Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

""" 

Plotting of Hyperplane Arrangements 

 

PLOT OPTIONS: 

 

Beside the usual plot options (enter ``plot?``), the plot command for 

hyperplane arrangements includes the following: 

 

- ``hyperplane_colors`` -- Color or list of colors, one for each 

hyperplane (default: equally spread range of hues). 

 

- ``hyperplane_labels`` -- Boolean, ``'short'``, ``'long'`` (default: 

``False``). If ``False``, no labels are shown; if 'short' or 'long', 

the hyperplanes are given short or long labels, respectively. If 

``True``, the hyperplanes are given long labels. 

 

- ``label_colors`` -- Color or list of colors, one for each hyperplane 

(default: black). 

 

- ``label_fontsize`` -- Size for hyperplane_label font (default: 

``14``). This does not work for 3d plots. 

 

- ``label_offsets`` -- Amount be which labels are offset from 

h.point() for each hyperplane h. The format is different for each 

dimension: if the hyperplanes have dimension 0, the offset can be a 

single number or a list of numbers, one for each hyperplane; if the 

hyperplanes have dimension 1, the offset can be a single 2-tuple, or 

a list of 2-tuples, one for each hyperplane; if the hyperplanes have 

dimension 2, the offset can be a single 3-tuple or a list of 

3-tuples, one for each hyperplane. (Defaults: 0-dim: ``0.1``, 

1-dim: ``(0,1)``, 2-dim: ``(0,0,0.2)``). 

 

- ``hyperplane_legend`` -- Boolean, ``'short'``, ``'long'`` (default: 

``'long'``; in 3-d: ``False``). If ``False``, no legend is shown; 

if ``True``, ``'short'``, or ``'long'``, the legend is shown with 

the default, long, or short labeling, respectively. (For 

arrangements of lines or planes, only.) 

 

- ``hyperplane_opacities`` -- A number or list of numbers, one for each 

hyperplane, between 0 and 1. Only applies to 3d plots. 

 

- ``point_sizes`` -- Number or list of numbers, one for each hyperplane 

giving the sizes of points in a zero-dimensional arrangement 

(default: ``50``). 

 

- ``ranges`` -- Range for the parameters or a list of ranges of 

parameters, one for each hyperplane, for the parametric plots of the 

hyperplanes. If a single positive number `r` is given for 

``ranges``, then all parameters run from -r to r. Otherwise, for a 

line in the plane, the range has the form ``[a,b]`` (default: 

``[-3,3]``), and for a plane in 3-space, the range has the form 

``[[a,b],[c,d]]`` (default: ``[[-3,3],[-3,3]]``). The ranges are 

centered around ``hyperplane_arrangement.point()``. 

 

EXAMPLES:: 

 

sage: H3.<x,y,z> = HyperplaneArrangements(QQ) 

sage: A = H3([(1,0,0), 0], [(0,0,1), 5]) 

sage: A.plot(hyperplane_opacities=0.5, hyperplane_labels=True, hyperplane_legend=False) 

Graphics3d Object 

 

sage: c = H3([(1,0,0),0], [(0,0,1),5]) 

sage: c.plot(ranges=10) 

Graphics3d Object 

sage: c.plot(ranges=[[9.5,10], [-3,3]]) 

Graphics3d Object 

sage: c.plot(ranges=[[[9.5,10], [-3,3]], [[-6,6], [-5,5]]]) 

Graphics3d Object 

 

 

sage: H2.<s,t> = HyperplaneArrangements(QQ) 

sage: h = H2([(1,1),0], [(1,-1),0], [(0,1),2]) 

sage: h.plot(ranges=20) 

Graphics object consisting of 3 graphics primitives 

sage: h.plot(ranges=[-1, 10]) 

Graphics object consisting of 3 graphics primitives 

sage: h.plot(ranges=[[-1, 1], [-5, 5], [-1, 10]]) 

Graphics object consisting of 3 graphics primitives 

 

sage: a = hyperplane_arrangements.coordinate(3) 

sage: opts = {'hyperplane_colors':['yellow', 'green', 'blue']} 

sage: opts['hyperplane_labels'] = True 

sage: opts['label_offsets'] = [(0,2,2), (2,0,2), (2,2,0)] 

sage: opts['hyperplane_legend'] = False 

sage: opts['hyperplane_opacities'] = 0.7 

sage: a.plot(**opts) 

