(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: wacg96.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Times-Roman Courier Times-Bold Helvetica Times-Italic %%EndComments %DVIPSCommandLine: dvips -D600 -o wacg96.ps wacg96.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1996.03.02:0129 %%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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6958 l gs col0 s gr n 7486.32 6834.52 m 7479.00 6958.00 l 7427.35 6845.61 l 7457.33 6840.56 l 7486.32 6834.52 l clp gs col7 1.00 shd ef gr gs col0 s gr % Polyline n 8851 8401 m 8851 8401 l gs col0 s gr % Polyline n 8851 8401 m 8851 8626 l gs col0 s gr n 8881.00 8506.00 m 8851.00 8626.00 l 8821.00 8506.00 l 8851.50 8506.50 l 8881.00 8506.00 l clp gs col7 1.00 shd ef gr gs col0 s gr % Polyline n 8551 6601 m 8551 7276 l gs col0 s gr n 8581.00 7132.00 m 8551.00 7276.00 l 8521.00 7132.00 l 8551.50 7156.50 l 8581.00 7132.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9151 7501 m 9151 6826 l gs col0 s gr n 9121.00 6970.00 m 9151.00 6826.00 l 9181.00 6970.00 l 9151.50 6946.50 l 9121.00 6970.00 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9376 6301 m 9901 6001 l gs col0 s gr n 9761.09 6046.40 m 9901.00 6001.00 l 9790.86 6098.49 l 9797.31 6061.04 l 9761.09 6046.40 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 9751 5401 m 9226 5701 l gs col0 s gr n 9365.91 5655.60 m 9226.00 5701.00 l 9336.14 5603.51 l 9330.69 5641.96 l 9365.91 5655.60 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 8626 5776 m 8101 5476 l gs col0 s gr n 8211.14 5573.49 m 8101.00 5476.00 l 8240.91 5521.40 l 8205.69 5536.04 l 8211.14 5573.49 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 7651 5926 m 8176 6226 l gs col0 s gr n 8065.86 6128.51 m 8176.00 6226.00 l 8036.09 6180.60 l 8072.31 6166.96 l 8065.86 6128.51 l clp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2684 8623 m 3081 8683 l gs col0 s gr n 2966.83 8635.40 m 3081.00 8683.00 l 2957.86 8694.73 l 2962.85 8665.57 l 2966.83 8635.40 l clp gs col7 1.00 shd ef gr gs col0 s gr % Polyline n 3621 8630 m 3216 8683 l gs col0 s gr n 3338.88 8697.18 m 3216.00 8683.00 l 3331.09 8637.68 l 3335.49 8667.93 l 3338.88 8697.18 l clp gs col7 1.00 shd ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 2252 5702 m 2241.29 5399.33 2222.54 5268.08 2177 5177 curveto 2149.67 5122.35 2093.42 5066.10 1952 4952 curveto gs col0 s gr [] 0 setdash n 2045.23 5065.77 m 1952.00 4952.00 l 2082.91 5019.07 l 2045.89 5027.85 l 2045.23 5065.77 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 1802 4952 m 1575.32 5056.68 1481.57 5112.93 1427 5177 curveto 1376.61 5236.17 1339.11 5329.92 1277 5552 curveto gs col0 s gr [] 0 setdash n 1344.68 5421.40 m 1277.00 5552.00 l 1286.89 5405.24 l 1309.82 5436.93 l 1344.68 5421.40 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 1352 5702 m 1585.42 5866.42 1697.92 5922.67 1802 5927 curveto 1885.56 5930.48 1979.31 5892.98 2177 5777 curveto gs col0 s gr [] 0 setdash n 2037.62 5823.99 m 2177.00 5777.00 l 2067.98 5875.74 l 2074.00 5838.22 l 2037.62 5823.99 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 2102 4652 m 2174.32 