# Curves and surfaces in computer aided geometric design - Yamaguchi F.

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interpolation of a sequence of points 247 to 250, 325-327 matrix representation 327-329 properties 311,312 B-spline, B-spline function approximation 233-336 definition 270-273 normalized 273 recursive calculation 285-291 relation to Bernstein basis 293, 294

properties 270-274,291-294 variation diminishing property 294 B-spline curve type (1) 281, 282 B-spline curve type (2) 283 B-spline curve type (3) 294-296

Cartesian product surface patch 132, 133 Cauchy’s relation 19 Chaikin’s algorithm 320-325 Characteristic net 217 Characteristic polygon 173 Circular arc, circle approximation 78-80, 177, 178, 243-245 by Bezier curve segments 177, 178 by cubic B-spline curve segments 243 to 245

by Ferguson curve segments 78-80 passing through 3 points 354, 355 approximation error 80, 178 Common perpendicular, length 352, 353 Condition for determining a tangent vector to a curve on a surface 48 Conic section 337-346 Connection

of curve segments 33-35 of Bezier curve segments 213,214 of surface patches 57-59 of Ferguson surface patches 89-91 of Coons bi-cubic surface patches 96, 97, 107-109

of Bezier surface patches 221-226 Continuity, class C- 20, 44 Control point 16 Convex combination 171,241,274 Convex hull property 241, 311 Crease 15 C-spline 139

Cross partial derivative vector 98, 112 -122 Cubic spline curves 145-160 using circular arc length 159, 160 Curvature 24-27 Gauss- 51 normal 50 radius 25 principal 51 total 51

average(mean) 51

376

Subject Index

Curvature of a surface 49-51 Curvature vector 24-27 Curve defining polygon 173 Curve generation by geometric processing 320-325 Curve segment increase of degree 85, 86, 204-209 Bezier- 169-232 B-spline 233-334 conic section 337-346 cubic Bezier 169-182 cubic B-spline 233-251 geometrical relations among derivatives 342, 343

cubic/cubic rational polynomial 346, 347 Ferguson- 73-80 Hermite- 72-86 Lagrange- 64-71 non-uniform B-spline- 296 rational polynomial 337-350 Ã-ñîøñ 347-350 uniform cubic B-spline 233-251 Cusp 15 by cubic B-spline curve segments 245

De Boor’s algorithm 312-315 Descartes’ sign rule 192 Degeneration of a surface patch 44 formation of triangular surface patch 59 to 61 Derivatives of Bernstein polynomials 198,199 of Bezier curve segments 198, 199 of B-spline curve segments 309 311 Determination of a point

on a cubic Bezier curve segment 178, 179 by linear operations 199 204 on a B-spline curve segment by linear operations 312-315 Divided difference 67 interpolation formula with remainder (Newton’s formula) 68

Ellipse 340 Equation

of a normal plane 23 of an osculating plane 23 of a plane passing through 3 points 353 of a straight line segment 351 of a tangent plane 46 Error in approximating a circle 80, 178

Finite difference computation 32,33 determination of a point on a curve 32, 33

on a surface 51-55 matrix 33, 52-55 operator 32

representation of a Bezier curve segment 194, 195

First fundamental matrix of a surface 46-48

FMILL 89

Forrest’s method 117

Frenet frame 29

Frenet-Serret formula 31

Functional determinant 44

Function

Bernstein basis- 182, 183, 189-193 blending- 77

B-spline- 236, 237, 270-273 Coons blending- 77 full spline basis- 291 Hermite- 72-83 truncated power- 138 uniform B-spline- 296

Gauss curvature 51 Gauss quadrature 42

Hermite interpolation 72-134 Hermite polynomial 72 -134 Hyperbola 340

Increase of degree

of a Bezier curve segment 204-209 of a Ferguson curve segment 85, 86 Independence of coordinate axes 12-14 Interpolation Hermite- 72-134 Lagrange- 64-71

of a sequence of points by a B-spline curve 247-250, 325-327 spline- 135-168 Intersection of 2 curves 43, 44 of a curve and a plane 43 of 3 planes 354 Invariance of shape under coordinate transformation 18, 19 Inverse transformation of a uniform cubic B-sphne curve 247-250, 325-327

Jacobian 44,60 62

Knot additional 272 extended 272 insertion 316-320 interior 272 pseudo 292 multiple 291 Kronecker delta 92

Subject Index

377

Lagrange’s formula 65 Leibnitz’ formula 356 Length of a curve 42 on a surface 47 of common perpendicular 352, 353 Line segment

by a Bezier curve segment 176, 177 by a cubic B-spline curve segment 243 perpendicular bisector 351 Local uniqueness 242

Matrix boundary condition- 111 finite difference- 33, 52-55 first fundamental- (of a surface) 46-48 second fundamental- (of a surface) 50 Matrix representation of a B-spline curve segment 327-329 Marsden’s identity 358, 359 Mathematical model 3, 4 Minimal interpolation property 140 144 Model mathematical 3,4 physical 3,4 Moving frame 29

Net, surface defining- 217 Normal, principal- 23 Normal plane 23 equation 23 Normal vector, unit- 46 Normalized B-spline 273

Offset surfaces 63 Order of spline 136 Ordinary point 20 Osculating plane 23

Parabola 342 Parametric cubic curve 31 Parametric representation of a curve 20 of a surface 44 Partitioning

of cubic Bezier curve segments 179-182 of curve segments 39 of Coons bi-cubic surface patches 122, 123

of Ferguson curve segments 84,85 Pentagonal surface 9, 10 Perpendicular bisector of a line segment 351,352 Physical model 3, 4 Plane curve, condition to be 29 Plane

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