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

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Index 929

linear system of, 895

solutions of, 894

Differential form

•

, 270, 537

closed, 548

continous, 270, 537

differentiable, 270, 537

exact, 548

integral of, 275, 544

on manifold, 536

on R

, 269

Differentiable function

rank at point of, 231, 471, 503, 509

Differential geometer, 557

Differential k-form

on manifold, 536

on R

, 269

Differential operator, 272, 539

Differential topology, 328

Digital image processing, 896

Dim, 854

Dimension

of cell, 389

of CW complex, 391

of linear polyhedron, 331

of plane, 15

of polyhedron, 387

of simplicial complex, 328

of variety, 792

of variety at point, 792

of variety at smooth point, 792

of vector space, 854

pure, 792

Dim

, 792

Direct analytic continuation, 749

Direct product

of groups, 830

Direct sum

of vector subspaces, 861

Direction cosine, 7

Directional derivative, 228

of vector ﬁeld, 651

Directrix

of conic section, 167

of ruled surface, 645

Discontinuity

essential, 216

removable, 216

Discrete metric, 284

Discrete topology, 290

Discriminant

of bilinear map, 45

of quadratic map, 46

Disjoint union, 301

Disk

n-dimensional, 814

Disk bundle, 517

Disk with handles, 354

Displacement, 83

Distance

between points, 864

between points of manifold, 589

between points of P

, 317

between sets, 286

from point to set, 286

oriented, 29

signed, 29

Distance-preserving map, 64

Distribution parameter

of ruled surface, 648

Divergence

of vector ﬁeld, 272

Divergence theorem, 545

Divide, 817

element in ring, 837

Division algebra, 859

Division algorithm

1-variable, 729, 848

multivariable, 735

Division ring, 847

Divisor, 817

in ring, 837

of ideal, 838

Dodecahedron, 322, 324

Domain

of function, 819

of rational function, 769

of relation, 818

Dominant rational function, 770, 811

Dot product

geometric interpretation of, 6

on C

, 863

on R

, 863

Double cone, 166

Double point

of plane curve, 705

Double torus, 23

Doubly ruled surface, 645

quadric, 192

Dual basis, 874

930 Index

Dual cell, 440

orientation of, 442

Dual cell complex, 442

Dual forms

for frame ﬁeld, 656

Dual map

of linear transformation, 874

Dual space, 873

Dual statement, 142

Duality principle

in projective plane, 142

Dunce hat, 397, 457

Dupin, Charles Pierre (1784–1873), 640, 641

Dupin cyclide, 640, 641

Dupin indicatrix, 609

classical interpretation of, 610

Edge loop

in simplicial complex, 330

Edge path

in simplicial complex, 330

Eigenvalue, 870

Eigenspace, 870

Eigenvector, 870

Eilenberg, Samuel (1913–1998)

