr n 8643.27 18944.39 m 8702.00 18752.00 l 8763.27 18943.60 l 8704.09 18992.49 l 8643.27 18944.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9607 19507 m 9602 18752 l gs col0 s gr n 9543.27 18944.39 m 9602.00 18752.00 l 9663.27 18943.60 l 9604.09 18992.49 l 9543.27 18944.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10507 19507 m 10502 18752 l gs col0 s gr n 10443.27 18944.39 m 10502.00 18752.00 l 10563.27 18943.60 l 10504.09 18992.49 l 10443.27 18944.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 12608 19508 m 12602 18752 l gs col0 s gr n 12543.53 18944.47 m 12602.00 18752.00 l 12663.52 18943.52 l 12604.40 18992.49 l 12543.53 18944.47 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13508 19508 m 13502 18752 l gs col0 s gr n 13443.53 18944.47 m 13502.00 18752.00 l 13563.52 18943.52 l 13504.40 18992.49 l 13443.53 18944.47 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14408 19508 m 14402 18752 l gs col0 s gr n 14343.53 18944.47 m 14402.00 18752.00 l 14463.52 18943.52 l 14404.40 18992.49 l 14343.53 18944.47 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 15308 19508 m 15302 18752 l gs col0 s gr n 15243.53 18944.47 m 15302.00 18752.00 l 15363.52 18943.52 l 15304.40 18992.49 l 15243.53 18944.47 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 10208 21308 m 10208 21458 l 10508 21458 l 10508 21308 l gs col0 s gr % Polyline n 9608 21308 m 9608 21458 l 9908 21458 l 9908 21308 l gs col0 s gr % Polyline n 12609 21309 m 12609 21459 l 12909 21459 l 12909 21309 l gs col0 s gr % Polyline n 13209 21309 m 13209 21459 l 13509 21459 l 13509 21309 l gs col0 s gr % Polyline n 14409 21309 m 14409 21459 l 14709 21459 l 14709 21309 l gs col0 s gr % Polyline n 15009 21309 m 15009 21459 l 15309 21459 l 15309 21309 l gs col0 s gr 15.000 slw % Polyline n 8702 18452 m 10052 17702 l gs col0 s gr n 9855.02 17742.79 m 10052.00 17702.00 l 9913.30 17847.69 l 9842.70 17819.05 l 9855.02 17742.79 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9602 18452 m 10202 17702 l gs col0 s gr n 10035.21 17814.45 m 10202.00 17702.00 l 10128.91 17889.41 l 10052.57 17889.91 l 10035.21 17814.45 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10502 18452 m 10352 17702 l gs col0 s gr n 10330.82 17902.04 m 10352.00 17702.00 l 10448.49 17878.50 l 10399.57 17937.84 l 10330.82 17902.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13502 18452 m 14852 17702 l gs col0 s gr n 14655.02 17742.79 m 14852.00 17702.00 l 14713.30 17847.69 l 14642.70 17819.05 l 14655.02 17742.79 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14402 18452 m 15002 17702 l gs col0 s gr n 14835.21 17814.45 m 15002.00 17702.00 l 14928.91 17889.41 l 14852.57 17889.91 l 14835.21 17814.45 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 15302 18452 m 15152 17702 l gs col0 s gr n 15130.82 17902.04 m 15152.00 17702.00 l 15248.49 17878.50 l 15199.57 17937.84 l 15130.82 17902.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10202 17097 m 10202 16652 l gs col0 s gr n 10142.00 16844.00 m 10202.00 16652.00 l 10262.00 16844.00 l 10202.50 16892.50 l 10142.00 16844.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 15002 17097 m 15002 16652 l gs col0 s gr n 14942.00 16844.00 m 15002.00 16652.00 l 15062.00 16844.00 l 15002.50 16892.50 l 14942.00 16844.