4915.64 2211.82 5028.14 2252 5102 curveto 2292.68 5176.78 2367.68 5270.53 2552 5477 curveto gs col0 s gr n 2478.48 5349.60 m 2552.00 5477.00 l 2433.72 5389.56 l 2472.58 5387.98 l 2478.48 5349.60 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 977 5552 m 1202.59 5299.61 1296.34 5187.11 1352 5102 curveto 1383.87 5053.27 1421.37 4978.27 1502 4802 curveto gs col0 s gr n 1414.82 4920.47 m 1502.00 4802.00 l 1469.38 4945.43 l 1452.58 4911.63 l 1414.82 4920.47 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 2327 6077 m 2026.97 6018.15 1895.72 5999.40 1802 6002 curveto 1734.42 6003.88 1640.67 6022.63 1427 6077 curveto gs col0 s gr n 1573.95 6070.56 m 1427.00 6077.00 l 1559.15 6012.42 l 1543.79 6047.91 l 1573.95 6070.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 4952 5702 m 4941.29 5399.33 4922.54 5268.08 4877 5177 curveto 4849.67 5122.35 4793.42 5066.10 4652 4952 curveto gs col0 s gr [] 0 setdash n 4745.23 5065.77 m 4652.00 4952.00 l 4782.91 5019.07 l 4745.89 5027.85 l 4745.23 5065.77 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 4502 4952 m 4275.32 5056.68 4181.57 5112.93 4127 5177 curveto 4076.61 5236.17 4039.11 5329.92 3977 5552 curveto gs col0 s gr [] 0 setdash n 4044.68 5421.40 m 3977.00 5552.00 l 3986.89 5405.24 l 4009.82 5436.93 l 4044.68 5421.40 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 4052 5702 m 4285.42 5866.42 4397.92 5922.67 4502 5927 curveto 4585.56 5930.48 4679.31 5892.98 4877 5777 curveto gs col0 s gr [] 0 setdash n 4737.62 5823.99 m 4877.00 5777.00 l 4767.98 5875.74 l 4774.00 5838.22 l 4737.62 5823.99 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 4802 4652 m 4874.32 4915.64 4911.82 5028.14 4952 5102 curveto 4992.68 5176.78 5067.68 5270.53 5252 5477 curveto gs col0 s gr n 5178.48 5349.60 m 5252.00 5477.00 l 5133.72 5389.56 l 5172.58 5387.98 l 5178.48 5349.60 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 3677 5552 m 3902.59 5299.61 3996.34 5187.11 4052 5102 curveto 4083.87 5053.27 4121.37 4978.27 4202 4802 curveto gs col0 s gr n 4114.82 4920.47 m 4202.00 4802.00 l 4169.38 4945.43 l 4152.58 4911.63 l 4114.82 4920.47 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 5027 6077 m 4726.97 6018.15 4595.72 5999.40 4502 6002 curveto 4434.42 6003.88 4340.67 6022.63 4127 6077 curveto gs col0 s gr n 4273.95 6070.56 m 4127.00 6077.00 l 4259.15 6012.42 l 4243.79 6047.91 l 4273.95 6070.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 2851 7051 m 2772.85 6788.19 2735.35 6675.69 2701 6601 curveto 2669.33 6532.14 2613.08 6438.39 2476 6226 curveto gs col0 s gr n 2528.88 6363.26 m 2476.00 6226.00 l 2579.29 6330.72 l 2541.57 6327.32 l 2528.88 6363.26 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 3976 6076 m 3752.03 6374.61 3658.28 6505.86 3601 6601 curveto 3571.22 6650.46 3533.72 6725.46 3451 6901 curveto gs col0 s gr n 3539.52 6783.53 m 3451.00 6901.00 l 3485.25 6757.95 l 3502.65 6792.95 l 3539.52 6783.53 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 2701 5551 m 2951.07 5658.06 3063.57 5695.56 3151 5701 curveto 3221.49 5705.39 3315.24 5686.64 3526 5626 curveto gs col0 s gr n 3379.32 5636.99 m 3526.00 5626.00 l 3395.91 5694.65 