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

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Index 939

Linear group

complex, 14

real, 13

Linear polyhedron, 331

dimension of, 331

Linear transformation, 854

alternating, 881

associated matrix of, 868

characteristic polynomial of, 871

determinant of, 869, 888

diagonalizable, 870

Hermitian, 40

image of, 854

kernel of, 854

k-fold exterior product of, 884

nonsingular, 854

normal, 44

rank of, 869

relation to matrix, 868, 869

singular, 854

symmetric, 40

trace of, 869

Linearly dependent

points, 860

vectors, 853

Linearly independent

points, 860

vectors, 853

Liouville, Joseph (1809–1882)

Liouville theorem, 905

Lipschitz, Rudolf (1832–1903), 217

Lipschitz condition, 217, 232

Listing, Johann B. (1806–1882), 22,

321

Local base, 291

Local coordinates

expressing functions in, 489

Local extremum, 240

Local isometry, 287

Local maximum, 240

Local minimum, 240

Local ring, 782

Locally compact space, 307

Locally ﬁnite cover, 315

Locally ﬂat curve, 574

Locally path-connected space, 429

Locally trivial bundle, 423

Lower bound

of set of real numbers, 889

Lower halfplane, 813

Lower hemisphere, 814

Lower sum, 257

Lpp, 734

Lt, 734, 738

LU-decomposition, 868

Lüroth, Jacob (1844–1910)

Lüroth’s problem, 773

Lüroth’s theorem, 773

Mainardi, Gaspare (1800–1879)

Mainardi-Codazzi equations, 619

Manifold

boundary of, 297, 466

closed, 297, 466

closed tubular neighborhood of,

527

combinatorial, 339

complex, 684

, 466, 501

differentiable, 466, 501

differentiable structure for, 501

hauptvermutung for, 340

homology, 441

interior of, 297

n-dimensional, 297, 466

normal bundle of, 525

orientable, 483, 522, 554

orientation of, 483, 522

oriented, 483, 522

parallelizable, 521

piecewise linear, 298

PL, 298

Riemannian, 521

smooth, 466, 501

spherical modiﬁcation of, 497

surgery of, 497

tangent bundle of, 519, 520

topological, 297, 339, 340

triangulation problem for, 339

tubular neighborhood of, 527

vector ﬁeld of, 521

with attached k-handle, 494

without boundary, 297, 466

Map (same as function), 819

Map coloring, 324

Matrix

adjoint of, 39, 866

940 Index

associated linear transformation of, 868,

869

characteristic polynomial of, 871

cofactor of, 866

column rank of, 867

condition number of, 908

determinant of, 866

diagonal, 814, 865

diagonalizable, 870

elementary, 48, 49, 181, 814

for linear transformation, 868

for projective transformation, 153

Hermitian, 865

ill-conditioned, 908

inverse of, 866

LU-decomposition of, 868

maximal rank of, 867

minor of, 866

nonsingular, 866

norm of, 907

orthogonal, 13

permutation, 868

pre vs post multiplication of, 868,

869

rank of, 867

singular, 866

singular value decomposition of,

singular values of, 57

special orthogonal, 13

special unitary, 14

Sylvester, 690

symmetric, 865

trace of, 867

unitary, 14

Max metric

on product, 289

on R

, 283

Maximal ideal

of functions, 719

of polynomials, 718

Maximum, 240

Maximum principle

for analytic functions, 906

Maxwell, James Clerk (1831–1879), 640,

641

Mean curvature, 607

Mean value theorem, 890

generalized, 229

Measure theory, 896

Measure zero set

in manifold, 528

in R

, 258

Membership test

for ideal, 743

Meromorphic function, 904

Metric, 282

bounded, 286

, 283

•

, 283

discrete, 284

for P

, 317

groups, 101

induced, 284

max, 283

properties, 101

standard Euclidean, 283

taxicab, 283

Metric coefﬁcients, 591, 667

determinant of, 591

Metric space, 283

bounded, 286

complete, 289

induced, 284

Metrics

topologically equivalent, 285

Metrizable, 291

Meusnier, Jean Babtiste (1754–1793)