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8552 18602 m 8852 17702 l gs col0 s gr n 8734.36 17865.17 m 8852.00 17702.00 l 8848.21 17903.12 l 8776.61 17930.18 l 8734.36 17865.17 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9452 18602 m 9150 17700 l gs col0 s gr n 9154.06 17901.12 m 9150.00 17700.00 l 9267.85 17863.02 l 9226.70 17928.08 l 9154.06 17901.12 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13352 18602 m 13652 17702 l gs col0 s gr n 13534.36 17865.17 m 13652.00 17702.00 l 13648.21 17903.12 l 13576.61 17930.18 l 13534.36 17865.17 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14252 18602 m 13950 17700 l gs col0 s gr n 13954.06 17901.12 m 13950.00 17700.00 l 14067.85 17863.02 l 14026.70 17928.08 l 13954.06 17901.12 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 9002 16502 m 9002 16652 l 8702 16652 l 8702 16502 l gs col0 s gr % Polyline n 13802 16502 m 13802 16652 l 13502 16652 l 13502 16502 l gs col0 s gr 15.000 slw % Polyline n 9002 17102 m 8852 16652 l gs col0 s gr n 8855.79 16853.12 m 8852.00 16652.00 l 8969.64 16815.17 l 8928.39 16880.18 l 8855.79 16853.12 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13802 17102 m 13652 16652 l gs col0 s gr n 13655.79 16853.12 m 13652.00 16652.00 l 13769.64 16815.17 l 13728.39 16880.18 l 13655.79 16853.12 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 12454 18304 m 12454 18604 l gs col0 s gr % Polyline n 8850 16425 m 8852 15602 l gs col0 s gr n 8791.53 15793.85 m 8852.00 15602.00 l 8911.53 15794.15 l 8851.92 15842.50 l 8791.53 15793.85 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9152 15827 m 9152 15602 l gs col0 s gr n 9092.00 15794.00 m 9152.00 15602.00 l 9212.00 15794.00 l 9152.50 15842.50 l 9092.00 15794.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9002 15002 m 8852 14252 l gs col0 s gr n 8830.82 14452.04 m 8852.00 14252.00 l 8948.49 14428.50 l 8899.57 14487.84 l 8830.82 14452.04 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 9002 14102 m 9002 14252 l 8702 14252 l 8702 14102 l gs col0 s gr 15.000 slw % Polyline n 7806 15006 m 7656 14256 l gs col0 s gr n 7634.82 14456.04 m 7656.00 14256.00 l 7752.49 14432.50 l 7703.57 14491.84 l 7634.82 14456.04 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 7806 14107 m 7806 14257 l 7506 14257 l 7506 14107 l gs col0 s gr 15.000 slw % Polyline n 7651 18601 m 7655 15605 l gs col0 s gr n 7594.74 15796.92 m 7655.00 15605.00 l 7714.74 15797.08 l 7655.18 15845.50 l 7594.74 15796.92 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 15001 16351 m 13351 15601 l gs col0 s gr n 13500.96 15735.07 m 13351.00 15601.00 l 13550.62 15625.83 l 13569.99 15700.81 l 13500.96 15735.07 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10201 16351 m 13051 15601 l gs col0 s gr n 12850.05 15591.84 m 13051.00 15601.00 l 12880.59 15707.89 l 12819.40 15662.58 l 12850.05 15591.84 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10505 15008 m 10500 14253 l gs col0 s gr n 10441.27 14445.39 m 10500.00 14253.00 l 10561.27 14444.60 l 10502.09 14493.49 l 10441.27 14445.39 l clp gs 0.00 setgray ef gr gs col0 s gr 30.000 slw % Polyline [100.0] 0 setdash n 9602 14402 m 15302 14402 l 15302 13802 l 9602 13802 l clp gs col0 s gr [] 0 setdash 15.000 slw % Polyline n 13206 15009 m 13201 14254 l gs col0 s gr n 13142.27 14446.39 m 13201.00 14254.00 l 13262.27 14445.60 l 13203.09 14494.49 l 13142.27 14446.