Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics

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Index 949

Semi-locally simply connected space, 430

Sense

of angle, 900

Sense preserving

linear transformation, 27

Sense reversing

linear transformation, 27

Separable extension ﬁeld, 857

Separable ﬁeld element, 857

Separable polynomial, 857

Separable space, 297

Sequential continuity, 296

Series

absolutely convergent, 892

conditionally convergent, 892

convergent, 892

divergent, 892

Serret, Joseph Alfred (1819–1885)

Serret-Frenet formulas, 576, 649, 666

generalized, 577, 654, 673

Shape operator

of surface, 605

Shear

in x-direction, 96

in y-direction, 96

Siebenmann, Laurence Carl, 339

Sign function

of permutation, 815, 823

of real number, 815

Signed curvature

of plane curve, 566, 569, 571

Similar matrices, 868

Similarity, 94

extended, 203

Similarity transformation, 94

Simple closed curve, 569

Simple closed curves

on combinatorial surface, 408

Simple extension

of ﬁeld, 850

Simple homotopy equivalence, 438

Simple point

of plane curve, 705

Simple root, 848

Simpler polynomial, 735

Simplex

abstract, 334

back k-face of, 449

face of, 31

front k-face of, 449

k-dimensional, 31

oriented, 362

standard, 450

vertex of, 31

Simplicial approximation, 380

Simplicial approximation theorem, 381, 417,

442

Simplicial complex, 328

abstract, 334

boundary of, 330

connected, 330

deﬁned by labeled ﬁgure, 336

determined by simplex, 332

dimension of, 328

ﬁnite, 328

inﬁnite, 331

labeled, 335

subcomplex of, 330

subdivision of, 328

underlying space of, 328

Simplicial complexes

combinatorially equivalent, 339

isomorphic, 332

Simplicial map, 37, 332

induced map of underlying spaces, 37,

333

Simply connected space, 420

Simpson, Thomas (1710–1761)