Meusnier theorem

for surface curvatures, 600,

606

Minding, Ernst Ferdinand Adolf

(1806–1885), 622

Minding theorem, 630

Minimal equation

for hypersurface, 699

Minimal polynomial

for hypersurface, 699

Minimal surface, 499, 609

Minimum, 240

Minimum handle decomposition theorem,

499

Minimun polynomial

of algebraic element, 856

Minor

of matrix, 866

Moduli, 790

Modulus

of complex number, 851

Index 941

Moebius, August Ferdinand (1790–1868), 22,

130, 137

Moebius strip, 22, 23, 140, 299, 317, 352,

355, 462, 673

area of, 548

meridian of, 513

open, 512, 534, 555

Moise, Edwin E., 339

Monge, Gaspard (1746–1818), 461, 638, 649

Monge patch, 461

Monodromy lemma, 427

Monoid, 790

Monomial, 843

coefﬁcient of 843

total degree of, 843

Monomial ordering

deglex, 732

degree lexicographic, 732

degree reverse lexicographic, 732

degrevlex, 732

lex, 732

lexicographic, 732

of N

, 733

of k[X

,...,X

], 733

reverse lexicographic, 732

Moore-Penrose inverse, 53

Moore-Penrose inverse matrix, 54, 255

Morphism, 214

Morse, Harold Calvin Marston (1892–1977)

Morse inequalities, 495

weak, 495

Morse lemma, 250

Morse-Sard theorem, 530, 531

Motion, 64

deﬁned by frame, 89, 111

equations for, 85, 105

orientation-preserving, 83

orientation-reversing, 83

Moving frames

of Cartan, 649

Moving trihedron, 574

Multilinear map, 862

alternating, 884

relation to linear map, 877

Multiple

of ideal, 838

Multiple factor, 848

Multiple root, 848

Multiplicity

of curve at point, 703, 704

of factor, 848

of function at root, 704

of root, 848

of variety at point, 801

Multiplicity of intersection, 800

Multipolynomial resultant, 695

Mutually orthogonal set, 7

Mutually orthogonal vectors, 7

Natural coordinate system

for conic section, 168

for quadric surface, 191

Natural inclusion

of projective spaces, 159

Natural isomorphism

between vector space and its dual, 874

Natural projection

of labeled complex, 336

N-cell

of CW complex, 390

N-cube, 256

Neighborhood, 209, 285, 291

Neighborhood base, 291

Net, 296

Newton, Sir Isaac (1642–1727)

Newton-Raphson method, 252

generalized 253

problems with, 253

NF, 736

Noether, Emmy (1882–1935)

Noether, Max (1844–1921)

