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Index

Aitken's algorithm, 65Akima, Hiroshi, 68, 85algebraic continuity, 96anti-cyclic spline, 102arc length, 38, 81, 83, 88arc length parameter limits, 82arc length parameterization, 39

B-spline, 151, 152, 157banded matrix, 128Barsky, Brian A., 136, 152Bartels, Richard H., 96, 152barycentric coordinates, 208basis, 8

for natural splines, 147B-spline, 151, 152cardinal, 148of vector space, 143

Beatty, John c., 152Bellman, Richard, xibent beam, 230Bernoulli, Daniel, 217Bernstein polynomials, 90

Bessel tangents, 141beta spline, 136bias, 97bicubic patch, 195bicubic, extended, 205bilinear interpolation, 201, 212bilinearly blended, 212binormal vector, 37, 40, 47biquadratic function, 201blending functions, 80,129,143,

196bivariate, 204cardinal, 204Hermite, 80

boundary conditions, 223boundary curve, 194,211Butland, J., 71Bezier polygon, 91, 99Bezier Polynomial, 90Bezier polytope, 91Bezier, P., 90

Ck continuous, 75

238 Index

ck curve, 133C k function, 75C programming language, 159cardinal basis, 148, 204cardinal functions, 148,215Carlson, R. E., 70Casteljau de, Paul, 91Casteljau's algorithm, 91centripetal parameter limits, 81chain rule, 33, 135Chou, Jin J., 89Chui, Charles K., 152circular arc parameter limits, 82Civilized Software, xiclamped spline, 101, 104clockwise rotation, 25closed curve, 32component function, 77concatenation, 53concavity, 70, 83conic, 158constant of integration, 226continuity

k-fold,75algebraic, 84, 96bicubic patch, 199curvature, 133geometric, 83, 96, 133tangent, 83

control points, 90, 154control vectors, 194convex combination, 141convex hull, 84convex-combination, 91convex-upward, 70, 99convexity, 70Coons patch, 214Coons, Steven, 211coordinate systems, 24coordinates, lefthanded, 24

coordinates, righthanded, 24comer twist vectors, 196counterclockwise rotation, 25cross product, 26cross validation, 127cubic polynomial, ix, 65cubically lofted, 213curvature, 39,133,138,140,219,

221directional, 46

curvature function, 39, 43curvature vector, 40curve, 31

Ck ,133Gk ,133arc length, 38central evolute, 44closed,32conic section, 158continuous, 35embedded, 53energy, 218envelope, 38evolute,43family of, 37helix, 42involute, 43minimal energy, 223normal,40offset, 33parabola, 139, 157parameterization, 34planar, 44, 139plane, 31quadratic, 139rectifiable, 38regular, 34semi-simple, 32simple, 32space, 31

tangent, 33trace, 34vertex of, 48

cusp, 35, 82cusp, in spline, 88cyclic spline, 102cycloid, 48

Darboux vector, 48Davis, M., 71DeBoor, Carl, 73, 152deformation, 194, 195,211density function, 132derivative

directional, 53, 205left, 135mixed partials, 196partial, 196right, 135

derivative of a function, 33, 55derivative, of spline, 101Descartes, Rene, 3differential equations, 108dimension, of a vector space, 8direction, of a vector, 4discontinuity, 96distance between two vectors, 11DoCarmo, Manfredo P., 41dot product, 12double tangent spline, 95, 133, 136Dowden, J., 71

elastic fiber, 217ellipse, 32elliptic integrals, 230end conditions, of spline, 101energy, of spline, 112, 114,217,

219,222engineering applications, 217entry tangent vector, 95, 137

Index 239

envelope curve, 38, 45equidegree, 79error, 72estimation, of slopes, 68estimation, of tangents, 200estimation, of twists, 200Euclid, 3euclidean distance, 11euclidean length, 11euclidean space, 3Euler equations, 222, 225evolute curve, 43exit tangent vector, 95, 137extended bicubic, 205extrapolation, 79

Feynman, Richard, 25flatness, 83Fletcher, G. Yates, 97-99, 205Fletcher-McAllister tangent esti-

mation,97Foley, Thomas A., 116force, 218Forrest, A. Robin, 211Fourier transform, 65Frenet equations, 42, 222Frenet-Mehlum equations, 228Fritsch, F. N., 70, 71function, 63