l 3411.18 5659.68 l 3379.32 5636.99 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 3601 8101 m 3590.29 7798.33 3571.54 7667.08 3526 7576 curveto 3498.67 7521.35 3442.42 7465.10 3301 7351 curveto gs col0 s gr [] 0 setdash n 3394.23 7464.77 m 3301.00 7351.00 l 3431.91 7418.07 l 3394.89 7426.85 l 3394.23 7464.77 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 3151 7351 m 2924.32 7455.68 2830.57 7511.93 2776 7576 curveto 2725.61 7635.17 2688.11 7728.92 2626 7951 curveto gs col0 s gr [] 0 setdash n 2693.68 7820.40 m 2626.00 7951.00 l 2635.89 7804.24 l 2658.82 7835.93 l 2693.68 7820.40 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 2701 8101 m 2934.42 8265.42 3046.92 8321.67 3151 8326 curveto 3234.56 8329.48 3328.31 8291.98 3526 8176 curveto gs col0 s gr [] 0 setdash n 3386.62 8222.99 m 3526.00 8176.00 l 3416.98 8274.74 l 3423.00 8237.22 l 3386.62 8222.99 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 3451 7051 m 3523.32 7314.64 3560.82 7427.14 3601 7501 curveto 3641.68 7575.78 3716.68 7669.53 3901 7876 curveto gs col0 s gr n 3827.48 7748.60 m 3901.00 7876.00 l 3782.72 7788.56 l 3821.58 7786.98 l 3827.48 7748.60 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 2326 7951 m 2551.59 7698.61 2645.34 7586.11 2701 7501 curveto 2732.87 7452.27 2770.37 7377.27 2851 7201 curveto gs col0 s gr n 2763.82 7319.47 m 2851.00 7201.00 l 2818.38 7344.43 l 2801.58 7310.63 l 2763.82 7319.47 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 3676 8476 m 3375.97 8417.15 3244.72 8398.40 3151 8401 curveto 3083.42 8402.88 2989.67 8421.63 2776 8476 curveto gs col0 s gr n 2922.95 8469.56 m 2776.00 8476.00 l 2908.15 8411.42 l 2892.79 8446.91 l 2922.95 8469.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 2551 8551 m 2427.53 8619.99 2390.03 8657.49 2401 8701 curveto 2454.99 8915.13 2954.46 8926.00 3151 8926 curveto 3347.54 8926.00 3791.05 8955.47 3901 8701 curveto 3918.66 8660.12 3899.91 8622.62 3826 8551 curveto gs col0 s gr n 3908.54 8672.75 m 3826.00 8551.00 l 3950.29 8629.66 l 3912.68 8635.01 l 3908.54 8672.75 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 4126 8401 m 4054.07 8716.19 3997.82 8847.44 3901 8926 curveto 3728.14 9066.26 3333.41 9076.00 3151 9076 curveto 2968.59 9076.00 2574.00 9069.90 2401 8926 curveto 2332.49 8869.01 2294.99 8775.26 2251 8551 curveto gs col0 s gr [] 0 setdash n 2249.28 8698.08 m 2251.00 8551.00 l 2308.16 8686.53 l 2274.60 8669.26 l 2249.28 8698.08 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 1276 6076 m 1128.82 6180.03 1072.57 6236.28 1051 6301 curveto 955.12 6588.66 1070.83 7106.94 1201 7351 curveto 1324.83 7583.18 1702.00 7930.45 1951 8026 curveto 1992.64 8041.98 2048.89 8041.98 2176 8026 curveto gs col0 s gr n 2029.38 8014.20 m 2176.00 8026.00 l 2036.87 8073.73 l 2057.44 8041.47 l 2029.38 8014.20 l clp gs 0.00 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682.39 5657.82 l 706.51 5628.02 l 681.63 5597.82 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline n 5326 5626 m 5571.71 5652.11 5665.46 5633.36 5701 5551 curveto 5876.34 5144.68 5432.42 4405.82 4951 4351 curveto 4896.87 4344.84 4859.37 4382.34 4801 4501 curveto gs