Eilenberg-Steenrod axioms, 384, 452

Elementary collapse, 395

Elementary diagonal matrix, 814

Elementary expansion, 395

Elementary P-reduction, 736

Elimination ideal, 745

Elimination theorem, 745

Ellipse, 167

center of, 169

major axis of, 169, 170

minor axis of, 169, 170

principal axes of, 170

string construction for, 167

Ellipsoid, 191, 192, 194

focal curve of, 194

focal ellipse of, 194

focal hyperbolá of, 194

string construction for, 192

Elliptic geometry, 205

Elliptic plane, 205

Elliptic point

of surface, 599, 610, 672

Envelope

of curves, 579

of surfaces, 638

Equality test

for ideals, 742

Equiafﬁne group, 101

Equiareal group, 101

Equivalence class, 819

Equivalence relation, 818

induced by relation, 819

Equivalent

analytic element representations, 751

bases, 25

differentiable structures, 503

knots, 420

parameterizations, 464, 756, 757

Essential discontinuity, 216

Euclidean algorithm, 729, 817

Euclidean metric

standard, 283

Euclidean space

n-dimensional, 852

Euclid of Alexandria (365–300 B.C.), 324

Euler, Leonhard (1707–1783), 598

Euler angles for rotation

X-Y-Z, 115

Z-Y-Z, 115

Euler characteristic, 323, 495, 659

and Euler number of manifold, 535

of surface, 349

Euler formula, 901

for homogeneous polynomial, 709, 846

for simple polyhedrom, 321

Euler number

for tangent bundle, 535

for vector bundle, 533

mod 2, 534

Euler-Poincaré characteristic

of CW complex, 394

of polyhedron, 389

of simplicial complex, 388

Euler-Poincaré formula, 388

Euler theorem

for surface curvatures, 599, 600

for homogeneous polynomials, 846

Evaluation map

for polynomial, 844

Evolute

of curve, 584, 666

of surface, 600

plane, 585

Exact sequence, 450

of homology groups, 450

of homotopy groups, 452

Index 931

Existence theorem

for motions, 81

Exp function, 815

Expansion

elementary, 395

of CW complex, 395

Expansion by minors, 866

Exponential function, 901

Exponential map

for manifold, 664

for surface, 637

Extended complex plane, 898

Extended plane, 136

Extended real numbers, 144

Extension ﬁeld, 849

algebraic, 856

degree of, 856

ﬁnite, 856

inﬁnite, 856

separable, 857

simple, 850

transcendental, 856

Extension theorem, 746

Exterior algebra, 883

Exterior form algebra, 265, 886

properties of, 266

Exterior k-form

on vector space, 265, 886

Exterior k-form bundle

of manifold, 536

Exterior product, 266, 882

of exterior forms, 886

of linear transformation, 884

of vector space, 882

External direct sum

of abelian groups, 831

Extremum, 240

with constraints, 245

Face

of rectangle, 256

of simplex, 31

of singular k-cube, 273, 539

of singular q-simplex, 451

Factor

in ring, 837

multiple, 848

multiplicity of, 848

Factor group, 827

Factor ring, 836

Fiber

of bundle, 422

of vector bundle, 510

Fiber bundle, 512

Fiber map

of vector bundle map, 511

Fibre bundle, 512

Field, 847

automorphism of, 849

algebraically closed, 858

characteristic of, 847

of rational functions, 768, 774

Field of quotients, 850

Fields

isomorphism of, 849

used in algebraic geometry, 674

Finite extension ﬁeld

of degree n, 856

Finite map, 775

between projective varieties, 781

Finite subcover, 212

First countable space, 297

First fundamental form, 590, 667

coefﬁcients of, 591, 667

First homotopy group, 416

First structural equations, 656, 658, 659

Fixed point, 820

Fixed set, 820

Flat point

of surface, 610, 672

Focal curve, 194

Focal point, 600

Focal surface, 600

Focus

of conic section, 167

of normal line, 600

Form

bilinear, 48

d-ic, 46

linear, 46

quadratic, 46

Formal linear combination, 274, 364, 406,

539, 855

Formal power series, 841

derivative of, 846

order of, 859

partial derivative of, 846

zero, 841

Formal power series ring

in several indeterminates, 843

over a ring, 842

932 Index

Four color theorem, 324

Four vertex theorem, 570

Fractional transformation, 136

Frame, 87, 110

as coordinate mapping, 88

as coordinate system, 87, 111

as motion, 89, 111

augmented, 197

for plane, 111

generalized, 197

nonorthogonal, 111

oriented, 87, 110

origin of, 87, 110, 197

standard, 87, 110

transformed, 110

-axis of, 110

x-axis of, 87, 110

y-axis of, 87, 110

z-axis of, 110

Frame coordinates, 88, 111

Frame ﬁeld, 651

adapted, 656

cylindrical, 655

standard, 651

Free abelian group, 833

Free cell, 395

Free group, 417, 830

generators of, 830

Free vector space, 876

with set as basis, 855

Frenet, Jean Frédéric (1816–1900), 576

Frenet basis, 578

Frenet frame, 574

Fresnel, Augustin Jean (1788–1827)

Fresnel integral, 892

Front k-face

of simplex, 449

Fubini, Guido (1879–1943)

