llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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12pt; mso-bidi-font-size: 10.0pt; mso-ascii-font-family: 宋体; mso-hansi-font-family: 'Courier New'">从以下网址下载:

http://www.csiam.edu.cn/mcm  

http://www.scicomput.com/  

http://www.163.com/网易教育频道

 

A题  血管的三维重建

    断面可用于了解生物组织、器官等的形态。例如,将样本染色后切成厚约1m m的切片,在显微镜下观察该横断面的组织形态结构。如果用切片机连续不断地将样本切成数十、成百的平行切片,可依次逐片观察。根据拍照并采样得到的平行切片数字图象,运用计算机可重建组织、器官等准确的三维形态。

假设某些血管可视为一类特殊的管道,该管道的表面是由球心沿着某一曲线(称为中轴线)的球滚动包络而成。例如圆柱就是这样一种管道,其中轴线为直线,由半径固定的球滚动包络形成。

现有某管道的相继100张平行切片图象,记录了管道与切片的交。图象文件名依次为0.bmp、1.bmp、 99.bmp,格式均为BMP,宽、高均为512个象素(pixel)。为简化起见,假设:管道中轴线与每张切片有且只有一个交点;球半径固定;切片间距以及图象象素的尺寸均为1。

    取坐标系的Z轴垂直于切片,第1张切片为平面Z=0,第100张切片为平面Z=99。Z=z切片图象中象素的坐标依它们在文件中出现的前后次序为

-256,-256,z),(-256,-255,z),-256,255,z),

-255,-256,z),(-255,-255,z),-255,255,z),

……

255,-256,z),( 255,-255,z),255,255,z)。

试计算管道的中轴线与半径,给出具体的算法,并绘制中轴线在XY、YZ、ZX平面的投影图。

2页是100张平行切片图象中的6张,全部图象请从网上下载

    关于BMP图象格式可参考:

1. 《Visual C++数字图象处理》第12页2.3.1节。何斌等编著,人民邮电出版社,2001年4月。

2. http://www.dcs.ed.ac.uk/home/mxr/gfx/2d/BMP.txt

 

B    公交车调度

公共交通是城市交通的重要组成部分,作好公交车的调度对于完善城市交通环境、改进市民出行状况、提高公交公司的经济和社会效益,都具有重要意义。下面考虑一条公交线路上公交车的调度问题,其数据来自我国一座特大城市某条公交线路的客流调查和运营资料。

该条公交线路上行方向共14站,下行方向共13站,第3-4页给出的是典型的一个工作日两个运行方向各站上下车的乘客数量统计。公交公司配给该线路同一型号的大客车,每辆标准载客100 人,据统计客车在该线路上运行的平均速度为20公里/小时。运营调度要求,乘客候车时间一般不要超过10分钟,早高峰时一般不要超过5分钟,车辆满载率不应超过 120%,一般也不要低于50%

试根据这些资料和要求,为该线路设计一个便于操作的全天(工作日)的公交车调度方案,包括两个起点站的发车时刻表;一共需要多少辆车;这个方案以怎样的程度照顾到了乘客和公交公司双方的利益;等等。

如何将这个调度问题抽象成一个明确、完整的数学模型,指出求解模型的方法;根据实际问题的要求,如果要设计更好的调度方案,应如何采集运营数据。

 

(第1页完)

 

A题示意图(第2页):

 

                                                   


 

B题数据(第3-4页):

某路公交汽车各时组每站上下车人数统计表     上行方向:A13开往A0

站名

 

A13

A12

A11

A10

A9

A8

A7

A6

A5

A4

A3

A2

A1

A0

站间距(公里)

 

 

1.6

0.5

1

0.73

2.04

1.26

2.29

1

1.2

0.4

1

1.03

0.53

5:00-6:00

371

60

52

43

76

90

48

83

85

26

45

45

11

0

 

0

8

9

13

20

48

45

81

32

18

24

25

85

57

6:00-7:00

1990

376

333

256

589

594

315

622

510

176

308

307

68

0

 

0

99

105

164

239

588

542

800

407

208

300

288

921

615

7:00-8:00

3626

634

528

447

948

868

523

958

904

259

465

454

99

0

 

0

205

227

272

461

1058

1097

1793

801

469

560

636

1871

1459

8:00-9:00

2064

322

305

235

477

549

271

486

439

157

275

234

60

0

 

0

106

123

169

300

634

621

971

440

245

339

408

1132

759

9:00-10:00

1186

205

166

147

281

304

172

324

267

78

143

162

36

0

 