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 7808 23408 m 8108 22808 l gs col0 s gr n 7968.47 22952.90 m 8108.00 22808.00 l 8075.80 23006.56 l 8001.17 23023.16 l 7968.47 22952.90 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8708 23408 m 8408 22808 l gs col0 s gr n 8440.20 23006.56 m 8408.00 22808.00 l 8547.53 22952.90 l 8515.83 23023.16 l 8440.20 23006.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8107 22207 m 7957 21457 l gs col0 s gr n 7935.82 21657.04 m 7957.00 21457.00 l 8053.49 21633.50 l 8004.57 21692.84 l 7935.82 21657.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8407 22207 m 8557 21457 l gs col0 s gr n 8460.51 21633.50 m 8557.00 21457.00 l 8578.18 21657.04 l 8510.43 21692.84 l 8460.51 21633.50 l clp gs 0.00 setgray ef gr gs col0 s gr 30.000 slw % Polyline [133.3] 0 setdash n 7506 21606 m 9006 21606 l 9006 21006 l 7506 21006 l clp gs col0 s gr [] 0 setdash 60.000 slw % Polyline n 8407 21307 m 8407 21457 l 8707 21457 l 8707 21307 l gs col0 s gr % Polyline n 7807 21307 m 7807 21457 l 8107 21457 l 8107 21307 l gs col0 s gr 15.000 slw % Polyline n 1802 12002 m 1652 11252 l gs col0 s gr n 1630.82 11452.04 m 1652.00 11252.00 l 1748.49 11428.50 l 1699.57 11487.84 l 1630.82 11452.04 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 1802 11102 m 1802 11252 l 1502 11252 l 1502 11102 l gs col0 s gr 15.000 slw % Polyline n 3001 12001 m 2851 11251 l gs col0 s gr n 2829.82 11451.04 m 2851.00 11251.00 l 2947.49 11427.50 l 2898.57 11486.84 l 2829.82 11451.04 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 3001 11101 m 3001 11251 l 2701 11251 l 2701 11101 l gs col0 s gr 15.000 slw % Polyline n 4202 12002 m 4052 11252 l gs col0 s gr n 4030.82 11452.04 m 4052.00 11252.00 l 4148.49 11428.50 l 4099.57 11487.84 l 4030.82 11452.04 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 4202 11102 m 4202 11252 l 3902 11252 l 3902 11102 l gs col0 s gr 15.000 slw % Polyline n 1951 13576 m 5550 13575 l gs col0 s gr % Polyline n 10500 13950 m 6750 12600 l gs col0 s gr n 6910.33 12721.49 m 6750.00 12600.00 l 6950.97 12608.58 l 6976.31 12681.79 l 6910.33 12721.49 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 10500 13950 m 9450 12600 l gs col0 s gr n 9520.52 12788.39 m 9450.00 12600.00 l 9615.24 12714.72 l 9597.85 12789.94 l 9520.52 12788.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 5850 13950 m 11850 12600 l gs col0 s gr n 11649.51 12583.61 m 11850.00 12600.00 l 11675.85 12700.68 l 11616.35 12653.18 l 11649.51 12583.61 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 6450 13950 m 14550 12600 l gs col0 s gr n 14350.75 12572.38 m 14550.00 12600.00 l 14370.48 12690.75 l 14313.77 12639.96 l 14350.75 12572.38 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13200 13950 m 14850 12600 l gs col0 s gr n 14663.41 12675.14 m 14850.00 12600.00 l 14739.39 12768.02 l 14664.75 12752.48 l 14663.41 12675.14 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 13200 13950 m 12150 12600 l gs col0 s gr n 12220.52 12788.39 m 12150.00 12600.00 l 12315.24 12714.72 l 12297.85 12789.94 l 12220.52 12788.