Simpson’s rule, 908

Sine function, 901

Singular cohomology group, 452

Singular homology group, 451

Singular k-chain, 274

boundary of, 274

in manifold, 539

Singular k-cube, 273

face of, 273

in manifold, 539

standard, 273

Singular point

neighborhoods of, 708

of analytic element, 749

of plane curve, 705

of variety, 798

Singular q-chains

group of, 450

Singular q-simplex, 450

ith face of, 451

Singular value decomposition

of matrix, 57

Singular values

of matrix, 57

950 Index

Singularity

blowing up, 760

isolated, 903

removable, 903

resolving, 753

Skew ﬁeld, 847

Skew lines, 21

Slope, 2, 3

Smale, Stephen, 497

Smallest element

of total order, 821

Smallest subﬁeld

containing ﬁeld and set, 849

of ﬁeld, 847

Smooth function, 224

Smooth manifold, 466, 501

Smooth point

of variety, 791

Smooth variety, 792

, 814

as regular CW complex, 453

SO(n), 13

Solid angle, 603

measure of, 603

Solving equations

using Gröbner bases, 743

Space curve, 573, 784

afﬁne, 783

fundamental theorem of, 577

projective, 783

twisted cubic, 670, 785

Span

of set, 853

of vectors, 853

Spanning

set, 853

vectors, 853

Special orthogonal group, 13

Special orthogonal matrix, 13

Special unitary group, 14

Special unitary matrix, 14

Spectral sequence, 453

Spectral theorem, 37

Spectrum, 37

Speed

of parametric curve, 475

Sphere

geodesics of, 627

homology, 438

homotopy, 438

n-dimensional, 814

pinched, 439

with crosscaps, 350

with handles, 350

with twisted handles, 353

Sphere bundle, 517

Sphere-preserving map, 202

Spherical coordinates, 552

Spherical modiﬁcation, 497

Spine curve

for cyclide, 640

Splitting ﬁeld, 857

S-polynomial, 740

Square knot, 421

Standard basis

of polynomial ideal, 738

of R

, 813

Standard C

•

structure

on R

, 501

Standard coordinate system

for P

, 147

for projective plane, 150

Standard coordinates

for P

, 147

for projective plane, 150

Standard CW structure

for P

, 453

for S

, 453

Standard Euclidean metric, 283

Standard frame ﬁeld, 651

Standard imbedding

of k

in P

(k), 685

of R

in P

, 140, 159, 549

Standard normal vector ﬁeld

induced by standard orientation,

486

Standard n-simplex, 450

Standard orientation

induced by parameterization, 484

of manifold, 484

of R

, 27, 51

Standard parameterization

of projective line, 147

of projective plane, 150

Standard projection

of P

onto R

, 160, 549

Standard topology

of R

, 290

Star

of simplex, 339

Index 951

Star-shaped region, 320

Staudt, Karl Georg von (1798–1867), 144

Steenrod, Norman Earl (1910–1971), 384,

452

Stereographic projection, 201, 202, 386, 502,

524

Stiefel manifold, 550

Stiefel variety, 550

Stokes, Sir George Gabriel (1819–1903)

Stokes theorem, 277, 541, 545

Straight line

properties of, 621

Striction curve

of ruled surface, 647, 673

String construction

for cyclide, 641

for ellipse, 167

for ellipsoid, 192

for involute, 584

Structural equations

ﬁrst, 656, 658, 659

second, 656, 658, 659

SU(n), 14

Subcomplex

of CW complex, 392

of simplicial complex, 330

Subcover, 304

Subdivision

of simplicial complex, 328

Subﬁeld, 847

Subgroup, 825

generated by elements, 831

normal, 826

Subinterval

of partition, 890

Submanifold, 473, 503

Subrectangle

of partition, 256

Subring, 835

Subspace

of topological space, 292

of vector space, 852

Substitution

of formal power series, 844

Sum

direct, 861

of abelian subgroups, 832

of ideals, 838

of subsets, 861

Sup, 889

Support

of cross-section, 511

of function, 218

of q-chain, 407

Support plane, 603

Supremum, 889

Surface, 298

area of, 548

bordered, 353

boundary of, 353

closed, 340, 353

combinatorial, 338, 353

combinatorial with boundary, 353

convex, 602

convexity at point, 602

developable, 648

doubly ruled, 192, 645

ﬁrst fundamental form of, 590

folded, 604

genus of, 350, 356

geodesically complete, 633

noncompact, 354

noncylindrical, 645

nonorientable, 23, 24 351

offset, 643, 669

one-sided, 23, 352

orientable, 23, 24, 350, 351

oriented, 30

parallel, 643

quadratic, 190

quadric, 190

ruled, 645

second fundamental form of, 605

topological, 298

two-sided, 22

with boundary, 353

without boundary, 338

Surface of centers, 600

Surface of revolution, 23, 192

circle of latitude of, 24

geodesics of, 628

meridian of, 23

quadric, 192

Surgery

on manifolds, 497

Surjection, 819

Suspension

homology groups of, 456

of space, 303

Sweep surface, 638

952 Index

Sylvester, James Joseph (1819–1897)

Sylvester matrix, 690

Sylvester resultant, 691

Symbol for surface, 343, 346

Symmetric difference

of sets, 814

Symmetric function, 823

Symmetric group

of degree n, 821

Symmetric polynomial, 845

elementary, 845

Symmetric linear transformation, 40

Symmetry

plane of, 672

Symmetry equation, 658

Synthetic geometry, 102

Tangent bundle

of manifold, 519, 520

Tangent bundle map

induced by map, 524

Tangent line

geometric intepretation of, 703

to conic, 184

to curve, 703, 705

to graph, 219

Tangent plane

at point of manifold, 476

to graph, 219

Tangent space

at point of manifold, 476, 506, 507

at point of R

, 268

caution for deﬁnition of, 476

Tangent surface

of curve, 583

Tangent vector

as contravariant tensor, 507

at point of curve, 475, 506

at point of manifold, 476, 506, 507

at point of R

, 268

caution for deﬁnition of, 476

ith component of, 506, 508

notation for, 650

Tangential vector ﬁeld, 484

Taxicab metric

on R

, 283

Taylor, Brook (1685–1731)

