[11] Gibson, C.G., Elementary Geometry of Algebraic Curves. Cambridge University

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1 References [1] Abhyankar, S S and Bajaj, C, Automatic parametrization of rational curves and surfaces I: Conics and conicoids. Computer-Aided Design Vol. 19, pp11 14, [2] Bézier, P, Style, mathematics and NC. Computer-Aided Design Vol. 22 No. 9, pp , [3] Boehm, W and Prautzsch, H, The insertion algorithm. Computer-Aided Design Vol. 17 No. 2, pp58 59, [4] Braid, I C, Hillyard, R C, and Stroud I A, Stepwise construction of polhedra in geometric modelling in Mathematical Methods in Computer Graphics and Design, ed. K W Brodlie, pp , Academic Press, [5] Coolidge, J L, A History of the Conic Sections and Quadric Surfaces. OUP, [6] Davis, P, B-splines and geometric design, SIAM News Vol. 29 No. 5, [7] Dill, J. An application of colour graphics to the display of surface curvature. Computer Graphics Vol. 15, pp , [8] Do Carmo, M P, Differential Geometry of Curves and Surfaces. Prentice- Hall, [9] Farin, G, Curves and Surfaces for Computer-Aided Geometric Design. Third Edition. Academic Press, [10] Forrest, A R, Interactive interpolation and approximation by Bézier polynomials, Computer-Aided Design Vol. 22 No. 9, pp , Originally published in The Computer Journal Vol. 15 No. 1, pp71 79,

2 346 References [11] Gibson, C.G., Elementary Geometry of Algebraic Curves. Cambridge University Press, [12] Haralick, R M and Shapiro, L G, Computer and Robot Vision. Addison- Wesley, [13] Hoschek, J and Lasser, D, Fundamentals of Computer Aided Geometric Design. A K Peters, [14] Howard, T L J, Hewitt, W T, Hubbold, R J, and Wyrwas, K M, APractical Introduction to PHIGS and PHIGS PLUS. Addison-Wesley, [15] Lane, J and Riesenfeld, R, A geometric proof for the variation diminishing property of B-spline approximation. J. of Approximation Theory Vol. 37, pp1 4, [16] Mäntylä, M, An Introduction to Solid Modeling, Computer Science Press, Maryland, [17] Munchmeyer, F, On surface imperfections. In R.Martin, editor, The Mathematics of Surfaces II, pp OUP, [18] Munchmeyer, F, Shape interrogation: a case study. In G.Farin, editor, Geometric Modelling: Algorithms and New Trends, pp SIAM, Philadelphia, [19] Phong, B-T, Illumination for computer-generated pictures. Comm. ACM, Vol. 18, No. 6, pp , June [20] Piegl, L and Tiller, W, The NURBS Book. Springer-Verlag, [21] Rogers, D F and Adams, J A, Mathematical Elements for Computer Graphics. Second Edition. McGraw-Hill, [22] Schoenberg, I, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. Vol. 4, pp45 99, [23] Sederberg, Th W, Anderson, D C and Goldman, R N, Implicit representation of parametric curves and surfaces. Computer Vision, Graphics and Image Processing Vol. 28, pp72 84, [24] Semple, J G and Kneebone, G T, Algebraic Projective Geometry. OUP, [25] Smith, G, Introductory Mathematics: Algebra and Analysis. Springer- Verlag, [26] Sommerville, D M Y, Analytical Conics. Bell and Sons, [27] Spivak, M, Calculus. W.A.Benjamin, 1967.

3 Index affine invariance 147, 177, 195, 214, 236 ambient light 299, 304 apparent contour 310 apparent cusps 319 attenuation 305 axonometric projection see projection B-rep 263 B-spline 194 basis 187, 192 closed 200, 235 curve 188 derivatives 207, 216, 238 integral 188 NURBS 212 open 196, 235 open uniform 202 periodic 200, 235 rational 213 surface 234 uniform 198 Bernstein polynomial 141, 144 Bézier control point 135 homogeneous 175 control polygon , 141 cubic 137 curve 136, 141, 161 curvature 283 torsion 283 derivatives 162 endpoint-interpolation 139, 147, 236 integral 141 linear 136 piecewise 168 properties 147 quadratic 136 rational 175 rendering 157 subdivision 154 surface 234 binomial 142 blend 233 Boehm algorithm 221 breakpoints 169, 192 C k -continuity 99, 170, 226 CAD see computer-aided design, 260 Cartesian plane 2 catenary 273 cavalier projection see projection centre of perspectivity 68 clip 76 clothoid 273 CMY 299 computer-aided design 49, 135 concatenation see transformation conic 109 applications 132 central 113 conversion 127 degenerate 109 directrix 110 discriminant 112 eccentricity 110 ellipse 109, 116, 177 focus 110 hyperbola 109, 116, 134, 177 irreducible 112,