Graphics3d Object 

sage: opts['hyperplane_labels'] = 'short' 

sage: a.plot(**opts) 

Graphics3d Object 

 

sage: H.<u> = HyperplaneArrangements(QQ) 

sage: pts = H(3*u+4, 2*u+5, 7*u+1) 

sage: pts.plot(hyperplane_colors=['yellow','black','blue']) 

Graphics object consisting of 3 graphics primitives 

sage: pts.plot(point_sizes=[50,100,200], hyperplane_colors='blue') 

Graphics object consisting of 3 graphics primitives 

 

sage: H.<x,y,z> = HyperplaneArrangements(QQ) 

sage: a = H(x, y+1, y+2) 

sage: a.plot(hyperplane_labels=True,label_colors='blue',label_fontsize=18) 

Graphics3d Object 

sage: a.plot(hyperplane_labels=True,label_colors=['red','green','black']) 

Graphics3d Object 

""" 

from __future__ import print_function 

 

from copy import copy 

from colorsys import hsv_to_rgb 

from sage.plot.plot3d.parametric_plot3d import parametric_plot3d 

from sage.plot.plot3d.shapes2 import text3d 

from sage.plot.graphics import Graphics 

from sage.plot.line import line 

from sage.plot.text import text 

from sage.plot.point import point 

from sage.plot.plot import parametric_plot 

from sage.symbolic.all import SR 

 

 

def plot(hyperplane_arrangement, **kwds): 

r""" 

Return a plot of the hyperplane arrangement.  

 

If the arrangement is in 4 dimensions but inessential, a plot of 

the essentialization is returned. 

 

.. NOTE:: 

 

This function is available as the 

:meth:`~sage.geometry.hyperplane_arrangement.arrangement.HyperplaneArrangementElement.plot` 

method of hyperplane arrangements. You should not call this 

function directly, only through the method. 

 

INPUT: 

 

- ``hyperplane_arrangement`` -- the hyperplane arrangement to plot 

 

- ``**kwds`` -- plot options: see 

:mod:`sage.geometry.hyperplane_arrangement.plot`. 

 

OUTPUT: 

 

A graphics object of the plot. 

 

EXAMPLES:: 

 

sage: B = hyperplane_arrangements.semiorder(4) 

sage: B.plot() 

Displaying the essentialization. 

Graphics3d Object 

""" 

N = len(hyperplane_arrangement) 

dim = hyperplane_arrangement.dimension() 

if hyperplane_arrangement.base_ring().characteristic() != 0: 

raise NotImplementedError('must be a field of characteristic 0') 

elif dim == 4: 

if not hyperplane_arrangement.is_essential(): 

print('Displaying the essentialization.') 

hyperplane_arrangement = hyperplane_arrangement.essentialization() 

elif dim not in [1,2,3]: # revise to handle 4d 

return # silently 

# handle extra keywords 

if 'hyperplane_colors' in kwds: 

hyp_colors = kwds.pop('hyperplane_colors') 

if not isinstance(hyp_colors, list): # we assume its a single color then 

hyp_colors = [hyp_colors] * N 

else: 

HSV_tuples = [(i*1.0/N, 0.8, 0.9) for i in range(N)] 

hyp_colors = [hsv_to_rgb(*x) for x in HSV_tuples] 

if 'hyperplane_labels' in kwds: 

hyp_labels = kwds.pop('hyperplane_labels') 

has_hyp_label = True 

if not isinstance(hyp_labels, list): # we assume its a boolean then 

hyp_labels = [hyp_labels] * N 

relabeled = [] 

for i in range(N): 

if hyp_labels[i] in [True,'long']: 

relabeled.append(True) 

else: 

relabeled.append(str(i)) 

hyp_labels = relabeled 

else: 

has_hyp_label = False 

if 'label_colors' in kwds: 

label_colors = kwds.pop('label_colors') 

has_label_color = True 

if not isinstance(label_colors, list): # we assume its a single color then 

label_colors = [label_colors] * N 

else: 

has_label_color = False 

if 'label_fontsize' in kwds: 

label_fontsize = kwds.pop('label_fontsize') 

has_label_fontsize = True 

if not isinstance(label_fontsize, list): # we assume its a single size then 

label_fontsize = [label_fontsize] * N 

else: 

has_label_fontsize = False 

if 'label_offsets' in kwds: 