Taylor polynomial, 230, 893

Taylor polynomial theorem, 230, 893

Taylor series, 894

complex, 903

T-closed set, 291

Tensor, 878

alternating, 881

contravariant, 507, 878

covariant, 507, 878

mixed, 878

of type (r,s), 878

Tensor algebra

of vector space, 878

Tensor product

of linear transformations, 879

of multilinear maps, 879

of vector space, 878

of vector spaces, 875

of vectors, 877

Tetrahedron, 322, 324

Theorema Egregrium, 619

Tietze, Heinrich (1880–1964), 399

Tietze Extension Lemma, 314

T-open set, 290

Topological invariant, 294, 358

Topological manifold, 297, 339, 340

boundary of, 297

importance of 2nd countability of, 298

interior of, 297

Topological property, 296

Topological space, 290

Topologically invariant function, 394

Topologist, 217, 296, 327

Topology

base for, 290

induced by metric, 290

on set, 289

what it is, 208, 290, 321, 326

Torsion

of space curve, 575

of regular space curve, 577

Torsion coefﬁcient

of polyhedron, 375

of simplicial complex, 374

Torsion coefﬁcients

of abelian group, 832

Torsion-free group, 831

Torsion subgroup, 831

Torus, 23, 24

geodesics of, 630

Torus knot

of type (p,q), 421, 708

Index 953

Total curvature

of plane curve, 571, 665

Total degree

of monomial, 843

of polynomial, 843

Total Gauss curvature, 659

Total order, 821

strict, 821

Total space

of bundle, 422

of vector bundle, 570

Trace

of linear transformation, 869

of matrix, 867

Traced out points

from function, 819

Transcendence basis, 857

Transcendence degree, 857

Transcendental element

over ﬁeld, 856

over subring, 841

Transcendental extension ﬁeld, 856

Transformation

barycentric coordinate preserving, 109

change of coordinates, 148, 151, 152,

464

change of parameters, 464

deﬁned by frame, 89, 111

quadratic, 759

Transformed curve, 761

Transformed frame, 110

Transformed parameterization, 464, 761

Translation, 67

Translation vector, 67

Transposition, 821

Transverse

planes, 21

Transverse intersection

of chains, 448

of homology classes, 448

Tranverse

to submanifold, 529

Trapezoidal rule, 908

Trefoil knot, 325

Triangle inequality, 4, 283, 864

Triangulation, 330

inﬁnite, 331

minimal, 356

of C

manifolds, 339, 340, 468

proper, 338, 339

Triangulation problem, 339

Triple point

of plane curve, 705

Triple product, 62

Trivial bundle, 423

Tube surface, 638

Tubular neighborhood, 527

closed, 527

Turning angle

of planar curve, 569

Turning point

of planar curve, 588

Twisted cubic, 670, 785

Twisted handle, 353

Tychonoff, Andrei Nikolaevich (1906–1993)

Tychonoff product theorem, 305

U(n), 14

UFD, 837

Umbilical point

of surface, 610

Unbounded set, 212

Underlying space, 37

of CW complex, 391

of parameterization, 460

of simplicial complex, 328

Uniformization, 751

Uniformization problem, 751

Uniformizing variable, 751

Uniformly continuous function, 216, 287

Uniformly convergent functions, 288

Unique factorization domain, 837

Unit

in ring, 835

Unit disk

closed, 814

open, 814

Unit sphere, 814

Unit tangent vectors

of curve, 475

Unit vector ﬁeld

on manifold, 484

Unitary group, 14

special, 14

Unitary matrix, 14

special, 14

Universal coefﬁcient theorem, 406

Universal cover, 430

Universal covering space, 430

954 Index

Universal factorization property

of exterior product, 884

of free abelian group, 834

of free group, 830

of free vector space, 855

of polynomial rings, 842

of tensor product, 875

u-parameter curve, 478

Upper bound

of set of real numbers, 889

Upper halfplane, 813

Upper hemisphere, 814

Upper sum, 257

Urysohn, Paul S. (1898–1924)