4 348 Index parabola 109, 116, 132, 177 parametrization 124 reducible 112, 114 spatial 130 continuity 99, 192, 195, 214, 226, 253 control point 187 conversion tobézier form 166 convex hull 146, 147, 177, 195, 214, 236 Coons surface 256 coordinate functions 96 coordinate curve 226 coordinates Cartesian 1 homogeneous 14, 20, 41 Plücker 54 viewplane see viewplane Cornu spiral 273 CSG 261 curvature 267, 275 Bézier curves 283 normal 286 principal 286 vector 275 curve algebraic 96 curvature 267, 275 implicit 96 non-parametric explicit 96 parametric 96 polynomial 96 rational 96 regular 99 segment 96 cusp 138 cycloid 273 de Boor algorithm 205 rational 218 de Casteljau 151, 152 rational 180 deformation 204 degree 96, 187 degree raising 146 Denavit Hartenberg 17 device coordinate transformation 37, 80 device window 76 diffuse reflection 300 dimetric projection see projection dual 40 elliptic point 291 equivalence relation 21 Euler angles 51 Euler Poincaré formula 265 evolute 274 flat point 291 flat shading 307 font design 203 foreshortening ratio 85 Frenet frame 276 Frenet Serret formulae 277 G k -continuity 172, 253 geometric continuity 172 geometric modelling 260 gimbal lock 51, 64 Gordon Coons surface 256 Gouraud shading 307 graphical primitive 1, 15 Hamilton 56 helix 279 Hermite 254 hidden line 318 homogeneous control point 213 coordinates 19, 21 equation 24, 38 Horner s method 98 hotspot 303 HSV 298 hue 298 hyperbolic point 291 identity see transformation image 3 implicit 2, 225 incident ray 300 inflection 138 instancing 1, 15, 36 intensity 301, 305 intersection line and Bézier curve 158 line and conic 121 three planes 53 twobézier curves 159 two lines 39 inverse see transformation isometric projection see projection knot insertion 221 knot vector 187 Lambert s Law 301

5 Index 349 Lambertian surfaces 301 light ambient 299 attenuation 305 directional 299 distributive 299 intensity 301 point source 299 specular 299 line 2 through two points 39 line coordinates 54 line vector 39, 52 local control 195, 214 local support 192 lofting 254 logarithmic spiral 273 Monge patch 227 monomial form 166 morphing 203 natural equation 272 normal line 101 vector 100 normal plane 276 numerically controlled machining 107, 232 NURBS see B-spline object 4 oblique projection see projection offset 107, 232, 296 order 187 orientation 50 orthogonal change of coordinates 33 orthographic projection see projection see projective space parabolic point 291 parallel curve 107 parallel projection see projection parameter curve 226 parametric 2, 226 parametrization 96 partition of unity 192 perspective projection see projection Phong 303, 309 picture elements 15 piecewise polynomial 170, 187, 192 plane Cartesian 2 projective 19, 23, 24 P 3 through three points 52 plane vector 52 point at infinity 23, 25, 26, 41 positivity 192 principal curvature 286 principal direction 286 projection centre of perspectivity 72 line 68 ofbézier curve 181 of NURBS curve 214 parallel 69, 72 axonometric 86 cavalier 88 dimetric 87 isometric 87 oblique 87 orthographic 86 trimetric 87 perspective 68, 72, 90 one-point 91 three-point 91 two-point 91 viewpoint 72 projective invariance 178, 236 projective plane 19, 23, 24 P 2 see projective plane projective space 41 quaternions 51, 56 algebraic properties 58 animation 65 conjugate 59 interpolation 65 inverse 59 polar form 60 rotations 62 unit 59 R 2 see Cartesian plane rational 175, 213 rectifying plane 276 reflected ray 300 reflection see transformation ambient 304 diffuse 300 specular 302 regular 99, 226 relation 21 rendering 98 reparametrization 104 RGB 298 right inverse 77 robotics 17

6 350 Index rolling-ball blend 233 rotation see transformation saturation 298 scaling see transformation self-occluding 302 shade 298 shading flat 307 Gouraud 307 Phong 309 shadow 320 shear see transformation shelling 232 silhouette 309 skinning 251 specular light 299 specular reflection 302 speed 99, 275 subdivision 154, 248 surface B-spline 235 Bézier 234 bilinear 243 constructions 241 curvature 285 Gaussian 291 mean 291 extruded 241 Gordon Coons 256 implicit 225 loft 254 non-parametric explicit 227 normal 227 NURBS 235 of revolution 49, 245 parametric 226 quadric 228 regular 226 ruled 242 singular 226 skin 251 subdivision 248 tangent vector 227 translational swept 244 tangent line 100, 101 vector 100, 275 thickening 233 tint 298 tone 298 torsion 275 vector 277 torus 232, 246, 295 trace 96 transformation 3 affine 21, 42 concatenation 13, 30 identity 6, 31 inverse 6, 31, 32 non-singular 4, 32 projective 20, 42 reflection 8 in arbitrary line 34 in arbitrary plane 47 rotation 9, 29, 43 about arbitrary line 45 about arbitrary point 33 scaling 7, 28, 43 shear 11 singular 4 translation 5, 27, 42 translation see transformation trimetric projection see projection umbilic point 291 variation diminishing property 147, 177 viewing pipeline 80 viewplane coordinates 76 window 76, 80 viewpoint see centre of perspectivity viewport window see device window visual tangent continuity 172 weight 175, 179

CHAPTER 1 Graphics Systems and Models 3

?????? 1 CHAPTER 1 Graphics Systems and Models 3 1.1 Applications of Computer Graphics 4 1.1.1 Display of Information............. 4 1.1.2 Design.................... 5 1.1.3 Simulation and Animation...........