has_offsets = True 

offsets = kwds.pop('label_offsets') 

else: 

has_offsets = False # give default values below 

hyperplane_legend = kwds.pop('hyperplane_legend', 'long' if dim < 3 else False) 

if 'hyperplane_opacities' in kwds: 

hyperplane_opacities = kwds.pop('hyperplane_opacities') 

has_opacity = True 

if not isinstance(hyperplane_opacities, list): # we assume a single number then 

hyperplane_opacities = [hyperplane_opacities] * N 

else: 

has_opacity = False 

point_sizes = kwds.pop('point_sizes', 50) 

if not isinstance(point_sizes, list): 

point_sizes = [point_sizes] * N 

if 'ranges' in kwds: 

ranges_set = True 

ranges = kwds.pop('ranges') 

if not type(ranges) in [list,tuple]: # ranges is a single number 

ranges = [ranges] * N 

# So ranges is some type of list. 

elif dim == 2: # arrangement of lines in the plane 

if not type(ranges[0]) in [list,tuple]: # a single interval 

ranges = [ranges] * N 

elif dim == 3: # arrangement of planes in 3-space 

if not type(ranges[0][0]) in [list,tuple]: 

ranges = [ranges] * N 

elif dim not in [2,3]: # ranges is not an option unless dim is 2 or 3 

ranges_set = False 

else: # a list of intervals, one for each hyperplane is given 

pass # ranges does not need to be modified 

else: 

ranges_set = False # give default values below 

# the extra keywords have now been handled 

# now handle the legend 

if dim in [1,2]: # points on a line or lines in the plane 

if hyperplane_legend in [True,'long']: 

hyps = hyperplane_arrangement.hyperplanes() 

legend_labels = [hyps[i]._latex_() for i in range(N)] 

elif hyperplane_legend == 'short' : 

legend_labels = [str(i) for i in range(N)] 

else: # dim==3, arrangement of planes in 3-space 

if hyperplane_legend in [True, 'long']: 

legend3d = legend_3d(hyperplane_arrangement, hyp_colors, 'long') 

elif hyperplane_legend == 'short': 

legend3d = legend_3d(hyperplane_arrangement, hyp_colors, 'short') 

## done handling the legend 

## now create the plot 

p = Graphics() 

for i in range(N): 

newk = copy(kwds) 

if has_hyp_label: 

newk['hyperplane_label'] = hyp_labels[i] 

if has_offsets: 

if not isinstance(offsets, list): 

newk['label_offset'] = offsets 

else: 

newk['label_offset'] = offsets[i] 

else: 

newk['hyperplane_label'] = False 

if has_label_color: 

newk['label_color'] = label_colors[i] 

if has_label_fontsize: 

newk['label_fontsize'] = label_fontsize[i] 

if has_opacity: 

newk['opacity'] = hyperplane_opacities[i] 

if dim == 1: 

newk['point_size'] = point_sizes[i] 

if dim in [1,2] and hyperplane_legend: # more options than T/F 

newk['legend_label'] = legend_labels[i] 

if ranges_set: 

newk['ranges'] = ranges[i] 

p += plot_hyperplane(hyperplane_arrangement[i], rgbcolor=hyp_colors[i], **newk) 

if dim == 1: 

if hyperplane_legend: # there are more options than T/F 

p.legend(True) 

return p 

elif dim == 2: 

if hyperplane_legend: # there are more options than T/F 

p.legend(True) 

return p 

else: # dim==3 

if hyperplane_legend: # there are more options than T/F 

return p, legend3d 

else: 

return p 

 

 

 

 

 

def plot_hyperplane(hyperplane, **kwds): 

r""" 

Return the plot of a single hyperplane. 

 

INPUT: 

 

- ``**kwds`` -- plot options: see below 

 

OUTPUT: 

 

A graphics object of the plot. 

 

.. RUBRIC:: Plot Options 

 

Beside the usual plot options (enter ``plot?``), the plot command for 

hyperplanes includes the following: 

 

- ``hyperplane_label`` -- Boolean value or string (default: ``True``). 

If ``True``, the hyperplane is labeled with its equation, if a 

string, it is labeled by that string, otherwise it is not 

labeled. 

 

- ``label_color`` -- (Default: ``'black'``) Color for hyperplane_label. 