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 5850 13875 m 5850 13950 l gs col0 s gr % Polyline n 8850 13425 m 8850 13950 l gs col0 s gr % Polyline n 1950 12975 m 1950 12600 l gs col0 s gr n 1890.00 12792.00 m 1950.00 12600.00 l 2010.00 12792.00 l 1950.50 12840.50 l 1890.00 12792.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1650 13275 m 1650 12600 l gs col0 s gr n 1590.00 12792.00 m 1650.00 12600.00 l 1710.00 12792.00 l 1650.50 12840.50 l 1590.00 12792.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1950 10500 m 1951 9751 l gs col0 s gr n 1890.74 9942.92 m 1951.00 9751.00 l 2010.74 9943.08 l 1951.18 9991.50 l 1890.74 9942.92 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1800 10500 m 1801 9751 l gs col0 s gr n 1740.74 9942.92 m 1801.00 9751.00 l 1860.74 9943.08 l 1801.18 9991.50 l 1740.74 9942.92 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1650 10500 m 1651 9751 l gs col0 s gr n 1590.74 9942.92 m 1651.00 9751.00 l 1710.74 9943.08 l 1651.18 9991.50 l 1590.74 9942.92 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 3225 10500 m 3526 9751 l gs col0 s gr n 3398.73 9906.78 m 3526.00 9751.00 l 3510.08 9951.53 l 3437.01 9974.19 l 3398.73 9906.78 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 3075 10500 m 3451 9751 l gs col0 s gr n 3311.24 9895.67 m 3451.00 9751.00 l 3418.48 9949.51 l 3343.83 9965.99 l 3311.24 9895.67 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2925 10500 m 3376 9676 l gs col0 s gr n 3231.18 9815.62 m 3376.00 9676.00 l 3336.45 9873.23 l 3261.27 9887.03 l 3231.18 9815.62 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 3901 10501 m 3676 9751 l gs col0 s gr n 3673.70 9952.14 m 3676.00 9751.00 l 3788.64 9917.66 l 3745.46 9981.38 l 3673.70 9952.14 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 4051 10501 m 3751 9751 l gs col0 s gr n 3766.60 9951.55 m 3751.00 9751.00 l 3878.02 9906.98 l 3840.63 9974.33 l 3766.60 9951.55 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 4201 10501 m 3826 9676 l gs col0 s gr n 3850.83 9875.62 m 3826.00 9676.00 l 3960.07 9825.96 l 3925.81 9894.99 l 3850.83 9875.62 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 13503 11103 m 13503 11253 l 15903 11253 l 15903 11103 l gs col0 s gr 15.000 slw % Polyline n 6605 12011 m 6600 11256 l gs col0 s gr n 6541.27 11448.39 m 6600.00 11256.00 l 6661.27 11447.60 l 6602.09 11496.49 l 6541.27 11448.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9306 12012 m 9301 11257 l gs col0 s gr n 9242.27 11449.39 m 9301.00 11257.00 l 9362.27 11448.60 l 9303.09 11497.49 l 9242.27 11449.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 12006 12012 m 12001 11257 l gs col0 s gr n 11942.27 11449.39 m 12001.00 11257.00 l 12062.27 11448.60 l 12003.09 11497.49 l 11942.27 11449.39 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14706 12012 m 14701 11257 l gs col0 s gr n 14642.27 11449.39 m 14701.00 11257.00 l 14762.27 11448.60 l 14703.09 11497.49 l 14642.27 11449.39 l clp gs 0.00 setgray ef gr gs col0 s gr 30.000 slw % Polyline [100.0] 0 setdash n 5102 11403 m 16502 11403 l 16502 10352 l 5102 10352 l clp gs col0 s gr [] 0 setdash 15.000 slw % Polyline n 6755 9155 m 6755 8405 l gs col0 s gr n 6695.00 8597.00 m 6755.00 8405.00 l 6815.00 