Urysohn Lemma, 313

Valuation ring, 763

Value

of polynomial, 763

of rational function, 769, 778

Vanishing line, 129

Vanishing point, 127, 129, 158

Variety, 468, 675, 676

afﬁne, 468, 675

as closed set, 724

codimension of, 792

complex, 675

degree of, 800

dimension at point, 792

dimension of, 792

in P

(k), 676

irreducible, 699

nonsingular, 798

nonsingular at point, 798

of a set of polynomials, 715

projective, 676, 807

quasiprojective, 724

rational afﬁne, 771

real, 675

reducible, 699

singular at point, 798

smooth, 792

smooth point of, 791

Varieties

properly intersecting, 799

Vect (M), 521

Vect (M,h), 660

Vector, 851

contravariant, 878

covariant, 878

from p to q, 864

orthogonal to hyperplane, 18

orthogonal to plane, 19

parallel to hyperplane, 18

parallel to plane, 19

unit, 863

Vector addition, 824, 851

Vector bundle, 510

base space of, 510

cross-section, 510

ﬁber of, 510

induced, 515

induced by map, 515

local coordinate chart of, 510

local triviality of, 510

nonzero cross-section of, 510

orientable, 517

orientation of, 517

oriented, 517

product, 510

projection, 510

pullback, 515

restriction of, 510

total space of, 510

trivial, 512

zero cross-section of, 510

Vector bundle isomorphism, 512

inverse of, 512

Vector bundle map, 511

orientation-preserving, 517

orientation-reversing, 518

Vector bundles

classiﬁcation of, 550, 551

isomorphic, 512

product of, 514

Whitney sum of, 516

Vector ﬁeld

along curve, 634

covariant derivative of, 634

, 484

deﬁned over manifold, 484

deﬁned over subset, 521

derivative of, 650

differentiable, 634

for R

, 269

gradient, 522, 556

nonzero, 484

normal to manifold, 484

of manifold, 484, 521

Index 955

parallel, 662

parallel along curve, 634

tangential to manifold, 484

unit, 484

Vector space, 851

codimension of, 854

dimension of, 854

oriented, 25

quotient, 855

Velocity

of parametric curve, 475

Veronese imbedding, 779

Veronese variety, 779, 780

Vertex

of abstract simplicial complex, 334

of cone, 166

of conic section, 168

of plane curve, 570

of polygon, 332

of simplex, 31

View direction

for generalized frame, 197

View plane

for generalized frame, 197

origin of, 197

View plane coordinate system, 197

Volume

of Jordan-measurable set, 259

of manifold, 545, 596, 667

of parallelotope, 263

of parameterization, 594

of rectangle, 256

Volume element, 542, 547, 882

v-parameter curve, 478

Weak Morse inequalities, 495

Weak topology, 302

Wedge

of maps, 435

of spaces, 303

Wedge product, 266, 882

of differential k-form, 536

of exterior forms, 886

Weierstrass, Karl Theodor Wilhelm

(1815–1897), 748, 749

Weight function, 910

Weingarten, Julius (1836–1910)