 

- ``label_fontsize`` -- Size for ``hyperplane_label`` font (default: 14) 

(does not work in 3d, yet). 

 

- ``label_offset`` -- (Default: 0-dim: 0.1, 1-dim: (0,1), 

2-dim: (0,0,0.2)) Amount by which label is offset from 

``hyperplane.point()``. 

 

- ``point_size`` -- (Default: 50) Size of points in a zero-dimensional 

arrangement or of an arrangement over a finite field. 

 

- ``ranges`` -- Range for the parameters for the parametric plot of the 

hyperplane. If a single positive number ``r`` is given for the 

value of ``ranges``, then the ranges for all parameters are set to 

`[-r, r]`. Otherwise, for a line in the plane, ``ranges`` has the 

form ``[a, b]`` (default: [-3,3]), and for a plane in 3-space, the 

``ranges`` has the form ``[[a, b], [c, d]]`` (default: [[-3,3],[-3,3]]). 

(The ranges are centered around ``hyperplane.point()``.) 

 

EXAMPLES:: 

 

sage: H1.<x> = HyperplaneArrangements(QQ) 

sage: a = 3*x + 4 

sage: a.plot() # indirect doctest 

Graphics object consisting of 3 graphics primitives 

sage: a.plot(point_size=100,hyperplane_label='hello') 

Graphics object consisting of 3 graphics primitives 

 

 

sage: H2.<x,y> = HyperplaneArrangements(QQ) 

sage: b = 3*x + 4*y + 5 

sage: b.plot() 

Graphics object consisting of 2 graphics primitives 

sage: b.plot(ranges=(1,5),label_offset=(2,-1)) 

Graphics object consisting of 2 graphics primitives 

sage: opts = {'hyperplane_label':True, 'label_color':'green', 

....: 'label_fontsize':24, 'label_offset':(0,1.5)} 

sage: b.plot(**opts) 

Graphics object consisting of 2 graphics primitives 

 

sage: H3.<x,y,z> = HyperplaneArrangements(QQ) 

sage: c = 2*x + 3*y + 4*z + 5 

sage: c.plot() 

Graphics3d Object 

sage: c.plot(label_offset=(1,0,1), color='green', label_color='red', frame=False) 

Graphics3d Object 

sage: d = -3*x + 2*y + 2*z + 3 

sage: d.plot(opacity=0.8) 

Graphics3d Object 

sage: e = 4*x + 2*z + 3 

sage: e.plot(ranges=[[-1,1],[0,8]], label_offset=(2,2,1), aspect_ratio=1) 

Graphics3d Object 

""" 

if hyperplane.base_ring().characteristic() != 0: 

raise NotImplementedError('base field must have characteristic zero') 

elif hyperplane.dimension() not in [0, 1, 2]: # dimension of hyperplane, not ambient space 

raise ValueError('can only plot hyperplanes in dimensions 1, 2, 3') 

# handle extra keywords 

if 'hyperplane_label' in kwds: 

hyp_label = kwds.pop('hyperplane_label') 

if not hyp_label: 

has_hyp_label = False 

else: 

has_hyp_label = True 

else: # default 

hyp_label = True 

has_hyp_label = True 

if has_hyp_label: 

if hyp_label: # then label hyperplane with its equation 

if hyperplane.dimension() == 2: # jmol does not like latex 

label = hyperplane._repr_linear(include_zero=False) 

else: 

label = hyperplane._latex_() 

else: 

label = hyp_label # a string 

if 'label_color' in kwds: 

label_color = kwds.pop('label_color') 

else: 

label_color = 'black' 

if 'label_fontsize' in kwds: 

label_fontsize = kwds.pop('label_fontsize') 

else: 

label_fontsize = 14 

if 'label_offset' in kwds: 

has_offset = True 

label_offset = kwds.pop('label_offset') 

else: 

has_offset = False # give default values below 

if 'point_size' in kwds: 

pt_size = kwds.pop('point_size') 

else: 

pt_size = 50 

if 'ranges' in kwds: 

ranges_set = True 

ranges = kwds.pop('ranges') 

else: 

ranges_set = False # give default values below 

# the extra keywords have now been handled 

# now create the plot 

if hyperplane.dimension() == 0: # a point on a line 

x, = hyperplane.A() 

d = hyperplane.b() 

p = point((d/x,0), size = pt_size, **kwds) 

if has_hyp_label: 

if not has_offset: 