col0 s gr n 4891.48 4385.03 m 4801.00 4501.00 l 4837.64 4358.55 l 4854.47 4393.82 l 4891.48 4385.03 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 2776 6001 m 2700.81 6204.42 2682.06 6298.17 2701 6376 curveto 2724.78 6473.74 2799.78 6567.49 3001 6751 curveto gs col0 s gr [] 0 setdash n 2914.82 6631.80 m 3001.00 6751.00 l 2874.39 6676.13 l 2912.84 6670.64 l 2914.82 6631.80 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 3151 6751 m 3331.92 6691.61 3406.92 6654.11 3451 6601 curveto 3522.75 6514.54 3560.25 6383.29 3601 6076 curveto gs col0 s gr [] 0 setdash n 3552.33 6214.81 m 3601.00 6076.00 l 3611.81 6222.69 l 3585.72 6195.46 l 3552.33 6214.81 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 3526 6001 m 3422.88 5853.28 3366.63 5797.03 3301 5776 curveto 3206.91 5745.85 3094.41 5764.60 2851 5851 curveto gs col0 s gr [] 0 setdash n 2996.74 5831.10 m 2851.00 5851.00 l 2976.67 5774.56 l 2964.59 5811.36 l 2996.74 5831.10 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 2176 8401 m 2047.68 8268.84 1991.43 8212.59 1951 8176 curveto 1770.41 8012.57 1269.71 7736.41 1126 7501 curveto 981.23 7263.85 927.12 6697.54 901 6451 curveto 895.53 6399.34 895.53 6324.34 901 6151 curveto gs col0 s gr [] 0 setdash n 866.47 6293.98 m 901.00 6151.00 l 926.44 6295.87 l 897.72 6271.44 l 866.47 6293.98 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 826 6001 m 565.92 5694.81 472.17 5544.81 451 5401 curveto 416.88 5169.25 526.52 4758.33 676 4576 curveto 822.57 4397.21 1173.16 4185.26 1426 4201 curveto 1487.15 4204.81 1543.40 4242.31 1651 4351 curveto gs col0 s gr [] 0 setdash n 1571.01 4227.56 m 1651.00 4351.00 l 1528.37 4269.77 l 1567.08 4266.22 l 1571.01 4227.56 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 1801 4351 m 2059.26 4258.18 2171.76 4220.68 2251 4201 curveto 2454.22 4150.52 2936.83 4052.34 3151 4051 curveto 3382.56 4049.55 3904.57 4115.04 4126 4201 curveto 4171.52 4218.67 4227.77 4256.17 4351 4351 curveto gs col0 s gr [] 0 setdash n 4255.17 4239.40 m 4351.00 4351.00 l 4218.58 4286.95 l 4256.40 4278.32 l 4255.17 4239.40 l clp gs 0.00 setgray ef gr gs col0 s gr % Interp Spline [66.7] 0 setdash n 4501 4351 m 4860.03 4195.63 5028.78 4158.13 5176 4201 curveto 5416.89 4271.14 5681.59 4601.77 5776 4801 curveto 5859.87 4977.97 5922.15 5351.67 5851 5551 curveto 5820.96 5635.15 5745.96 5710.15 5551 5851 curveto gs col0 s gr [] 0 setdash n 5685.29 5790.99 m 5551.00 5851.00 l 5650.16 5742.35 l 5648.77 5781.23 l 5685.29 5790.99 l clp gs 0.00 setgray ef gr gs col0 s gr F2psBegin10setmiterlimit0.060000.06000sc7.500slwF2psEnd restore %%EndDocument @endspecial 1992 3335 a(Figure)13 b(1:)25 b Fo(A)14 b(tr)q(iangulation)9 b(\(top\))k(and)g(its)h(corresponding)c(represen-) 1992 3427 y(tations)24 b(with)g(quad-edge)e(and)i(tr)q(iangular)f(data) h(str)q(uctures)o(.)43 b(Each)1992 3518 y(quad-edge)17 b(and)i(each)h(tr)q(iangle)e(contains)h(six)i(pointers)o(.)1992 3784 y Fv(three)g(pointers)g(to)g(v)o(ertices.)33 b(Hence,)22 b(both)f(structures)f(contain)h(six)1992 3883 y(pointers.)2282 3853 y Fl(1)2364 3883 y Fv(A)28 b(triangulation)d(contains)i(roughly)e (three)h(edges)h(for)1992 3983 