Abstract
This paper offers a unifying survey of some of the most relevant computational issues appearing in geometric processing (such as blending, trimming, intersection of curves and surfaces, offset curves and surfaces, NC milling machines and implicitization). Applications of these topics to industrial environments are also described.
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K. Abdel-Malek and H. J. Yeh: On the determination of starting points for parametric surface intersections. CAD 29 (1997) 21–35
S. Aomura and T. Uehara: Self-intersection of an offset surface. CAD 22 (1990) 417–422
C. Asteasu: Intersection of arbitrary surfaces. CAD 20 (1988) 533–538
L. Bardis and N. M. Patrikalakis: Blending rational B-spline surfaces. Eurographics’89 (1989) 453–462
R. E. Barnhill: Geometry Processing for Design and Manufacturing, SIAM, Philadelphia, PA (1992)
R. E. Barnhill and S. N. Kersey: A marching method for parametric surface/surface intersection. CAGD 7 (1990) 257–280
E. L. Brechner: General tool offset curves and surfaces. In: R. E. Barnhill (ed.): Geometry Processing for Design and Manufacturing, SIAM (1992) 101–121
B. Buchberger: Gröbner bases: an algorithmic method in polynomial ideal theory. In: N. K. Rose (ed.): Multidimensional Systems theory, Reidel Publishing Co. (1985) 184–232
W. R. Carlson: An algorithm and data structure for 3D object synthesis using surface patch intersections. Computer Graphics 16 (1982) 255–263
E. W. Chionh and R. N. Goldman: Using multivariate resultants to find the implicit equation of a rational surface. The Visual Computer 8 (1992) 171–180
K. P. Cheng: Using plane vector fields to obtain all the intersection curves of two general surfaces. In: W. Strasser and H. P. Seidel (ed.): Theory and Practice in Geometric Modeling, Springer, New York (1989) 187–204
B. K. Choi and S. Y. Ju: Constant-radius blending in surface modeling. CAD 21 (1989) 213–220
E. Cohen, T. Lyche and R. F. Riesenfeld: Discrete B-splines and subdivision techniques in CAGD and computer graphics. Computer Graphics and Image Processing 14 (1980) 87–111
D. P. Dobkin, S. V. F. Levy, W. P. Thuston and A. R. Wilks: Contour tracking by piecewise linear approximations. ACM Trans. on Graph. 9 (1990) 389–423
G. Elber, I. Lee and M. S. Kim: Comparing offset curve approximation methods. IEEE Comp. Graph. and Appl. 17(3) (1997) 62–71
G. Farin: Trends in curve and surface design. CAD 21(5) (1989) 293–296
G. Farin: An ISS bibliography. In: R. E. Barnhill (ed.): Geometry Processing for Design and Manufacturing, SIAM (1992) 205–207
G. Farin: Curves and Surfaces for Computer Aided Geometric Design, Fourth Edition, Academic Press, San Diego (1996)
G. Farin: NURB Curves and Surfaces: from Projective Geometry to Practical Use, Second Edition, AK Peters, Wellesley, MA (1999)
G. Farin and D. Hansford: The Essentials of CAGD, AK Peters, Wellesley, MA (2000)
R. T. Farouki: The approximation of non-degenerate offset surfaces. CAGD 3 (1986) 15–43
R. T. Farouki: Pythegorean-hodograph curves in practical use. In: R. E. Barnhill (ed.): Geometry Processing for Design and Manufacturing, SIAM (1992) 3–33
I. D. Faux and M. J. Pratt: Computational Geometry for Design and Manufacture, Ellis Horwood, Chichester (1979)
D. J. Filip: Blending parametric surfaces. ACM Trans. on Graph. 8(3) (1989) 164–173
G. Glaeser and E. Gröller: Efficient volume-generation during the simulation of NC-milling. In: H. C. Hege and K. Polthier (ed.): Mathematical Visualization. Algorithms, Applications and Numerics, Springer Verlag, Berlin (1998) 89–106
R. N. Goldman: Subdivision algorithms for Bézier triangles. CAD 15 (1983) 159–166
J. G. Griffiths: A data structure for the elimination of hidden surfaces by patch subdivision. CAD 7 (1975) 171–178
E. Hartmann: Blending of implicit surfaces with functional splines. CAD 22 (1990) 500–506
M. Held: On the computational geometry of pocket machining. Lectures Notes in Computer Science, 500, Springer Verlag, Berlin, New York (1991)
C. M. Hoffmann: Geometric and Solid Modeling, Morgan Kaufmann, San Mateo, CA (1989)