Fubini theorem, 260

Function, 819

absolutely integrable, 891

analytic, 893, 899

bijective, 820

•

, 224, 890

, 223, 224, 471, 502, 890

conformal, 900

continuous, 215

continuous at point, 215

continuously differentiable, 224

differentiable, 218, 471, 503

ﬁxed point of, 820

ﬁxed set of, 820

from X to Y, 819

graph of, 820

holomorphic, 899

injective, 819

integrable, 257, 259

invariant set of, 820

inverse of, 820

monotonic, 889

one-to-one, 819

onto, 819

periodic, 901

polynomial, 765

regular, 464, 777

regular at point, 777

smooth, 224

surjective, 819

symmetrio, 823

Function ﬁeld

for afﬁne variety, 768

for projective variety, 778

Function space, 852

metric for, 283

Functor, 452

Fundamental group, 416

homotopy invariance of, 418

signiﬁcance of, 420

Fundamental homology class, 524, 804

of oriented manifold, 487

Fundamental lemma

of Riemannian geometry, 663

Fundamental theorem

of algebra, 906

of calculus, 277, 562, 891

of conformal mappings, 906

of ﬁnitely generated abelian groups, 832

of plane curves, 568

of projective geometry, 135, 161

of space curves, 577

of surfaces, 620

of symmetric polynomials, 845

Fundamental theorem of algebra

relation to Bézout’s theorem, 711

Fundamental theorem of calculus

generalized, 277

Gauss-Bonnet theorem, 659

Gauss, Carl Friedrich (1777–1885)

Index 933

Gauss curvature, 601, 607, 659

for polygonal curve, 572

for polygonal surface, 603

total, 659

Gauss elimination, 867

Gauss equation, 619, 658

Gauss map

derivative of, 613

for oriented surface, 601, 667

for planar curve, 565, 571, 573,

589

Gauss (or Gaussian) quadrature, 912

Gaussian points, 913

Gaussian weights, 913

Gaussian zeros, 913

Gcd, 817, 849

Gelfand, Israil Moiseevic (1913– )

Gelfand–Kolmogoroff theorem, 719

General position, 21

General topology, 281

Generalized frame, 197

base plane of, 197

view direction of, 197

view plane of, 197

Generalized inverse, 53

Generalized inverse matrix, 54

Generalized mean value theorem,

229

Generator

for abelian group, 831

for free group, 830

Generic point

for curve, 774

Genus,

of plane curve, 789

of surface, 350, 356

Geodesic, 624, 625, 632, 635

and mechanics, 621

in manifold, 664

maximal domain, 633

Geodesic convex closure, 602

Geodesic curvature

of curve in surface, 622, 623

Geodesic path, 624

generated by curve, 624

Geodesically complete, 633

Geodesically convex, 602

Geometria situs, 321

Geometric interpretation

of degree, 443, 702

of dot product, 6

of tangent line, 703

Geometric realization

of abstract simplicial complex, 334

of labeled simplicial complex, 336

GL(n,C), 14

GL(n,R), 13

Glide reﬂection, 86

Global analytic function, 749

Global maximum, 240

Global minimum, 240

n+k

), 797

n+k

), 797

n+k

(C)), 797

Graded algebra, 878

Graded ring, 411, 878

Gradient, 227

of function on manifold, 522, 556

Gram, Jorgen Pedersen (1850–1916)

Gram-Schmidt algorithm, 8–10, 43, 92, 118,

164, 649, 910

Granny knot, 421

Grassmann, Hermann (Günther)

(1809–1877)

Grassmann algebra, 265, 883

Grassmann manifold, 550

Grassmann variety, 550

generalized, 797

Grassmannian, 550

Great circle, 627

Greatest common divisor

of integers, 817

of polynomials, 849

Greatest lower bound, 889

Green, George (1793–1841)