y(e)n(v)o(ery)g(tw)o(o)h(triangles.)53 b(Hence,)29 b(the)g(triangular)d(data)i(structure)g(is)1992 4082 y(more)19 b(space-ef)n(\256cient.)2091 4182 y(It)h(is)g(dif)n (\256cult)f(to)g(ascertain)g(with)h(certainty)e(why)h(the)g(triangular) 1992 4282 y(data)e(structure)f(is)j(superior)d(in)h(time)h(as)g(well)g (as)g(space,)g(b)n(ut)f(one)g(can)1992 4381 y(mak)o(e)f(educated)f (inferences.)27 b(When)17 b(a)g(program)e(mak)o(es)h(structural)1992 4481 y(changes)c(to)h(a)h(triangulation,)f(the)g(amount)f(of)h(time)g (used)g(depends)f(in)1992 4581 y(part)h(on)g(the)g(number)e(of)i (pointers)g(that)g(ha)n(v)o(e)g(to)g(be)h(read)e(and)h(written.)p 1992 4652 805 4 v 2082 4707 a Fk(1)2111 4730 y Fr(Both)29 b(the)g(quad-edge)h(and)e(triangle)i(data)g(structures)f(must)f(store)h (not)f(only)1992 4809 y(pointers)c(to)e(their)i(neighbors,)h(b)o(ut)d (also)h(the)g Fj(orientations)i Fr(of)e(their)g(neighbors,)i(to)1992 4888 y(mak)o(e)30 b(clear)g(ho)n(w)g(the)o(y)g(are)g(connected.)63 b(F)o(or)29 b(instance,)34 b(each)d(pointer)g(from)e(a)1992 4967 y(triangle)d(to)e(a)g(neighboring)j(triangle)g(has)d(an)g (associated)j(orientation)g(\(a)d(number)1992 5046 y(between)18 b(zero)f(and)g(tw)o(o\))f(that)i(indicates)h(which)e(edge)g(of)f(the)h (neighboring)i(triangle)1992 5125 y(is)25 b(contacted.)54 b(An)25 b(important)j(space)f(optimization)i(is)c(to)h(store)h(the)f (orientation)1992 5203 y(of)c(each)i(quad-edge)h(or)d(triangle)j(in)e 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pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind 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def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save 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(non-rob)n(ust)d(v)o(ersion)1992 1197 y(is)f(less)g(lik)o(ely)f(to)g(f) o(ail.)2091 1296 y(Of)g(course,)g(adapti)n(v)o(e)e(tests)j(do)f(not)g (solv)o(e)f(all)i(rob)n(ustness)e(prob-)1992 1396 y(lems.)65 b(Geometric)31 b(computations)f(that)i(produce)e(ne)n(w)i(v)o(ertices,) 1992 1496 y(including)19 b(circumcenters)h(and)h(se)o(gment)f (intersections,)h(could)g(be)1992 1595 y(performed)29 b(e)o(xactly)i(in)h(principle,)h(b)n(ut)f(the)g(results)g(w)o(ould)f (ha)n(v)o(e)1992 1695 y(lar)o(ge)f(bit)i(comple)o(xity)d(and)i(w)o (ould)g(be)h(incon)m(v)o(enient)c(to)k(manip-)1992 1794 y(ulate)d(and)f(e)o(xpensi)n(v)o(e)f(to)i(store.)56 b(W)-7 b(orse,)32 b(v)o(ertices)c(of)h(arbitrarily)1992 1894 y(lar)o(ge)24 b(bit)i(comple)o(xity)e(could)h(e)n(v)o(entually)f(be)h (produced)e(in)j(a)g(cas-)1992 1994 y(cading)13 b(ef)n(fect)i(when)f (the)h(Delaunay)e(re\256nement)h(algorithm)f(inserts)1992 2093 y(circumcenters)29 b(of)i(triangles)g(whose)g(v)o(ertices)g(were)g (themselv)o(es)1992 2193 y(circumcenters.)j(Hence,)22 b(it)i(is)f(infeasible)f(to)h(mak)o(e)f(the)g(algorithm)1992 2293 y(perfectly)e(rob)n(ust.)33 b(F)o(ortunately)-5 b(,)20 b(the)i(Delaunay)e(re\256nement)g(algo-)1992 2392 y(rithm)14 b(is)i(naturally)d(stable)i(with)g(re)o(gard)e(to)i (\257oating-point)e(roundof)n(f)1992 2492 y(error)-5 b(.)51 b(Problems)27 b(arise)h(only)f(when)g(triangles)g(are)g (re\256ned)g(to)h(so)1992 2591 y(small)17 b(a)g(size)g(that)g(it)h(is)f (no)f(longer)g(possible)g(to)h(construct)f(a)h(circum-)1992 2691 y(center)i(that)i(is)g(distinct)f(from)f(its)i(triangle')-5 b(s)20 b(v)o(ertices.)2091 2791 y(I)k(ha)n(v)o(e)e(not)h(produced)e(a)j (rob)n(ust)e(v)o(ersion)g(of)h(the)h(sweepline)e(al-)1992 2890 y(gorithm)32 b(for)i(a)h(some)n(what)e(technical)h(reason.)70 b(The)34 b(sweepline)1992 2990 y(algorithm)17 b(maintains)i(a)h (priority)e(queue)g(\(normally)f(implemented)1992 3090 y(as)k(a)h(heap\))d(containing)g(tw)o(o)i(types)g(of)g(e)n(v)o(ents:)30 b Fm(site)22 b(e)o(vents)p Fv(,)e(where)1992 3189 y(the)26 b(sweepline)h(passes)g(o)o(v)o(er)e(an)i(input)f(point,)h(and)f Fm(cir)m(cle)h(e)o(vents)p Fv(,)1992 3289 y(where)h(the)i(sweepline)f (reaches)g(the)g(top)g(of)g(a)h(circle)g(de\256ned)e(by)1992 3388 y(three)d(consecuti)n(v)o(e)e(v)o(ertices)i(on)g(the)g(boundary)e (of)i(the)g(triangula-)1992 3488 y(tion.)44 b(Unfortunately)-5 b(,)23 b(the)i Fq(y)s Fv(-coordinate)d(of)j(such)g(a)h(circle)f(top)g (is)1992 3588 y(e)o(xpensi)n(v)o(e)h(to)j(compute)d(e)o(xactly)-5 b(,)29 b(may)f(be)h(irrational,)g(and)f(has)g(a)1992 3687 y(some)n(what)17 b(complicated)f(e)o(xact)i(representation.)26 b(A)19 b(rob)n(ust)e(imple-)1992 3787 y(mentation)i(must)h(k)o(eep)g (the)g(e)n(v)o(ents)g(correctly)f(ordered,)g(and)g(hence)1992 3887 y(must)g(replace)f(the)h(simple)g(comparisons)e(normally)g(used)i (to)g(main-)1992 3986 y(tain)k(a)h(priority)e(queue)h(with)g(a)h(test)g (that)g(correctly)e(compares)g(tw)o(o)1992 4086 y(circle)e(tops.)30 b(Ev)o(en)19 b(a)i(f)o(ast)g(adapti)n(v)o(e)e(v)o(ersion)g(of)h(such)g (a)h(test)g(w)o(ould)1992 4185 y(be)d(so)h(much)e(slo)n(wer)i(than)f (simple)g(comparisons)f(that)h(e)n(v)o(ent)g(queue)1992 4285 y(maintenance,)25 b(which)g(is)h(a)g(dominant)e(cost)i(of)g(the)f (sweepline)g(al-)1992 4385 y(gorithm,)18 b(w)o(ould)i(become)f (prohibiti)n(v)o(ely)e(e)o(xpensi)n(v)o(e.)1992 4571 y Ft(A)99 b(Additional)25 b(Implementation)h(Notes)2091 4717 y Fv(The)13 b(sweepline)g(and)g(incremental)g(Delaunay)g(triangu)o (lation)f(im-)1992 4816 y(plementations)26 b(compared)f(by)j(Su)f(and)h (Drysdale)e([18)o(])i(each)f(use)1992 4916 y(some)h(v)n(ariant)f(of)h (uniform)e(b)n(uck)o(eting)h(to)h(locate)g(points.)53 b(Buck-)1992 5016 y(eting)24 b(yields)g(f)o(ast)i(implementations)c(on) i(uniform)f(point)h(sets,)j(b)n(ut)1992 5115 y(is)c(easily)f(defeated;) g(a)g(small,)h(dense)f(cluster)g(of)f(points)h(in)g(a)h(lar)o(ge,)1992 5215 y(sparsely)28 b(populated)e(re)o(gion)h(may)h(all)h(f)o(all)f (into)g(a)h(single)f(b)n(uck)o(et.)1992 5314 y(I)c(ha)n(v)o(e)f(not)g (used)g(b)n(uck)o(eting)g(in)g(T)m(riangle,)h(preferring)d(algorithms) 1881 5564 y(8)p eop %%Page: 9 9 9 8 bop -198 -98 a Fv(that)22 b(e)o(xhibit)g(good)f(performance)e(with) k(an)o(y)e(distrib)n(ution)h(of)g(input)-198 1 y(points.)48 b(As)27 b(a)g(result,)h(T)m(riangle)e(may)g(be)g(slo)n(wer)h(than)f (necessary)-198 101 y(when)d(triangulating)f(uniformly)f(distrib)n (uted)i(point)g(sets,)i(b)n(ut)f(will)-198 200 y(not)h(e)o(xhibit)g (asymptotically)g(slo)n(wer)g(running)f(times)i(on)f(dif)n(\256cult) -198 300 y(inputs.)