8597.00 l 6755.50 8645.50 l 6695.00 8597.00 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 5555 8255 m 5555 8405 l 7955 8405 l 7955 8255 l gs col0 s gr % Polyline n 10803 11103 m 10803 11253 l 13203 11253 l 13203 11103 l gs col0 s gr 15.000 slw % Polyline n 11850 10500 m 12150 6450 l gs col0 s gr n 12075.98 6637.04 m 12150.00 6450.00 l 12195.65 6645.91 l 12132.77 6689.84 l 12075.98 6637.04 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 12150 10500 m 12150 6450 l gs col0 s gr n 12090.00 6642.00 m 12150.00 6450.00 l 12210.00 6642.00 l 12150.50 6690.50 l 12090.00 6642.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 12450 10500 m 12150 6450 l gs col0 s gr n 12104.35 6645.91 m 12150.00 6450.00 l 12224.02 6637.04 l 12168.23 6689.84 l 12104.35 6645.91 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14550 10500 m 12300 6450 l gs col0 s gr n 12340.79 6646.98 m 12300.00 6450.00 l 12445.69 6588.70 l 12417.05 6660.30 l 12340.79 6646.98 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 14850 10500 m 12300 6450 l gs col0 s gr n 12351.53 6644.45 m 12300.00 6450.00 l 12453.07 6580.51 l 12428.38 6653.60 l 12351.53 6644.45 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 15150 10500 m 12300 6450 l gs col0 s gr n 12361.43 6641.55 m 12300.00 6450.00 l 12459.56 6572.49 l 12438.62 6646.77 l 12361.43 6641.55 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9150 10500 m 12000 6450 l gs col0 s gr n 11840.44 6572.49 m 12000.00 6450.00 l 11938.57 6641.55 l 11862.38 6646.77 l 11840.44 6572.49 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9450 10500 m 12000 6450 l gs 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8100 m 4651 6151 l gs col0 s gr n 4499.19 6282.98 m 4651.00 6151.00 l 4601.36 6345.93 l 4525.59 6355.82 l 4499.19 6282.98 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 5250 6450 m 4951 6151 l gs col0 s gr n 5044.34 6329.19 m 4951.00 6151.00 l 5129.19 6244.34 l 5121.21 6321.21 l 5044.34 6329.19 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 4802 5552 m 4651 5101 l gs col0 s gr n 4655.06 5302.12 m 4651.00 5101.00 l 4768.85 5264.02 l 4727.70 5329.08 l 4655.06 5302.12 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 4801 4951 m 4801 5101 l 4501 5101 l 4501 4951 l gs col0 s gr 15.000 slw % Polyline n 1802 9152 m 1652 8402 l gs col0 s gr n 1630.82 8602.04 m 1652.00 8402.00 l 1748.49 8578.50 l 1699.57 8637.84 l 1630.82 8602.04 l clp gs 0.00 setgray ef gr gs col0 s gr 60.000 slw % Polyline n 1802 8252 m 1802 8402 l 1502 8402 l 1502 8252 l gs col0 s gr 15.000 slw % Polyline n 3602 9152 m 3452 8402 l gs col0 s gr n 3430.82 8602.04 m 3452.00 8402.00 l 3548.49 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b(I)g(tested)f(them)g(in)h(tw)o(o)f(of)g(my)g(Delaunay)f(triangulation) -150 4399 y(codes.)i(T)m(riangle)17 b([14)o(])i(is)g(a)f(2D)h(Delaunay) e(triangulator)f(and)i(mesh)-150 4492 y(generator)m(,)h(publicly)h(a)n (v)n(ailable)h(from)f(Netlib,)h(that)g(uses)h(a)g(di)n(vide-)-150 4584 y(and-conquer)i(algorithm)i([11)o(,)i(8)o(].)51 b(Pyramid)27 b(is)h(a)g(3D)g(Delaunay)-150 4677 y(tetrahedralizer)22 b(that)i(uses)g(an)g(incremental)e(algorithm)h([15)n(].)41 b(F)o(or)-150 4770 y(both)22 b(2D)i(and)f(3D,)g(three)f(types)i(of)f (inputs)f(were)i(tested:)36 b(uniform)-150 4862 y(random)14 b(points,)i(points)g(lying)f(\(approximately\))e(on)i(the)h(boundary) -150 4955 y(of)30 b(a)h(circle)g(or)f(sphere,)i(and)e(a)h(square)f(or)g (cubic)g(grid)g(of)g(lattice)-150 5048 y(points,)18 b(tilted)g(so)g(as) h(not)e(to)h(be)g(aligned)f(with)h(the)g(coordinate)e(ax)o(es.)