Green’s theorem, 545

Gröbner basis, 738

minimal, 742

reduced, 742

Gröbner basis algorithm

Buchberger, 740

Group, 824

abelian, 824

abelianization of, 829

commutative, 824

cyclic, 828

ﬁnite, 829

free, 830

linear, 13, 14

of covering transformations, 431

934 Index

of ﬁnite order, 829

of integers mod n, 825

of knot, 421

subgroup of, 825

trivial, 824

Groups

direct product of, 830

isomorphic, 826

Hairy billiard ball problem, 488

Hairy circle, 488

Halﬂine, 20

Halfplane, 20

lower, 813

upper, 813

Handle, 490, 491

Handle decomposition

of manifold, 495

Hauptvermutung

for manifolds, 340

Hausdorff, Felix (1868–1942)

Hausdorff space, 291

H-cobordant manifolds, 498

H-cobordism, 498

H-cobordism theorem, 498

Heine, Edward (1821–1881)

Heine-Borel theorem, 305

Heine-Borel-Lebesgue theorem, 212, 213,

306

Helicoid, 646

Helix, 575, 669

generalized, 670

Hemisphere

lower, 814

upper, 814

Hermite, Charles (1822–1901)

Hermite interpolation, 909

Hermitian linear transformation, 40

Hertz, H., 621

Hessian, 249, 255

Hessian matrix, 249

Hilbert, David (1862–1943)

Hilbert basis theorem, 716, 846

Hilbert Nullstellensatz, 698, 717,

723

weak form of, 717

Hill climbing problem, 255

Hironaka, Heisuke (1931– )

Hironaka theorem, 804

Hole

in a space, 359, 374, 412

in polygon, 332

Holomorphic function, 899

Hom, 834

Homeomorphic spaces, 294

Homeomorphism, 217, 294

orientation-preserving, 420, 445

orientation-reversing, 445

relative, 389

Homogeneous component

of polynomial, 843

of power series, 843

Homogeneous coordinates, 137, 138, 139,

154, 158

for projective line, 147

for projective plane, 150

relative to matrix, 686

Homogeneous ideal, 723, 808

Homogeneous polynomial, 843

of degree d, 843

rational, 776

Homogenization

of ideal, 724

of polynomial, 688

Homologous

q-chains, 368

Homology class, 368

and imbedded sphere, 374

Homology functor, 452

Homology group

homotopy invariance of, 384, 385

mod 2, 407

of CW complex, 393

of polyhedron, 375

of simplicial complex, 366

relative, 450

singular, 451

with coefﬁcients in group, 406

Homology manifold, 441

Homology n-sphere, 438

Homology sequence of (K,L), 450

Homology theory

axiomatized, 384

fails to classify spaces, 385, 399

for CW complexes, 393

motivation for, 360

summary, 384

Homomorphism

image of, 827

Index 935

kernel of, 827

of groups, 826

of rings, 836

Homotopic maps, 310

relative to subspace, 312

Homotopy, 309

relative to subspace, 312

Homotopy class, 311

Homotopy equivalence, 311

simple, 438

Homotopy functor, 452

Homotopy group

“0th”, 435

ﬁrst, 416

nth, 434

relative, 452

Homotopy lifting theorem, 427

Homotopy n-sphere, 438

Homotopy theory

fails to classify spaces, 399

Homotopy type, 311

Hopf, Heinz (1894–1971)

Hopf-Rinow theorem, 633

Horizon, 129

Horizon plane, 129

Hurewicz, Witold (1904–1956)