label_offset = 0.1 

p += text(label, (d/x,label_offset), 

color=label_color,fontsize=label_fontsize) 

p += text('',(d/x,label_offset+0.4)) # add space at top 

if 'ymax' not in kwds: 

kwds['ymax'] = 0.5 

elif hyperplane.dimension() == 1: # a line in the plane 

pnt = hyperplane.point() 

w = hyperplane.linear_part().matrix() 

x, y = hyperplane.A() 

d = hyperplane.b() 

t = SR.var('t') 

if ranges_set: 

if type(ranges) in [list,tuple]: 

t0, t1 = ranges 

else: # ranges should be a single positive number 

t0, t1 = -ranges, ranges 

else: # default 

t0, t1 = -3, 3 

p = parametric_plot(pnt+t*w[0], (t,t0,t1), **kwds) 

if has_hyp_label: 

if has_offset: 

b0, b1 = label_offset 

else: 

b0, b1 = 0, 0.2 

label = text(label,(pnt[0]+b0,pnt[1]+b1), 

color=label_color,fontsize=label_fontsize) 

p += label 

elif hyperplane.dimension() == 2: # a plane in 3-space 

pnt = hyperplane.point() 

w = hyperplane.linear_part().matrix() 

a, b, c = hyperplane.A() 

d = hyperplane.b() 

s,t = SR.var('s t') 

if ranges_set: 

if type(ranges) in [list,tuple]: 

s0, s1 = ranges[0] 

t0, t1 = ranges[1] 

else: # ranges should be a single positive integers 

s0, s1 = -ranges, ranges 

t0, t1 = -ranges, ranges 

else: # default 

s0, s1 = -3, 3 

t0, t1 = -3, 3 

p = parametric_plot3d(pnt+s*w[0]+t*w[1],(s,s0,s1),(t,t0,t1),**kwds) 

if has_hyp_label: 

if has_offset: 

b0, b1, b2 = label_offset 

else: 

b0, b1, b2 = 0, 0, 0 

label = text3d(label,(pnt[0]+b0,pnt[1]+b1,pnt[2]+b2), 

color=label_color,fontsize=label_fontsize) 

p += label 

return p 

 

 

 

 

 

def legend_3d(hyperplane_arrangement, hyperplane_colors, length): 

r""" 

Create plot of a 3d legend for an arrangement of planes in 3-space. The 

``length`` parameter determines whether short or long labels are used in 

the legend. 

 

INPUT: 

 

- ``hyperplane_arrangement`` -- a hyperplane arrangement 

 

- ``hyperplane_colors`` -- list of colors 

 

- ``length`` -- either ``'short'`` or ``'long'`` 

 

OUTPUT: 

 

- A graphics object. 

 

EXAMPLES:: 

 

sage: a = hyperplane_arrangements.semiorder(3) 

sage: from sage.geometry.hyperplane_arrangement.plot import legend_3d 

sage: legend_3d(a, list(colors.values())[:6],length='long') 

Graphics object consisting of 6 graphics primitives 

 

sage: b = hyperplane_arrangements.semiorder(4) 

sage: c = b.essentialization() 

sage: legend_3d(c, list(colors.values())[:12], length='long') 

Graphics object consisting of 12 graphics primitives 

 

sage: legend_3d(c, list(colors.values())[:12], length='short') 

Graphics object consisting of 12 graphics primitives 

 

sage: p = legend_3d(c, list(colors.values())[:12], length='short') 

sage: p.set_legend_options(ncol=4) 

sage: type(p) 

<class 'sage.plot.graphics.Graphics'> 

""" 

if hyperplane_arrangement.dimension() != 3: 

raise ValueError('arrangements must be in 3-space') 

hyps = hyperplane_arrangement.hyperplanes() 

N = len(hyperplane_arrangement) 

if length == 'short': 

labels = [' ' + str(i) for i in range(N)] 

else: 

labels = [' ' + hyps[i]._repr_linear(include_zero=False) for i in 

range(N)] 

p = Graphics() 

for i in range(N): 

p += line([(0,0),(0,0)], color=hyperplane_colors[i], thickness=8, 

legend_label=labels[i], axes=False) 

p.set_legend_options(title='Hyperplanes', loc='center', labelspacing=0.4, 

fancybox=True, font_size='x-large', ncol=2) 

p.legend(True) 

return p