Weingarten equations, 614, 643, 668

Weingarten map, 604

Well-ordering, 821

Whitehead, John Henry Constantine

(1904–1960), 391, 399

Whitney, Hassler (1907–1989), 504

Whitney imbedding theorem, 504, 804

Whitney sum

of vector bundles, 516

Winding number, 572

World coordinates, 88, 111

Wu-Ritt method

for implicitization, 744

Yaw angle

of rotation, 114

Zariski topology, 723, 724, 767, 770, 800,

808, 811

Zero

of polynomial, 844

Zero polynomial, 841

Zero power series, 841

Zeros

ﬁnding, 252

[AbdY96], 255, 919

[Abhy90], 786, 788, 789, 916

[Abik81], 751, 917

[AdaL94], 728, 735, 737, 738, 739, 742, 744,

916

[AgoM05], v, vi, 86, 331, 687, 896, 908,

919

[AgoM76], 326, 342, 346, 348, 349, 354, 363,

375, 381, 382, 383, 387, 394, 439, 440, 444,

488, 572, 916

[AgoM76b], 460, 918

[Ahlf66], 898, 900, 901, 902, 903, 905, 906,

917

[Alex96], 253, 919

[AppH77], 324, 919

[Arms83], 916

[Arno83], 916

[AusM63], 875, 876, 877, 880, 881, 882, 883,

884, 885, 918

[Ayre67], 126, 920

[BaGM82], 589, 918

[Baja92a], 915

[Baja92b], 915

[BanW83], 919

[Barr72], 920

[BehS62], 917

[Berb66], 886, 920

[Berg03], 918

[BisC64], 508, 918

[Blin98], 150, 920

[Boeh89], 917

[Boeh90], 640, 917

[BoeP94], 36, 919

[Brad75], 919

[Brec92], 582, 917

[BriK81], 708, 710, 711, 713, 714, 752, 753,

775, 804, 916

[Buck78], 216, 229, 231, 241, 243, 256, 258,

259, 260, 261, 265, 890, 891, 892, 893,

915

[BurM71], 355, 397, 920

[Cair68], 399, 404, 411, 417, 419, 441, 442,

916

[Call86], 573, 640, 917

[ChDH89], 640, 641, 642, 917

[ChiS66], 920

[CodL55], 894, 895, 896, 915

[CoLO97], 717, 718, 723, 724, 728, 735, 739,

742, 744, 745, 746, 767, 768, 771, 772, 785,

916

[CoLO98], 695, 728, 916

[ConD72], 254, 908, 910, 911, 913, 919

[CooF67], 393, 394, 433, 455, 916

[CroF65], 420, 916

[Crom97], 604, 920

[DahB74], 919

[Dean66], 829, 831, 832, 842, 848, 850, 854,

858, 915

[Dieu89], 438, 916

[Divi75], 192, 919

[DoCa76], 569, 570, 571, 605, 607, 609, 610,

618, 620, 917

[Dupi22], 640, 641, 917

Bibliographic Index

Bibliographic Index 957

[Eise39], 167, 168, 176, 917

[Eise74], 216, 217, 281, 287, 289, 290, 291,

294, 296, 297, 299, 300, 301, 302, 305, 306,

307, 308, 309, 313, 920

[Fari87], 916, 919

[Fari95], 199, 920

[FarN90a], 587, 588, 589, 920

[Fink72], 48, 49, 919

[Flan63], 265, 915

[ForM67], 57, 920

[Fral67], 830, 836, 838, 915

[Free82], 498, 920

[FreF67], 916

[FreL89], 503, 918

[Full73], 168, 917

[Fult69], 808, 916

[Gans69], 66, 81, 99, 101, 126, 134, 135, 136,

152, 199, 920

[Gluc71], 570, 918

[Gott96], 324, 916

[Gray98], 569, 570, 571, 572, 578, 600, 609,

639, 918

[Greg86], 917, 918, 919

[GrüS94], 324, 916

[GuiP74], 265, 536, 548, 918

[Harr92], 775, 776, 789, 793, 796, 804,

916

[Hick65], 633, 918

[HilC99], 166, 167, 168, 192, 200, 201, 202,

203, 204, 205, 557, 600, 604, 640, 642,

919