Ck,75B-spline, 152bicubically blended, 213bilinear, 201biquadratic, 201complex-valued, 81component, 77Lagrange multiplier, 223spline segment, 78surface interpolation, 208

240 Index

functionscardinal, 148

Gk curve, 133G 2 joined, 136G 2 oined, 134generalized bicubic, 198,200geometric continuity, 83, 96, 133global cubic spline, 66, 101, 103global spline end conditions, 101gradient, 52graph of a function, 31grid, 204, 207grid rectangle, 199grid, regular, 199,208

helix,42,45Hermite blending functions, 80,

129,143,198Hermite cubic spline, 66, 68, 76,

77,80,194differential equation for, 68

Hermite, Charles, 66, 68Hilbert space, 7Hooke's law, 217Homer's rule, 91,196Hyman, James M., 72hypotenuse of a triangle, 1

inflection point, 37, 41, 47crossing, 47simple, 47touching, 47

inflection point, spline, 98inner product, 12integration by parts, 224interpolation, 59

bicubic patch, 195bilinear, 201biquadratic, 201constraints, 59

function, 65functional, 63Lagrange, 63linear scheme, 84monotonic, 69oscillation, 64polynomial, 63quadratic polynomial, 64two-point, 60Whittaker's formula, 65

intersection of two surfaces, 32involute curve, 43

Jacobian matrix, 56join order, 76,109,149join points, 75, 78, 134Jou, Emory D., 230

Kallay, Michael, 230Kiefer, James E., 48knot values, 199,204knot values, 76Knuth, Donald E., xiKochanek, Doris H. U., 96Kochanek-Bartels tangent estima-

tion,96

Lagrange interpolation, 142Lagrange multiplier, 117, 223Lagrange, Joseph L., 63Lancaster, Peter, 110, 124Lee, E., 81lefthand screw rotation, 25lefthanded coordinates, 24length of a vector, 11linear combinability, 60linear combinations, 6linear dependence, 8linear independence, 8linear transformation, 55, 56lines, 6

local cubic spline, 66lofted surface, 212, 213, 215loop, in spline, 88

Malcolm, Michael A., 217maximum norm, 81McAllister, David E, 97-99, 205Mehlum's equations, 222, 228Mehlum, Even, 222minimal energy, 223minimization, 131, 155mixed quadratic spline, 142MLAB program, ximodulus of elasticity, 218moment of inertia, 219monotonic interpolation, 69monotonic smoothing spline, 130monotonic spline, 71, 72monotonicity region, 70Moore-Penrose inverse, 130moving trihedral, 37, 41

natural spline, 101, 107basis, 147

nearest neighbor method, 206Nielson, Gregory M., 90,116normal plane, 37, 44normal space, 55not-a-knot condition, 102nu-spline, 116nullspace, 56NURB-spline, 157

offset curve, 33optimal smoothing spline, 124orientation of coordinate systems,

24origin, 4orthonormal coordinates, 24osculating circle, 39, 40osculating plane, 36, 90

Index 241

osculating sphere, 43Overhauser spline, 141overshoot, 70

parabola, 139, 157parallelogram, 5parameter limits, 77, 81, 87, 199

arc length, 82bicubic patch, 198centripetal, 81circular are, 82

parameterization,39arc length, 39

parametric representation, 31,157parametric spline, 77patch,193

bicubic, 195adjacent, 200bicubically blended, 213boundary curves, 193, 205,

211continuity, 199Coons, 214degenerate, 207derivatives, 196diagonal,195extended, 205generalized bicubic, 198network, 206normal vector, 197parameter limits, 198tangent vectors, 195triangular, 205, 206twist vectors, 196

perspective projection, 158physical spline, 217, 221, 229

computation, 222curvature, 222energy, 221planar, 230

242 Index

torsion, 222piecewise-defined, 66, 75, 78,193planar curve, 44plane, 6

osculating, 36plane curve, 31point

inflection, 37non-regular, 36regular, 35singular, 35

points, 3polygon, Bezier, 91polynomial

cubic, 65, 67oscillation, 66quadratic, 157segment, 78

polytope, Bezier, 91potential energy, 218principal normal, 37, 40product space, 204program