-98 400 y(F)o(ortune')-5 b(s)34 b(sweepline)i (algorithm)e(uses)i(tw)o(o)g(nontri)n(vial)e(data)-198 499 y(structures)19 b(in)g(addition)f(to)i(the)f(triangulation:)27 b(a)20 b(priority)e(queue)g(to)-198 599 y(store)28 b(e)n(v)o(ents,)h (and)e(a)i(balanced)d(tree)i(data)g(structure)f(to)h(store)g(the)-198 699 y(sequence)e(of)h(edges)f(on)h(the)g(boundary)d(of)j(the)g(mesh.)49 b(F)o(ortune')-5 b(s)-198 798 y(o)n(wn)14 b(implementation,)g(a)n(v)n (ailable)g(from)g(Netlib,)h(uses)g(b)n(uck)o(eting)f(to)-198 898 y(perform)22 b(both)h(these)h(functions;)h(hence,)f(an)g Fs(O)r Fe(\()p Fq(n)14 b Fv(log)f Fq(n)p Fe(\))25 b Fv(running)-198 997 y(time)e(is)g(not)g(guaranteed,)d(and)j(Su)f(and)g(Drysdale)g([18)o (])h(found)e(that)-198 1097 y(the)14 b(original)e(implementation)g(e)o (xhibits)h Fs(O)r Fe(\()p Fq(n)1152 1067 y Fl(3)p Fc(=)p Fl(2)1249 1097 y Fe(\))h Fv(performance)d(on)-198 1197 y(uniform)20 b(random)h(point)h(sets.)36 b(By)23 b(modifying)d(F)o (ortune')-5 b(s)21 b(code)h(to)-198 1296 y(use)g(a)f(heap)g(to)h(store) f(e)n(v)o(ents,)g(the)o(y)g(obtained)f Fs(O)r Fe(\()p Fq(n)14 b Fv(log)f Fq(n)p Fe(\))22 b Fv(running)-198 1396 y(time)c(and)g(better)g(performance)d(on)j(lar)o(ge)f(point)h (sets)h(\(ha)n(ving)e(more)-198 1496 y(than)h(50,000)f(points\).)28 b(Ho)n(we)n(v)o(er)m(,)17 b(b)n(uck)o(eting)h(outperforms)e(a)k(heap) -198 1595 y(on)g(small)g(point)g(sets.)-98 1695 y(T)m(riangle')-5 b(s)27 b(implementation)f(uses)j(a)g(heap)e(as)i(well,)i(and)d(also) -198 1794 y(uses)c(a)h(splay)e(tree)h([17)o(])g(to)g(store)g(mesh)f (boundary)e(edges,)k(so)f(that)-198 1894 y(an)d Fs(O)r Fe(\()p Fq(n)14 b Fv(log)f Fq(n)p Fe(\))21 b Fv(running)e(time)h(is)i (attained,)e(re)o(gardless)f(of)h(the)h(dis-)-198 1994 y(trib)n(ution)h(of)h(points.)38 b(Not)23 b(all)h(boundary)c(edges)j (are)g(stored)g(in)g(the)-198 2093 y(splay)e(tree;)i(when)e(a)h(ne)n(w) f(edge)g(is)i(created,)e(it)h(is)h(inserted)e(into)g(the)-198 2193 y(tree)g(with)g(probability)e(0)p Fq(:)p Fv(1.)30 b(\(The)21 b(v)n(alue)f(0)p Fq(:)p Fv(1)g(w)o(as)i(chosen)e(empiri-) -198 2293 y(cally)i(to)g(minimize)f(the)h(triangulation)e(time)i(for)f (uniform)f(random)-198 2392 y(point)29 b(sets.\))57 b(At)30 b(an)o(y)e(time,)k(the)d(splay)g(tree)g(contains)g(a)h(random)-198 2492 y(sample)d(of)f(roughly)f(one)h(tenth)h(of)f(the)h(boundary)d (edges.)49 b(When)-198 2591 y(the)24 b(sweepline)g(sweeps)h(past)g(an)f (input)g(point,)g(the)h(point)f(must)g(be)-198 2691 y(located)19 b(relati)n(v)o(e)g(to)h(the)g(boundary)d(edges;)i(this)h(point)f (location)g(in-)-198 2791 y(v)n(olv)o(es)h(a)h(search)f(in)g(the)h (splay)f(tree,)g(follo)n(wed)f(by)h(a)h(search)f(on)g(the)-198 2890 y(boundary)d(of)j(the)g(triangulation)f(itself.)-98 2990 y(Splay)k(trees)h(adjust)f(themselv)o(es)g(so)h(that)g(frequently) d(accessed)-198 3090 y(items)16 b(are)f(near)g(the)g(top)g(of)h(the)f (tree.)28 b(Hence,)15 b(a)h(point)f(set)h(or)o(ganized)-198 3189 y(so)26 b(that)f(man)o(y)f(ne)n(w)h(v)o(ertices)g(appear)f(at)i (roughly)d(the)i(same)g(loca-)-198 3289 y(tion)d(on)g(the)h(boundary)d (of)i(the)g(mesh)h(is)g(lik)o(ely)g(to)f(be)h(triangulated)-198 3388 y(quickly)-5 b(.)26 b(This)17 b(ef)n(fect)g(partly)f(e)o(xplains)g (why)g(T)m(riangle')-5 