C. M. Hoffmann: Algebraic and numerical techniques for offsets and blends. In: S. Micchelli, M. Gasca and W. Dahmen (ed.): Computations of Curves and Surfaces, Kluwer Academic (1990) 499–528
T. van Hook: Real time shaded NC milling display. Computer Graphics 20(4) (1986) 15–20 (Proc. SIGGRAPH’86)
J. Hoschek and F. J. Schneider: Approximate spline conversion for integral and rational Bézier and B-spline surfaces. In: R. E. Barnhill (ed.): Geometry Processing for Design and Manufaturing, SIAM (1992) 45–86
J. Hoschek and D. Lasser: Fundamentals of Computer Aided Geometric Design, A.K. Peters, Wellesley, MA (1993)
E. G. Houghton, R. F. Emnett, J. D. Factor and C. L. Sabharwal: Implementation of a divide-and-conquer method for intersection of parametric surfaces. CAGD 2 (1985) 173–183
Y. Huang and J. H. Oliver: NC milling error assessment and tool path correction. Computer Graphics Proceedings (1994) 287–294 (Proc. SIGGRAPH’94)
K. C. Hui: Solid sweeping in image space-application in NC simulation. The Visual Computer 10 (1994) 306–316
R. B. Jerard, S. Z. Hussaini, R. L. Drysdale and B. Schaudt: Approximate methods for simulation and verification on NC machining programs. The Visual Computer 5 (1989) 329–348
D. Kapur and Y. N. Lakshman: Elimination methods. In: B. Donald, D. Kapur and J. Mundy (ed.): Symbolic and Numerical Computing for Artificial Intelligence, Academic Press (1992)
M. Kalkbrener: Implicitization of rational parametric curves and surfaces. Technical Report, Kepler Universität, Linz, Austria, RISC, Linz (1990)
Y. Kawashima, K. Itoh, T. Ishida, S. Nonaka and K. Ejiri: A flexible quantitative method for NC machining verification using a space-division based solid model. The Visual Computer 7 (1991) 149–157
T. L. Kay and J. T. Kajiya: Ray tracing complex scenes. Computer Graphics 20 (1986) 269–278
K. J. Kim and M. S. Kim: Torus/sphere intersection based on configuration space approach. Graphical Models and Image Processing 60(1) (1998) 77–92
R. Klass and P. Schramm: NC milling of CAD surface data. In: H. Hagen and D. Roller (ed.): Geometric Modeling. Methods and Applications, Springer Verlag, Berlin Heidelberg (1991) 213–226
R. Klass and B. Kuhn: Fillet and surface intersections defined by rolling balls. CAGD 9 (1992) 185–193
P. A. Koparkar: Designing parametric blends: surface model and geometric correspondence. The Visual Computer 7 (1991) 39–58
D. Lasser: Intersection of parametric surfaces in the Bernstein-Bézier representation. CAGD 3 (1986) 186–192
W. Lü: Offset-rational parametric plane curves. CAGD 12 (1995) 601–616
W. Ma and Y. S. Lee: Detection of loops and singularities of surface intersections. CAD 30 (1998) 1059–1067
T. Maekawa: An overview of offset curves and surfaces. CAD 31 (1999) 165–173
D. Manocha and J. F. Canny: Algorithm for implicitizing rational parametric surfaces. CAGD 9 (1992) 25–50
D. Manocha and J. F. Canny: Implicit representations of rational parametric surfaces. J. of Symbolic Computation 13 (1992) 485–510
J. Miller and R. N. Goldman: Geometric algorithms for detecting and calculating all conic sections in the intersection of any two natural quadric surfaces. Graphical Models and Image Processing 57(1) (1995) 55–66
J. C. Owen and A. P. Rockwood: Intersection of general implicit surfaces. In: G. E. Farin (ed.): Geometric Modeling: Algorithms and New Trends, SIAM (1987) 335–345
N. M. Patrikalakis and L. Bardis: Offsets of curves on rational B-spline surfaces. Engineering with Computers 5 (1989) 39–46
J. Pegna and D. J. Wilde: Spherical and circular blending of functional surfaces. Trans. of ASME, Journal of Offshore Mechanics and Artic Engineering 112 (1990) 134–142
H. Persson: NC machining of arbitrarily shaped pockets. CAD 10 (1978) 169–174
G. Petrie and T. K. M. Kennie: Terrain modeling in surveying and civil engineering. CAD 19 (1987) 171–187
B. Pham: Offset curves and surfaces: a brief survey. CAD 24 (1992) 223–229
L. Piegl: Key developments in Computer-Aided Geometric Design, CAD 21(5) (1989) 262–273
L. Piegl and W. Tiller: The NURBS Book, Second Edition, Springer Verlag, Berlin Heidelberg (1997)
L. Piegl and W. Tiller: Computing offsets of NURBS curves and surfaces. CAD 31 (1999) 147–156