0

81

75

120

181

407

411

551

250

136

187

233

774

483

10:00-11:00

923

151

120

108

215

214

119

212

201

75

123

112

26

0

 

0

52

55

81

136

299

280

442

178

105

153

167

532

385

11:00-12:00

957

181

157

133

254

264

135

253

260

74

138

117

30

0

 

0

54

58

84

131

321

291

420

196

119

159

153

534

340

12:00-13:00

873

141

140

108

215

204

129

232

221

65

103

112

26

0

 

0

46

49

71

111

263

256

389

164

111

134

148

488

333

13:00-14:00

779

141

103

84

186

185

103

211

173

66

108

97

23

0

 

0

39

41

70

103

221

197

297

137

85

113

116

384

263

14:00-15:00

625

104

108

82

162

180

90

185

170

49

75

85

20

0

 

0

36

39

47

78

189

176

339

139

80

97

120

383

239

15:00-16:00

635

124

98

82

152

180

80

185

150

49

85

85

20

0

 

0

36

39

57

88

209

196

339

129

80

107

110

353

229

16:00-17:00

1493

299

240

199

396

404

210

428

390

120

208

197

49

0

 

0

80

85

135

194

450

441

731

335

157

255

251

800

557

17:00-18:00

2011

379

311

230

497

479

296

586

508

140

250

259

61

0

 

0

110

118

171

257

694

573

957

390

253

293

46

47

160

117

21:00-22:00

209

37

32

26

53

55

29

47

52

16

28

27

6

0

 

0

14

14

21

33

78

63

125

62

30

40

41

128

92

22:00-23:00

19

3

3

2

5

5

3

5

5

1

3

2

1

0

 

0

3

3

5

8

18

17

27

12

7

9

9

32

21

 

某路公交汽车各时组每站上下车人数统计表   下行方向:A0开往A13

站名

 

A0

A2

A3

A4

A5

A6

A7

A8

A9

A10

A11

A12

A13

站间距(公里)

 

 

1.56

1

0.44

1.2

0.97

2.29

1.3

2

0.73

1

492

224

404

532

333

345

354

120

153

46

0

 

0

266

158

149

756

827

856

529

367

428

237

376

1167

9:00-10:00

1556

204

274

125

235

308

162

203

198

76

99

27

0

 

0

157

100

80

410

511

498

336

199

276

136

219

556

10:00-11:00

902

147

183

82

155

206

120

150

143

50

59

18

0

 

0

103

59

59

246

346

320

191

147

185

96

154

438

11:00-12:00

847

130

132

67

127

150

108

104

107

41

48

15

0

 

0

94

48

48

199

238

256

175

122

143

68

128

346

12:00-13:00

706

90

118

66

105

144

92

95

88

34

40

12

0

 

0

70

40

40

174

215

205

127

103

119

65

98

261

13:00-14:00

770

97

126

59

102

133

97

102

104

36

43

13

0

 

0

75

43

43

166

210

209

136

90

127

60

115

309

14:00-15:00

839

133

156

69

130

165

101

118

120

42

49

15

0

 

0

84

48

48

219

238

246

155

112

153

78

118

346

15:00-16:00

1110

170

189

79

169

194

141

152

166

54

64

19

0

 

0

110

73

63

253

307

341

215

136

167

102

144

425

16:00-17:00

1837

260

330

146

305

404

229

277

253

95

122

34

0

 

0

175

96

106

459

617

549

401

266

304

162

269

784

17:00-18:00

3020

474

587

248

468

649

388

432

452

157

205

56

0

 

0

330

193

194

737

934

1016

606

416

494

278

448

1249

18:00-19:00

1966

350

399

204

328

471

289

335

342

122

132

40

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0

223

129

150

635

787

690

505

304

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246

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19:00-20:00

939

130

165

88

138

187

124

143

147

48

56

17

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0

113

59

59

266

306

290

201

147

155

86

154

398

20:00-21:00

640

107

126

69

112

153

87

102

94

36

43

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0

75

43

43

186

230

219

146

90

127

70

95

319

21:00-22:00

636

110

128

56

105

144

82

95

98

34

40

12

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0

73

41

42

190

243

192

132

107

123

67

101

290

22:00-23:00

294

43

51

24

46

58

35

41

42

15

17

5

0

 

0

35

20

20

87

108

92

69

47

60

33

49

136

AB题题目完)