Hurewicz homomorphism, 419, 437

Hurewicz isomorphism theorem,

437

Hyperbola, 167

asymptote of, 170

center of, 169

conjugate axis of, 169, 170

transverse axis of, 169, 170

Hyperbolic geometry, 205

Hyperbolic plane, 205

Hyperbolic point

of surface, 599, 610, 672

Hyperboloid

of one sheet, 191, 192, 194

of two sheets, 191, 192, 194

Hyperelliptic curve, 789

Hyperplane, 15

equation for, 15

in P

(k), 676

in projective space, 160

orthogonal vector for, 18

parallel vector for, 18

point-normal equation for, 16

projective, 160

Hyperplanes

angle between, 28

orthogonal, 18

parallel, 18

Hypersurface, 675

degree of, 699

in P

(k), 676

minimal equation for, 699

minimal polynomial of, 699

order of, 699

Icosahedron, 322, 324

Ideal, 836

alternate deﬁnition for, 838

colon, 722

divisor of, 838

elimination, 745

generated by elements, 837

homogeneous, 723, 808

homogenization of, 724

irreducible, 840

maximal, 718, 719, 837

multiple of, 838

of set of points, 715

primary, 839

prime, 838

principal, 837

radical, 839

reducible, 840

Ideal line

in P

, 142

Ideal point, 136, 137, 138, 140, 159

with respect to coordinate neighborhood,

686

Ideal quotient, 722

Ideals

product of, 838

sum of, 838

Identity

of group, 824

Identity matrix, 814

Ill-conditioned

linear system, 908

matrix, 908

Im, 827, 836, 854

Image

of group homomorphism, 827

of linear transformation, 854

of rational function, 769

936 Index

of ring homomorphism, 836

Imaginary part

of complex number, 851

Imbedding, 296, 473, 503

of manifolds in R

, 504

of varieties in P

, 804

Immersion, 473, 503

Implicit deﬁnition, 3

Implicit function theorem, 238

complex version of, 791

Implicit parametrization theorem, 470

Implicitization algorithm

via Gröbner bases, 743, 747

Implicitization problem, 677

Implicitization theorem

for polynomials, 746

Incidence matrix, 400

mod, 2, 409

normalized, 402, 405

with respect to basis, 401

Incidence number, 400

Indeterminate, 841

Index

of nondegenerate critical point, 250,

490

of map at point, 533

of intersection point, 533

of subgroup, 829

Induced abstract simplicial complex,

336

Induced equivalence relation, 819

Induced homomorphism

on cochain groups, 410

on cohomology groups, 410

on fundamental group, 418

on homology groups, 379, 383, 452

on homotopy groups, 436

Induced map

between simplices, 37

of simplicial map, 37, 333

on differential forms, 271, 538

on exterior forms, 266

on quotient space, 299

on tangent spaces, 269

Induced metric, 284

Induced orientation, 365, 483

for induced vector bundle, 518

Induced topology, 290

Induced vector bundle, 515

Inf, 889

Inﬁmum, 889

Inﬁnite extension ﬁeld

of degree •, 856

Inﬂection point, 241, 574, 665

Injection, 819

Inner product, 862

Inner product space, 863

Integrable function, 257, 259

absolutely, 891

Integral, 257

improper, 891

Fresnel, 892

iterated, 260

Lebesgue, 896

of differential form, 275

of n-form over n-manifold, 544

over ring, 775

over set, 257, 259

over singular k-chain, 276, 539

over singular k-cube, 275, 539

over unbounded domain, 891

Riemann, 891

Integral domain, 837

Interior

of manifold, 297

of set, 211, 292

of closed planar curve, 570

Interior point

of manifold, 297

of set, 292

Intermediate value theorem, 309, 890

Internal direct sum

of abelian groups, 832

Intersect properly, 799

Intersect transversally, 530

Intersection multiplicity, 801, 802

along component, 802

at point, 713

of curve and line, 703

Intersection number

for submanifolds, 533

mod 2, 534

of map, 533

of oriented cells, 446

of two curves, 713

Intersection pairing

in manifolds, 448

Intersection problem, 677

Intersection product

of varieties, 802

Index 937

Interval

closed, 4, 889

distinction between it and segment, 58

half-open, 889

open, 889

Invariance of boundary theorem, 387

Invariance of dimension theorem, 387

Invariance of domain theorem, 387