-150 5140 y(The)25 b(latter)g(tw)o(o)h(were)f(chosen)f(for)h(their)g (nastiness.)45 b(The)25 b(lattices)-150 5233 y(ha)n(v)o(e)15 b(been)f(tilted)i(using)f(approximate)e(arithmetic,)i(so)h(the)o(y)f (are)g(not)-150 5326 y(perfectly)g(cubical,)i(and)f(the)h(e)o(xponents) e(of)h(their)h(coordinates)e(v)n(ary)p 2086 35 1927 4 v 2084 120 4 85 v 2350 95 a Fd(2D)k(di)n(vide-and-conquer)j(Delaunay)e (triangulation)p 4011 120 V 2086 124 1927 4 v 2084 209 4 85 v 2816 209 V 2888 183 a(Uniform)p 3214 209 V 125 w(Perimeter)p 3613 209 V 165 w(T)m(ilted)p 4011 209 V 2084 294 V 2816 294 V 2890 268 a(Random)p 3214 294 V 141 w(of)f(Circle)p 3613 294 V 198 w(Grid)p 4011 294 V 2086 297 1927 4 v 2084 382 4 85 v 2136 356 a(Input)g(points)p 2816 382 V 373 w(1,000,000)p 3214 382 V 102 w(1,000,000)p 3613 382 V 101 w(1,000,000)p 4011 382 V 2086 385 1927 4 v 2136 445 a(O)p Fc(RIENT)p Fd(2D)f(calls)p 2086 473 V 2084 558 4 85 v 2136 533 a(Adapti)n(v)o(e)h(A,)g(approx.)p 2816 558 V 125 w(9,497,314)p 3214 558 V 102 w(6,291,742)p 3613 558 V 101 w(9,318,610)p 4011 558 V 2084 643 V 2136 618 a(Adapti)n(v)o(e)g(B)p 2816 643 V 3214 643 V 3613 643 V 1245 w(121,081)p 4011 643 V 2084 728 V 2136 703 a(Adapti)n(v)o(e)g(C)p 2816 728 V 3214 728 V 3613 728 V 1376 w(118)p 4011 728 V 2084 813 V 2136 788 a(Adapti)n(v)o(e)g(D,)g (e)o(xact)p 2816 813 V 3214 813 V 3613 813 V 1253 w(3)p 4011 813 V 2086 816 1927 4 v 2084 901 4 85 v 2136 876 a(A)-6 b(v)o(erage)20 b(time,)e Fe(\026)p Fd(s)p 2816 901 V 389 w(0.32)p 3214 901 V 269 w(0.38)p 3613 901 V 268 w(0.33)p 4011 901 V 2086 905 1927 4 v 2086 925 V 2084 1009 4 85 v 2136 984 a(LN)g(approximate)p 2816 1009 V 241 w(9,497,314)p 3214 1009 V 102 w(2,112,284)p 3613 1009 V 309 w(n/a)p 4011 1009 V 2084 1094 V 2136 1069 a(LN)g(e)o(xact)p 2816 1094 V 3214 1094 V 856 w(4,179,458)p 3613 1094 V 309 w(n/a)p 4011 1094 V 2086 1098 1927 4 v 2084 1183 4 85 v 2136 1157 a(LN)g(a)o(v)o(erage)h(time,)g Fe(\026)p Fd(s)p 2816 1183 V 287 w(0.35)p 3214 1183 V 269 w(3.16)p 3613 1183 V 308 w(n/a)p 4011 1183 V 2086 1186 1927 4 v 2136 1245 a(I)p Fc(N)p Fd(C)p Fc(IRCLE)e Fd(calls)p 2086 1274 V 2084 1359 4 85 v 2136 1334 a(Adapti)n(v)o(e)i (A,)g(approx.)p 2816 1359 V 125 w(7,596,885)p 3214 1359 V 102 w(3,970,796)p 3613 1359 V 101 w(7,201,317)p 4011 1359 V 2084 1444 V 2136 1419 a(Adapti)n(v)o(e)g(B)p 2816 1444 V 3214 1444 V 884 w(50,551)p 3613 1444 V 157 w(176,470)p 4011 1444 V 2084 1529 V 2136 1504 a(Adapti)n(v)o(e)g(C)p 2816 1529 V 3214 1529 V 978 w(120)p 3613 1529 V 324 w(47)p 4011 1529 V 2084 1614 V 2136 1588 a(Adapti)n(v)o(e)g(D,)g(e)o(xact)p 2816 1614 V 3214 1614 V 3613 1614 V 1253 w(4)p 4011 1614 V 2086 1617 1927 4 v 2084 1702 4 85 v 2136 1677 a(A)-6 b(v)o(erage)20 b(time,)e Fe(\026)p Fd(s)p 2816 1702 V 389 w(0.65)p 3214 1702 V 269 w(1.11)p 3613 1702 V 268 w(1.67)p 4011 1702 V 2086 1706 1927 4 v 2086 1725 V 2084 1810 4 85 v 2136 1785 a(LN)g(approximate)p 2816 1810 V 241 w(6,077,062)p 3214 1810 V 363 w(0)p 