Noether normalization theorem,

782

Noether theorem, 789

Noetherian ring, 840

Noncollapsible cell complex, 397

Non-Euclidean geometry, 205

Nonhomogeneous coordinates, 136

Nonsingular plane curve, 705

Nonsingular point

of variety, 798

Norm

of matrix, 907

of partition, 256

Normal

inward-pointing, 486

outward-pointing, 486

Normal angle

at point on surface, 603

942 Index

Normal bundle

of manifold, 525

Normal curvature, 599

in a direction, 606

of curve in surface, 599, 605, 623, 668

of normal section, 599

Normal form

of surface, 348

with respect to polynomials, 736

Normal neighborhood

at point on surface, 603

Normal plane, 574

Normal section

of surface, 598

Normal space, 313

Normal subgroup, 826

Normal transformation, 44

Normal vector

for hyperplane, 16

for subspace, 11

Normal vector ﬁeld

associated to orientation, 486

of manifold, 484

Nowhere dense set, 292

N-cube, 256

N-plane bundle, 510

N-rectangle, 256

face of, 256

Nullstellensatz, 698, 717, 723

O(n), 13

Oblique parallel projection, 103

Octahedron, 322, 324

Offset curve, 586

Offset surface, 643

One-point compactiﬁcation, 201, 308

One-point union

of spaces, 303

One-point wedge

of spaces, 303

One-sided derivative, 467

One-sided limit, 232, 467

One-to-one, 819

Onto, 819

Open

in set, 213

Open map, 293

Open set, 209, 285, 290

relative, 213

Ord, 844

Order

of curve, 699

of differential equation, 894

of formal power series, 859

of group, 829

of group element, 829

of hypersurface, 699

of pole, 904

of polynomial, 844

of polynomial at point, 703

of power series, 752

smallest element of, 821

Order of contact, 713

Ordering

monomial, 733

Ordinary line

in P

, 142

Ordinary point

of plane curve, 705

Ord

, 703

Orientability, 375

compatibility issue, 522

determination of, 375, 440

Orientation, 22, 29

at point, 22, 24

determined by ordered basis,

for pseudomanifold, 440

global, 29

induced, 365, 483

induced by isomorphism, 483

induced by ordered basis, 25

induced on boundary, 528

local, 24

of angle, 900

of dual cell, 442

of manifold, 483, 522

of plane, 28

of R

, 25

of simplex, 362

of surface, 29

of vector space, 25

opposite, 25

standard, 27, 484

Orientation preserving, 83

homeomorphism, 420, 445

linear transformation, 27

manifold map, 524

vector bundle map, 517

Index 943

Orientation reversing, 83

homeomorphism, 445

linear transformation, 27

manifold map, 524

vector bundle map, 518

Orientations

continuously varying, 483, 517, 553

Oriented cell, 393

Origin

of view plane, 197

Orthogonal basis, 7

Orthogonal matrix, 13

special, 13

Orthogonal complement

of vector subspace, 11

with respect to plane, 19

with respect to subspace, 11, 12

with respect to vector, 12

Orthogonal direct sum

of vector spaces, 11

Orthogonal group, 13

special, 13

Orthogonal polynomials

associated to inner product, 909

Orthogonal projection, 103, 637

of vector, 12

on plane, 19

on vector, 6, 8, 12

Orthogonal vectors, 7

Orthographic projection, 103

Orthonormal basis, 7

Osculating circle

for planar curve, 564

Osculating plane

for space curve, 574

Parabola, 167

axis of, 169

Parabolic point

of surface, 600, 610, 672

Paraboloid

elliptic, 191, 192

hyperbolic, 191, 192

Paracompact space, 315

Parallel curve, 586, 666

nondegenerate, 587

Parallel projection, 103

as a perspectivity, 136

Parallel surface, 643, 669

Parallel translate vector

along curve, 636

Parallel vector ﬁeld, 634, 662

history of, 634, 635, 638

Parallel vectors, 6

Parallelizable manifold, 521

Parallelogram, 263

Parallelopiped, 263

volume of, 263

Parallelotope, 263

volume of, 263

Parametric curve, 474

closed, 475

differentiable, 474

length of, 558

path of, 474

proper, 475

Parameterization, 3

afﬁne, 756

•

assumption of, 461

, 460

center of, 755

irreducible, 756, 757

local, 466

of projective curve, 756

of projective line, 147

of projective plane, 150

projective, 754

proper, 466, 475

reducible, 756, 757

regular, 464, 567

regular at point, 464

second derivative assumption of, 574

transformed, 464, 761

via implicit function theorem, 236, 237

Parameterization problem, 677

Parameterizations

equivalent, 464, 756, 757

Partial derivative, 223

as tangent vector, 508

mixed, 223

of formal power series, 846

of order k, 223

of polynomial, 846

Partial order, 821

strict, 821

Partition

norm of, 256, 890

of interval, 890

of rectangle, 256

944 Index

reﬁnement of, 256, 890

subinterval of, 890

subrectangle of, 256

Partition of unity, 314

subordinate to cover, 314

Pascal, Blaise (1623–1662)