b(s)16 b(sweepline)-198 3488 y(implementation)k (triangulates)h(points)g(on)h(the)g(boundary)d(of)j(a)h(cir)n(-)-198 3588 y(cle)e(more)f(quickly)f(than)i(the)f(other)g(point)g(sets,)i(e)n (v)o(en)d(though)g(there)-198 3687 y(are)32 b(man)o(y)e(more)h (boundary)e(edges)j(in)g(the)g(cocircular)e(point)h(set)-198 3787 y(and)20 b(the)g(splay)g(tree)g(gro)n(ws)g(to)g(be)g(much)g(lar)o (ger)f(\(containing)f Fs(O)r Fe(\()p Fq(n)p Fe(\))-198 3887 y Fv(boundary)f(edges)j(instead)g(of)g Fs(O)r Fe(\()792 3827 y Fs(p)p 862 3827 50 4 v 862 3887 a Fq(n)p Fe(\))p Fv(\).)-98 3986 y(T)m(riangle')-5 b(s)30 b(incremental)f(insertion)h (algorithm)f(for)h(Delaunay)-198 4086 y(triangulation)35 b(uses)k(the)e(point)g(location)g(method)f(proposed)f(by)-198 4185 y(M)-117 4184 y(\310)-124 4185 y(uck)o(e,)28 b(Saias,)h(and)e(Zhu) f([14)o(].)50 b(Their)26 b Fm(jump-and-walk)e Fv(method)-198 4285 y(chooses)19 b(a)g(random)f(sample)h(of)g Fs(O)r Fe(\()p Fq(n)909 4255 y Fl(1)p Fc(=)p Fl(3)1006 4285 y Fe(\))h Fv(v)o(ertices)f(from)f(the)h(mesh)-198 4385 y(\(where)14 b Fq(n)i Fv(is)g(the)f(number)e(of)i(nodes)g Fm(curr)m(ently)g Fv(in)g(the)g(mesh\),)g(deter)n(-)-198 4484 y(mines)f(which)g(of)g(these)g(v)o(ertices)g(is)h(closest)g(to)g (the)f(query)f(point,)h(and)-198 4584 y(w)o(alks)28 b(through)e(the)i (mesh)f(from)g(the)h(chosen)f(v)o(erte)o(x)f(to)n(w)o(ard)h(the)-198 4684 y(query)c(point)h(until)h(the)g(triangle)f(containing)e(that)j (point)f(is)i(found.)-198 4783 y(M)-117 4782 y(\310)-124 4783 y(uck)o(e)i(et)i(al.)f(sho)n(w)f(that)h(the)g(resulting)f (incremental)f(algorithm)-198 4883 y(tak)o(es)e(e)o(xpected)e Fs(O)r Fe(\()p Fq(n)466 4853 y Fl(4)p Fc(=)p Fl(3)562 4883 y Fe(\))j Fv(time)e(on)g(uniform)f(random)f(point)i(sets.)-198 4982 y(T)-7 b(able)16 b(1)h(appears)f(to)g(con\256rm)g(this)h (analysis.)28 b(T)m(riangle)15 b(uses)i(a)g(sam-)-198 5082 y(ple)25 b(size)h(of)e(0)p Fq(:)p Fv(45)p Fq(n)377 5052 y Fl(1)p Fc(=)p Fl(3)472 5082 y Fv(;)k(the)d(coef)n(\256cient)f(w) o(as)i(chosen)e(empirically)-198 5182 y(to)d(minimize)f(the)g (triangulation)f(time)i(for)f(uniform)f(random)g(point)-198 5281 y(sets.)42 b(T)m(riangle)24 b(also)g(checks)g(the)g(pre)n(viously) e(inserted)i(point,)g(be-)1992 -98 y(cause)15 b(in)g(man)o(y)f (practical)h(point)f(sets,)j(an)o(y)d(tw)o(o)i(consecuti)n(v)o(e)d (points)1992 1 y(ha)n(v)o(e)19 b(a)i(high)e(lik)o(elihood)g(of)h(being) f(near)h(each)g(other)-5 b(.)2091 101 y(A)22 b(more)f(elaborate)g (point)g(location)f(scheme)i(such)f(as)h(that)g(sug-)1992 200 y(gested)17 b(by)g(Guibas,)h(Knuth,)f(and)h(Sharir)f([9)o(])h (could)f(be)g(used)h(\(along)1992 300 y(with)29 b(randomization)d(of)i (the)h(insertion)f(order\))f(to)i(obtain)f(an)g(e)o(x-)1992 400 y(pected)16 b Fs(O)r Fe(\()p Fq(n)e Fv(log)g Fq(n)p Fe(\))k Fv(triangulation)d(algorithm,)h(b)n(ut)h(the)h(data)f(struc-) 1992 499 y(ture)27 b(used)g(for)f(location)g(is)j(lik)o(ely)e(to)g(tak) o(e)g(up)g(as)h(much)e(memory)1992 599 y(as)c(the)f(triangulation)f (itself,)i(and)e(unlik)o(ely)h(to)g(surpass)h(the)f(perfor)n(-)1992 699 y(mance)i(of)g(the)h(di)n(vide-and-conquer)19 b(algorithm;)24 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