L. Piegl and W. Tiller: Geometry-based triangulation of trimmed NURBS surfaces. CAD 30 (1998) 11–18
H. Pottmann: Rational curves and surfaces with rational offsets. CAGD 12 (1995) 175–192
H. Pottmann: General offset surfaces. Neural, Parallel and Scientific Computations 5 (1997) 55–80
A. Rockwood: The displacement method for implicit blending of surfaces in solid modeling. ACM Trans. on Graph. 8(4) (1989) 279–297
A. Rockwood, K. Heaton and T. Davis: Real-time rendering of trimmed surfaces. Computer Graphics 23 (1989) 107–116 (Proc. SIGGRAPH’89)
D. F. Rogers: An Introduction to NURBS: with Historical Perspective, Morgan Kaufmann, San Mateo, CA (2000)
J. R. Rossignac and A. A. G. Requicha: Offsetting operations in solid modeling. CAGD 3 (1986) 129–148
S. E. O. Saeed, A. de Pennington and J. R. Dodsworth: Offsetting in geometric modeling. CAD 20 (1988) 67–74
G. Salmon: Lessons Introductory to the Modern Higher Algebra, G.E. Stechert & Co., New York (1885)
T. Saito and T. Takahashi: NC machining with G-buffer method. Computer Graphics 25(4) (1991) 207–216 (Proc. SIGGRAPH’91)
T. W. Sederberg: Implicit and parametric curves and surfaces for computer aided geometric design. Ph.D. thesis, Purdue Univ., West Lafayette, IN (1983) 29–42
T. W. Sederberg, D. C. Anderson and R. N. Goldman: Implicit representation of parametric curves and surfaces. Computer Vision, Graphics and Image Processing 28 (1984) 72–74
T. W. Sederberg: Algebraic geometry for surface and solid modeling. In: G. E. Farin (ed.): Geometric Modeling: Algorithms and New Trends, SIAM (1987) 29–42
T. W. Sederberg and R. J. Meyers: Loop detection in surface patch intersections. CAGD 5 (1988) 161–171
M. Shantz and S. L. Chang: Rendering trimmed NURBS with adaptive forward differences. Computer Graphics 22 (1988) 189–198 (Proc. SIGGRAPH’88)
C. K. Shene and J. Johnstone: On the lower degree intersections of two natural quadrics. ACM Trans. on Graphics 13(4) (1994) 400–424
X. Sheng and B. E. Hirsch: Triangulation of trimmed surfaces in parametric space. CAD 24(8) (1992) 437–444
A. I. Sourin and A. A. Pasko: Function representation for sweeping by a moving solid. IEEE Trans. on Visualization and Computer Graphics 2(2) (1996) 11–18
D. C. Sutcliffe: Contouring over rectangular and skewed rectangular grids. In: K. Brodlie (ed.): Mathematical Methods in Computer Graphics and Design, Academic Press (1980) 39–62
M. Sweeney and R. Bartels: Ray tracing free-form B-spline surfaces. IEEE Comp. Graph. and Appl. 6 (1986) 41–49
A. E. Vries-Baayens: Conversion of a Composite Trimmed Bézier Surface into Composite Bézier Surfaces. In: P. J. Laurent, Le Mehaute and L. L. Schumaker (ed.): Curves and Surfaces in Geometric Design, Academic Press, Boston, USA (1991) 485–489
A. E. Vries-Baayens and C. H. Seebregts: Exact Conversion of a Composite Trimmed Nonrational Bézier Surface into Composite or Basic Nonrational Bézier Surfaces. In: H. Hagen (ed.): Topics in Surface Modeling, SIAM, Philadelphia, USA (1992) 115–143
Y. Wang: Intersections of offsets of parametric surfaces. CAGD 13 (1996) 453–465
J. Warren: Blending algebraic surfaces. ACM Trans. on Graph. 8(4) (1989) 263–278
D. B. Welborun: Full three-dimensional CAD/CAM. CAE Journal 13 (1996) 54–60, 189-192
J. R. Woodwark: Blends in geometric modeling. In: R. R. Martin (ed.): The Mathematics of Surfaces II, Oxford Univ. Press (1987) 255–297
W. T. Wu: Basic principles of mechanical theorem proving in geometries. J. of Systems Sciences and Mathematical Sciences 4 (1986) 207–235
C. G. Yan: On speeding up ray tracing of B-spline surfaces. CAD 19 (1987) 122–130
J. Yen, S. Spach, M Smith and R. Pulleyblank: Parallel boxing in B-spline intersection. IEEE Comp. Graph. and Appl. 11 (1991) 72–79
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Iglesias, A., Gálvez, A., Puig-Pey, J. (2001). Computational Methods for Geometric Processing. Applications to Industry. In: Alexandrov, V.N., Dongarra, J.J., Juliano, B.A., Renner, R.S., Tan, C.J.K. (eds) Computational Science — ICCS 2001. ICCS 2001. Lecture Notes in Computer Science, vol 2073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45545-0_81
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