Pascal theorem, 199, 714

Path, 474

between points, 309

geodesic, 624

Path lifting theorem, 425

Path-component, 309

Path-connected space, 217, 309

Periodic function, 901

period of, 901

Permutation, 821

even, 823

odd, 823

Permutation matrix, 868

Perpendicular vectors, 6

Perspective transformation, 127

generalized, 196

Perspectivity, 127

generalized, 196

PID, 837

Pinched sphere, 439

P-irreducible, 736

Pitch angle

of rotation, 114

Place

center of, 757

of curve, 757

of ﬁeld, 763

of space curve, 783

Place at inﬁnity, 788

Planar point

of surface, 610, 672

Plane

basis for, 15

complex, 60

homogeneous deﬁnition of, 34

k-dimensional, 15

k-dimensional projective, 160

oriented, 28

oriented by basis, 28

orthogonal to plane, 19

parallel to plane, 19

point-normals equation for, 18

projective, 160

punctured, 632

Plane at inﬁnity, 685, 686

Plane curve, 675, 676

algebraic, 675

equation of tangent line for, 706, 709

fundamental theorem of, 568

Plane of symmetry

for surface, 672

Planes

angle between, 18

in general position, 21

parallel, 19

transverse, 21

Plateau, Joseph A.F. (1801–1883)

Plateau’s problem, 609

Plücker, Julius (1801–1868)