Invariance of pseudomanifold theorem,

439

Invariant

topological, 294, 358

under a map, 820

Invariant set, 820

Inverse

generalized, 53

of function, 820

of group element, 824

of matrix, 866

of vector bundle isomorphism, 512

Inverse function theorem, 232

Inverse image, 820

Inversion, 204

in a sphere, 204

Involute, 670

of curve, 583, 666

of pair of surfaces, 642

string construction for, 584

Irreducible

ring element, 837

Irreducible component

of variety, 701, 721

Irredundant intersection

of ideals, 840

Irredundant union, 721

Isolated point, 210, 296

Isolated singularity, 903

Isometry, 64, 287

local, 287

Isomorphic points, 774

Isomorphism

of afﬁne varieties, 767

of groups, 826

of projective varieties, 778

of rings, 836

of simplicial complexes, 332

of vector spaces, 854

over ﬁeld, 849

Isoperimetric inequality, 571

Isoperimetric problem, 570

Isotopic imbeddings, 420

Isotopy

between imbeddings, 420

Jacobi, Carl G.J. (1804–1851), 621

Jacobian, 224

Jacobian matrix, 224

Jordan, Camille (1838–1922)

Jordan curve theorem, 326, 332,

570

Jordan-measurable set, 259

K-form

on manifold, 536

on R

, 269

k-cell

closed, 390

open, 389

Ker, 827, 836, 854

Kernel

of group homomorphism, 827

of linear transformation, 854

of ring homomorphism, 836

K-fold exterior product

of linear transformation, 884

Kirby, Robion, 339

Klein, Felix Christian (1849–1925), 23, 63,

351

Klein bottle, 351, 352

Knot, 420, 708

complement of, 420

granny, 421

group of, 421

polygonal, 420

square, 421

tame, 420

torus, 421, 708

trefoil, 325

trivial, 420

Knot theory, 325

Knot type, 420

Knots

equivalent, 420

Kolmogoroff, A. N. (1903–1987), 719

Königsberg bridges problem, 325

Kronecker, Leopold (1823–1891)

Kronecker delta, 813

Künneth theorem, 436

938 Index

Labels

for simplicial complex, 335

Lagrange, Joseph Louis (1736–1813)

Lagrange basis functions, 913

Lagrange interpolation, 909

Lagrange multipliers

method of, 245

Lagrange theorem, 829

Laurent, Pierre Alphonse (1813–1854)

Laurent series, 905

Law of cosines, 6

Lc, 734

Lcm, 817, 849

Leading coefﬁcient

of polynomial, 734, 842

Leading power product

of polynomial, 734

Leading term

of polynomial, 734, 842

Leading term set, 738

Least common multiple

of integers, 817

of polynomials, 849

Least squares approximation

a special case, 56

linear case, 55, 56

Least upper bound, 889

Lebesgue, Henry (1875–1941), 212

Lebesgue covering lemma, 307

Lebesgue integral, 896

Lebesgue number, 307

Left distributive, 835

Left-handed limit, 890

Legendre, Adrien Marie (1752–1833)

Legendre polynomials, 910

Legendre-Gauss quadrature, 912

Leibniz, Gottfried Wilhelm (1646–1716), 321

Leibnitz formula, 890

Length

of curve, 561, 596

of curve in manifold, 589

of parametric curve, 558

of parameterization, 594

of surface curve, 667

of symbol for surface, 346

of vector, 863

Lens space, 398, 433

properties of, 399

Levi-Civita, Tullio (1873–1941), 634, 635

Levi-Civita connection, 663

Lex order, 732

Lexicographic order, 732

Lie, Marius Sophus (1842–1899)

Lie bracket

of vector ﬁelds, 662

Lifting

of curve, 425

of map, 425

Limit function, 288

Limit point, 210, 296

in set, 214

of function, 214

of sequence, 214, 288

relative, 214

Line

asymptotic, 611

complex, 60

directed, 28, 108

direction vector of, 2

equation deﬁnition of, 2

horizontal, 2

ideal, 142

in P

, 141

in P

(k), 676

ordinary, 142

parametric equations for, 3

point-direction-vector deﬁnition of, 3

point-slope form of, 2

projective, 141, 160

slope-intercept form of, 2

slope of, 2

two-point form of, 2

vertical, 2

Line at inﬁnity, 686

Line bundle, 510

canonical, 513, 549

Line conic, 189

Line integral, 276

of complex function, 902

Line of curvature

on surface, 617

Line of striction

of ruled surface, 647, 673

Linear combination

of vectors, 853

Linear form, 46

Linear functional, 873