3613 1810 V 308 w(n/a)p 4011 1810 V 2084 1895 V 2136 1870 a(LN)g(e)o(xact)p 2816 1895 V 457 w(1,519,823)p 3214 1895 V 102 w(4,021,467)p 3613 1895 V 309 w(n/a)p 4011 1895 V 2086 1899 1927 4 v 2084 1984 4 85 v 2136 1958 a(LN)g(a)o(v)o(erage)h(time,)g Fe(\026)p Fd(s)p 2816 1984 V 287 w(7.36)p 3214 1984 V 232 w(32.78)p 3613 1984 V 308 w(n/a)p 4011 1984 V 2086 1987 1927 4 v 2136 2046 a(Program)g(running)h(time,)e(seconds)p 2086 2075 V 2084 2160 4 85 v 2136 2135 a(Approximate)i(v)o(ersion)p 2816 2160 V 268 w(57.3)p 3214 2160 V 269 w(59.9)p 3613 2160 V 268 w(48.3)p 4011 2160 V 2084 2245 V 2136 2220 a(Rob)o(ust)f(v)o(ersion)p 2816 2245 V 452 w(61.7)p 3214 2245 V 269 w(64.7)p 3613 2245 V 268 w(62.2)p 4011 2245 V 2084 2330 V 2136 2304 a(LN)f(rob)o(ust)h(v)o(ersion)p 2816 2330 V 322 w(116.0)p 3214 2330 V 232 w(214.6)p 3613 2330 V 308 w(n/a)p 4011 2330 V 2086 2333 1927 4 v 2049 2480 a FG(T)-7 b(able)16 b(3:)27 b Fj(Statistics)16 b(f)n(or)g(2D)f (divide-and-conquer)f(Delauna)n(y)g(tr)q(iangulation)2049 2554 y(of)19 b(se)n(v)n(er)o(al)g(point)g(sets)o(.)p 2086 2702 V 2084 2786 4 85 v 2395 2761 a Fd(3D)g(incremental)g (Delaunay)h(tetrahedralization)p 4011 2786 V 2086 2790 1927 4 v 2084 2875 4 85 v 2816 2875 V 2888 2849 a(Uniform)p 3214 2875 V 157 w(Surf)o(ace)p 3613 2875 V 197 w(T)m(ilted)p 4011 2875 V 2084 2960 V 2816 2960 V 2890 2934 a(Random)p 3214 2960 V 129 w(of)f(Sphere)p 3613 2960 V 186 w(Grid)p 4011 2960 V 2086 2963 1927 4 v 2084 3048 4 85 v 2136 3022 a(Input)g(points)p 2816 3048 V 467 w(10,000)p 3214 3048 V 194 w(10,000)p 3613 3048 V 195 w(10,000)p 4011 3048 V 2086 3051 1927 4 v 2136 3111 a(O)p Fc(RIENT)p Fd(3D)f(counts)p 2086 3139 V 2084 3224 4 85 v 2136 3199 a(Adapti)n(v)o(e)h(A,)g(approx.)p 2816 3224 V 125 w(2,735,668)p 3214 3224 V 102 w(1,935,978)p 3613 3224 V 101 w(5,542,567)p 4011 3224 V 2084 3309 V 2136 3284 a(Adapti)n(v)o(e)g(B)p 2816 3309 V 3214 3309 V 3613 3309 V 1245 w(602,344)p 4011 3309 V 2084 3394 V 2136 3369 a(Adapti)n(v)o(e)g(C)p 2816 3394 V 3214 3394 V 3613 3394 V 1189 w(1,267,423)p 4011 3394 V 2084 3479 V 2136 3454 a(Adapti)n(v)o(e)g(D,)g(e)o(xact)p 2816 3479 V 3214 3479 V 3613 3479 V 1085 w(28,185)p 4011 3479 V 2086 3483 1927 4 v 2084 3567 4 85 v 2136 3542 a(A)-6 b(v)o(erage)20 b(time,)e Fe(\026)p Fd(s)p 2816 3567 V 389 w(0.72)p 3214 3567 V 269 w(0.72)p 3613 3567 V 268 w(4.12)p 4011 3567 V 2086 3571 1927 4 v 2086 3591 V 2084 3676 4 85 v 2136 3650 a(LN)g(approximate)p 2816 3676 V 241 w(2,735,668)p 3214 3676 V 102 w(1,935,920)p 3613 3676 V 309 w(n/a)p 4011 3676 V 2084 3761 V 2136 3735 a(LN)g(e)o(xact)p 2816 3761 V 3214 3761 V 1080 w(58)p 3613 3761 V 308 w(n/a)p 4011 3761 V 2086 3764 1927 4 v 2084 3849 4 85 v 2136 3823 a(LN)g(a)o(v)o(erage)h(time,)g Fe(\026)p Fd(s)p 2816 3849 V 287 w(0.99)p 3214 3849 V 269 w(1.00)p 3613 3849 V 308 w(n/a)p 4011 3849 V 2086 3852 1927 4 v 2136 3912 a(I)p Fc(N)p Fd(S)p Fc(PHERE)e Fd(counts)p 2086 3940 V 2084 4025 4 85 v 2136 4000 a(Adapti)n(v)o(e)i (A,)g(approx.)p 2816 4025 V 181 w(439,090)p 3214 4025 V 158 w(122,273)p 3613 4025 V 101 w(3,080,312)p 4011 4025 V 2084 4110 V 2136 4085 a(Adapti)n(v)o(e)g(B)p 2816 4110 V 3214 4110 V 847 w(180,383)p 3613 4110 V 157 w(267,162)p 4011 4110 V 2084 4195 V 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