Plücker coordinates, 796

Plücker relations, 797

, 139, 316

as regular CW complex, 453

(k), 675

P-normal form, 736

Poincaré, Jules Henri (1854–1912), 205,

498

Poincaré conjecture 498

generalized, 498

Poincaré duality, 496

Poincaré duality theorem, 447, 449

mod 2 version, 447

Poincaré space, 438

Point

nonsingular, 798

r-fold, 704

singular, 798

smooth, 791

Point at inﬁnity, 686, 787, 898

of one-point compactiﬁcation, 308

Point of continuability, 749

Point of multiplicity r, 704

Point of noncontinuability, 749

Point set topology, 281, 282

Pointed space, 303

Point-normal form

equation for hyperplane, 16

Point-normals form

equation for plane, 18

Points at inﬁnity, 696, 788

Pointwise addition, 852

Pointwise convergent functions, 288

Pointwise scalar multiplication, 852

Polar coordinates, 68, 261

Index 945

Polar form

of complex number, 898

Pole

of analytic function, 903

order of, 904

Polygon, 332

vertices of, 332

with holes, 332

Polygonal Gauss curvature, 572, 603

Polyhedron

convex linear, 31

ﬁnite, 330

inﬁnite, 331

linear, 321, 331

regular, 321

simple, 321

Polynomial

associated to hypersurface, 699

cubic, 843

degree of, 841, 843

derivative of, 846

evaluation map of, 844

homogeneous, 843

order at point of, 703

order of, 844

over ring, 841

partial derivative of, 846

root of, 844

quadratic, 843

rational homogeneous, 776

resultant of, 693

separable, 857

symmetric, 845

total degree of, 843

zero, 841

zero of, 844

Polynomial function, 765

between varieties, 765

representative for, 765

Polynomial implitization theorem, 746

Polynomial ring

generated by subset, 840

in several indeterminates, 843

over a ring, 842

over subring, 840

Power product, 843

Power series, 892

interval of convergence of, 893

obtained by substitution, 844

radius of convergence of, 893

P-reduces, 736

Prime ﬁeld, 847

in a ﬁeld, 847

Prime number, 817

Primitive element

of extension, 850

theorem of, 856

Principal axes theorem, 37, 42

complex, 44

for quadratic forms, 48

real, 41

Principal axis theorem, 107, 112

Principal ideal, 837

Principal ideal domain, 837

Principal normal

of plane curve, 566, 665

of regular curve, 567

of space curve, 574

Principal normal curvatures

of surface, 599, 607, 664, 668

equation for, 614

Principal normal directions, 599, 607, 668

Principal radii, 599

Principle of duality

in projective plane, 142

Principle of least action, 621

Principle of least curvature, 621

Product

of ideals, 838

of permutations, 821

Product bundle, 422

Product C

manifold, 502

Product C

structure, 502

Product map, 819

Product n-plane bundle, 510

Product topology, 302

Product vector bundle, 510

Projection

center of, 779

central, 127, 137

of bundle, 422

of projective set, 779

of projective space, 779

of vector bundle, 510

orthogonal, 6, 8, 12, 19, 103, 637

Projection operator, 861

Projective closure

of afﬁne variety, 724

Projective completion

of afﬁne variety, 688

946 Index

Projective conic, 173

Projective invariant, 129

Projective line, 137, 139, 160

Projective plane, 138, 139

characterizations of, 140

principle of duality in, 142

Projective properties, 129

Projective space

characterizations of, 315, 316, 317

n-dimensional, 139, 315

over ﬁeld, 675

Projective transformation, 129, 134, 152, 160

matrix of, 152

Projectively equivalent ﬁgures, 160

Projectivity, 134, 152, 160

Properly discontinuous, 432

Pseudo-inverse matrix, 54

Pseudomanifold, 438

closed, 439

invariance of, 439

n-dimensional, 438

nonorientable, 440

orientable, 440

orientation of, 440

Pseudometric, 319

Puiseux, Victor Alexandre (1820–1883)