cinterpolate,159smoothspline, 159

projection, 48, 150, 215perspective, 158

pseudo-inverse, 130pythagorean theorem, 2

quad-patch, 193, 199,206quadratic curve, 139, 157

mixing, 141quadratic end condition spline, 102

random variable, 132range dimension, 75, 76rational spline, 157rectifiable, 38rectifying plane, 37

regular grid, 199, 208regular parameterization, 34, 36,

133retrograde motion, 34, 35righthand screw rotation, 25righthanded coordinates, 24rotation, 81, 89

clockwise, 25counterclockwise, 25

rotation operation, 25rowspace, 55ruled surface, 45, 211, 220

Salkauskas, K~stutis, 110, 124scalar, 3scalar vector product, 4Schoenberg, Issac, 114Schumaker, L. L., 69second derivative conditions, 102second derivative

ofspline,108segment function, 78segment polynomial, 80semi-simple curve, 32similar triangles, 1simple curve, 32single-valued surface, 51slope, 60, 68

Akima's estimation, 68Davis-Dowden estimation, 71estimation, 68, 87Fritsch-Butland estimation, 71Hyman estimation, 72quadratic estimation, 68

small support, 146smoothing parameter, 124, 127

estimation, 128smoothing spline, 91, 123, 154,

155,209best fit, 129

computation of, 124monotonic, 130

smoothness functional, 114space curve, 31space-curve spline, 77span

of a set of vectors, 7spin vector, 48Spivak, Michael, 39spline, ix, 66

Ck ,133G2 joined, 134, 136Gk ,133v,116anti-cyclic, 102arc length, 88B-spline, 151, 157beta spline, 136bias, 97Bezier,90cardinal basis, 148clamped, 101, 104complex representation, 81component function, 77,149cubic, 217cubic polynomial, 65cusp,82,88,99,114cyclic, 102derivative, 101differential equation, 68double tangent, 95, 133, 136end conditions, 101energy, 112, 114error in, 72first derivative, 66, 68, 79, 85generalized bicubic, 200global, 66, 101, 103Hermite cubic, 66, 68inflection point, 98Lambda, 75

Index 243

linear, 60local,66loop, 88, 99mixed quadratic, 142monotonic, 69-72, 107natural, 101, 107normal vector, 89NURB,157optimal,124osculating plane, 90Overhauser, 141parametric, 77physical, 217piecewise-defined, 66, 68, 78quadratic end condition, 102rational, 157second derivative, 102, 108segment polynomial, 80shape,87,138smoothing, 91, 123, 155space curve, 77tension, 84, 97third derivative, 112third derivative-constrained,

102vector space dimension, 76vector space of, 143weighted,116

subspace, 7orthogonal complement, 55

support, 146surrace, 51, 193, 199

bilinear interpolation, 201curvature of, 55energy of, 220lofted, 212, 213normal vector, 53parametric, 193patch,193representations of, 52

244 Index

ruled, 211, 220single-valued, 51 193, 207,

209tangent vectors, 200, 206twist vectors, 200

sweep, 194, 195,211

tangent vector, 33Akima's estimation, 85, 86Bessel's estimation, 141eatmull-Rom estimation, 86chordal estimation, 86cross boundary, 200curve, 33entry, 95, 137estimation, 85, 200exit, 95, 137Fetcher-McAllister estimation,

97Kochanek-Bartels estimation,

96line, 32,43,44,55plane, 54space, 54

tension, 84, 88, 97tensor product, 195, 198tent function, 110, 124third derivative, of spline, 112third derivative-constrained spline,

102torsion, 139,219torsion function, 41-43, 47

trace of a curve, 34trace, non-retrograde, 34triangular patch, 206triangulation, 207tridiagonal systems, 104triple product, 28triple product identity, 226twist vectors, 196

estimation, 200

undershoot, 70unit vectors, 12

Vandermonde matrix, 64,151vector difference, 5vector product, 26vector space, 6, 76

cardinal basis, 148of splines, 143

vector sum, 4, 5vector triple product, 28vectors, 3vertices, 48

Wahba, Grace, 127Wardle, K. L., 43weighted spline, 116Whittaker's interpolation formula,

152work,218

Yamaguchi, E, 82