[Hild87], 908, 920

[HilP94], 324, 916

[HilP96], 324, 916

[HilW60], 399, 433, 916

[Hirs76], 504, 519, 527, 528, 531, 532, 533,

534, 535, 918

[Hobb91], 728, 916

[Hoff89], 789, 790, 919

[Hoff93], 744, 916

[HofK71], 873, 919

[Huse66], 515, 551, 916

[IpsM95], 18, 919

[Jaco64], 856, 857, 858, 915

[Jaco66], 840, 845, 846, 915

[Jäni84], 313, 314, 315, 392, 427, 428, 429,

430, 431, 434, 920

[John67], 854, 865, 867, 868, 869, 919

[Kend77], 678, 683, 688, 689, 764, 791, 792,

793, 795, 798, 799, 800, 801, 802, 803,

916

[KirS69], 339, 920

[KleL72], 796, 916

[KobN63], 875, 880, 918

[Laug65], 649, 918

[Lips65], 281, 288, 920

[Lips68], 14, 44, 108, 861, 865, 866, 919

[Lips69], 576, 577, 584, 585, 598, 618, 620,

649, 670, 918

[Livi93], 420, 916

[LunW69], 391, 392, 438, 920

[Lyus64], 621, 919

[ManC92b], 916

[MaPS86], 917

[Mark58], 453, 916

[Mart82], 917

[Mass67], 354, 417, 418, 420, 428, 429, 430,

431, 433, 500, 917

[Matv03], 452, 917

[Maxw86], 640, 917

[McCl85], 453, 917

[McCl97], 633, 918

[Mill58], 817, 839, 847, 848, 849, 850, 856,

915

[Miln58], 504, 918

[Miln63], 251, 491, 492, 495, 918

[Miln65a], 491, 497, 498, 918

[Miln65b], 532, 535, 918

[Miln03], 499, 918

[MilP77], 609, 610, 632, 663, 918

[MilS74], 449, 487, 535, 551, 917

[Mish93], 790, 916

[Mois52], 339, 920

[Mois77], 452, 917

[Munk61], 231, 468, 504, 551, 918

[Nata61], 896, 920

[Need98], 898, 900, 917

[NobD77], 14, 865, 868, 919

[ONei66], 605, 656, 658, 918

[Osse69], 609, 918

958 Bibliographic Index

[PalB98], 917

[PenP86], 126, 183, 184, 185, 190, 920

[Penr55], 55, 919

[PetR98], 196, 920

[Prat90], 640, 918

[Radó25], 339, 920

[RaoM71], 55, 57, 919

[Reid35], 339, 917

[RogA90], 176, 919

[Rolf76], 420, 917

[SakZ71], 906, 917

[Schu68], 315, 920

[Sede87], 744, 916

[Seid68], 695, 754, 757, 758, 762, 763, 774,

775, 785, 842, 916

[SeiT80], 399, 438, 441, 442, 443, 447, 917

[Shaf94], 689, 766, 768, 769, 771, 772, 773,

775, 776, 778, 780, 781, 782, 789, 790, 793,

794, 796, 804, 916

[Smal61], 497, 919

[Smal62], 500, 919

[Span66], 437, 438, 450, 455, 456, 487, 917

[Spie69], 896, 897, 920

[Spiv65], 221, 223, 224, 226, 256, 258, 259,

260, 261, 263, 265, 271, 272, 273, 277, 539,

541, 875, 883, 886, 915

[Spiv70a], 265, 273, 507, 516, 536, 538, 539,

541, 545, 548, 875, 918

[Spiv70b], 571, 572, 574, 577, 578, 607, 618,

659, 660, 918

[Spiv75], 578, 579, 581, 618, 620, 638, 640,

649, 663, 918

[Stee51], 456, 551, 917

[Stok69], 571, 572, 581, 620, 635, 649, 672,

918

[Taba00], 570, 918

[TayM72], 915

[Thom98], 499, 919

[Thor79], 632, 633, 636, 637, 664, 918

[Tiet08], 399, 917

[Waer50], 915

[Walk50], 754, 575, 762, 763, 772, 773, 774,

775, 782, 784, 785, 789, 844, 845, 856, 859,

916

[Wall68], 497, 919

[Wein00], 670, 918

[Weyl55], 917

[Whit41], 399, 917

[Will59], 649, 918

[ZarS60], 764, 840, 844, 846, 858, 916