Puiseux pairs, 708

Pullback

of differential form, 271

Pullback map, 766, 771

of polynomial function, 766

of rational function, 771

Pullback vector bundle, 515

Punctured plane, 632

Pure dimension, 792

q-boundary, 366

and boundary of imbedded (q+1)-disk,

374

mod 2, 407

with coefﬁcients in group, 406

q-chain, 362

elementary, 363

mod 2, 406

singular, 451

with coefﬁcients in group, 405

q-coboundary, 410

q-cochain, 410

q-cocycle, 410

q-cycle, 366

and imbedded q-sphere, 374

mod 2, 407

with coefﬁcients in group, 406

Quadratic form, 46

degenerate, 48

discriminant of, 48

nondegenerate, 48

positive deﬁnite, 48

relation to quadratic map, 47

signature of, 48

Quadratic map, 46

degenerate, 46

discriminant of, 46

nondegenerate, 46

positive deﬁnite, 46

Quadratic nonresidue, 817

Quadratic residue, 817

Quadratic surface, 190

classiﬁcation of, 196

Quadratic transformation, 759

center of, 759

Quadric surface, 190

classiﬁcation of, 194

focal curves of, 194

natural coordinate system for,

191

principle planes of, 192

properties of, 192

tangent plane to, 196

Quasiprojective variety, 724

Quotient ﬁeld, 847, 850

Quotient group, 827

Quotient map, 299

Quotient ring, 836

Quotient space, 299

by collapsing subspace, 300

modulo a group, 432

of relation, 819

Quotient topology, 299

Quotient vector space, 855

Radial transformation, 94

Radical

of ideal, 839

Radius function

for canal surface, 638

Radius of convergence

of power series, 893

Index 947

Radius of curvature

for planar curve, 564, 565

Radó, Tibor (1895–1965), 338

Range

of function, 818

of relation, 818

Rank

maximal, 867

of abelian group, 832

of bilinear map, 46

of differentiable map, 231, 471, 503,

509

of linear transformation, 869

of matrix, 867

Raphson, Joseph (1648–1715), 252

Ratio of division, 100

Rational afﬁne variety, 771

Rational function, 804, 904

between afﬁne varieties, 769

between projective varieties, 780

domain of, 769, 778

dominant, 770

image of, 770

on P

(k), 776

on afﬁne variety, 768, 770

on hypersurface, 770

on projective variety, 778, 779

regular, 776

regular at point, 769, 776, 778

regular point of, 769

value of, 769, 778

Rational map

see rational function

Ray, 5

Real part

of complex number, 851

Real point, 140, 159

in complex space, 898

Real variety, 675

Rectangle, 256

Rectiﬁable curve, 559

Rectifying plane

for space curve, 574

Reduced word, 830

Reduction

of polynomial, 731

Reﬁnement

of partition, 256, 890

Reﬂection

about hyperplane, 105

about line, 72

axis of, 72

Region

bounded by closed planar curve, 570

Regular function, 464, 776

at point, 464

Regular k-gon

standard, 341

Regular point, 529, 777

of power series, 752

of rational function, 769, 776, 778

Regular value, 529

Reidemeister, Kurt Werner Friedrick

(1893–1971), 399

Relation

antisymmetric, 818

between sets, 818

domain of, 818

equivalence, 818

on set, 818

one-to-one, 818

onto, 818

range of, 818

reﬂexive, 818

symmetric, 818

transitive, 818

well-deﬁned, 818

Relative boundary, 214

Relative closed set, 213

Relative closure, 214

Relative homeomorphism, 389

Relative homotopy, 312

Relative limit point, 214

Relative open set, 213

Relative topology, 292

Relatively prime, 817

Remainder

of polynomial division, 729, 735

Removable discontinuity, 216

Removable singularity, 903

Reparameterization

of singular k-cube, 540

orientation-preserving, 540

orientation-reversing, 540

regular, 464

Representative

of polynomial function, 765

Resolution of singularities, 804

Resolving

a singularity, 760

948 Index

Resultant, 691

multipolynomial, 695

of polynomial, 693

Sylvester, 691

Retract, 311

Retraction, 311

Reverse lexicographic order, 732

R-fold point, 704

Riemann, Bernhard (1826–1866), 460, 660

Riemann integral, 891

versus Lebesgue integral, 896

Riemann mapping theorem, 906

Riemann sphere, 898

Riemann surface, 748, 750

Riemannian manifold, 521

Riemannian metric

for manifold, 521

for vector bundle, 516

induced from R

, 521

Right distributive, 835

Right hand rule, 112

Right-handed limit, 890

Rigid motion, 83, 107

Ring, 835

commutative, 835

graded, 411, 878

ideal of, 836

local, 782

Noetherian, 839

of continuous functions, 719

of polynomial functions, 766

with unity, 835

Rings

isomorphic, 836

Rinow, Willi (1907–1978), 633

, 852

Rodrigues, Benjamin Olinde (1794–1851)

Rodrigues formula, 617

Roll angle

of rotation, 114

Roll-pitch-yaw representation, 114

uniqueness of, 114

Root

multiplicity of, 848

of polynomial, 844

simple, 848

Root of unity, 901

Roots

ﬁnding, 252

Rotation, 107

about directed line through angle, 112, 113

about line, 107, 108

about origin, 68

about point, 70, 107, 108

about x-axis, 113

about y-axis, 113

about z-axis, 107, 113

axis-angle representation of, 112

center of, 70, 107

compact axis-angle representation of, 113

equations for, 69, 71

Euler angle representation of, 115

general, 70

orientation of angle for, 112

roll-pitch-yaw representation of, 114

Rotation about line

ambiguity in, 112

Row rank

of matrix, 867

Rubber sheet geometry, 296, 309, 327

Rubinstein, J.H., 499

Ruled surface, 645

base curve of, 645

central point of, 647

directrix of, 645

distribution parameter of, 648

line of striction of, 647

quadric, 192

ruling of, 645

striction curve of, 647, 673

Saddle point, 242

Sard, A., 530

Scalar multiplication, 852

Scalar product, 862

Scaling transformation

global, 96, 110

local, 96, 110

Schmidt, Erhard (1876–1959), 8, 9

Schoenﬂies, Arthur Moritz (1853–1928)

Schoenﬂies theorem, 332, 421

Schwarz, Hermann Amandus (1843–1921),

864

Screw motion, 125

Second countable space, 297

Second fundamental form, 605, 667

Second structural equation, 658

Second structural equations, 656, 659

Sectional curvature, 663

Segment, 4

Self-adjoint, 40