Издательство Springer, 2002, -521 pp.
Computer aided geometric design (CAGD) is a discipline dealing with computational aspects of geometric objects. It is best explained by a brief historical sketch.^
Renaissance naval architects in Italy were the first to use drafting techniques that involved conic sections. Prior to that, ships were built "hands on" without any mathematics being involved. These design techniques were refined through the centuries, culminating in the use of splines—wooden beams that were bent into optimal shapes. In the beginning of the twentieth century, airplanes made their first appearance. Their design (or rather, the design of the outside fuselage) was streamlined by the use of conic sections, as pioneered by R. Liming [390]. He devised methods that went beyond traditional drafting with conies—for the first time, certain conic coefficients could be used to define a shape—thus numbers could be used to replace blueprints!
The automobile, one of the defining cultural icons of the twentieth century, also needed new design approaches as mass production started. In the late 1950s, hardware became available that allowed the machining of 3D shapes out of blocks of wood or steel.^ These shapes could then be used as stamps and dies for products such as the hood of a car. The bottleneck in this production method was soon found to be the lack of adequate software. In order to machine a shape using a computer, it became necessary to produce a computer-compatible description of that shape. The most promising description method was soon identified to be in terms of parametric surfaces. An example of this approach is provided in Color Plates I and III: Color Plate I shows the actual hood of a car; Color Plate III shows how it is represented inteally as a collection of parametric surfaces.
The major breakthroughs in CAGD were the theory of Bezier curves and surfaces, later combined with B-spline methods. Bezier curves and surfaces were independently developed by P. de Casteljau at Citroen and by P. Bezier at Renault. De Casteljau's development, slightly earlier than Bezier's, was never published, and so the whole theory of polynomial curves and surfaces in Bestein form now bears Bezier's name. CAGD became a discipline in its own right after the 1974 conference at the University of Utah (see Bahill and Riesenfeld [34]). Several other disciplines have emerged and interacted with CAGD. Computational geometry is conceed with the analysis of geometric algorithms. An example would be finding a bound on the time it takes to triangulate a set of points. Knowledge of such bounds allows a comparison and evaluation of different algorithms. The literature includes Prepata and Shamos [497] and de Berg et al. [135]. Ironically, another book with the term computational geometry in it is the one by Faux and Pratt [228]. It was a very influential text, but today, it would be classified as a CAGD text.
Another related discipline is solid modeling. It is conceed with the representation of objects that are enclosed by an assembly of surfaces, mostly very elementary ones such as planes, cylinders, or tori. The literature includes Hoffmann [327] and Mantyla [416]. CAGD has also influenced fields such as medical imaging, geographic information systems, computer gaming, and scientific visualization. It should go without saying that computer graphics is one of the earliest and most important applications of CAGD; see [238] or [9].
How a Simple System Was Bo
Introductory Material
Linear Interpolation
The de Casteljau Algorithm
The Bestein Form of a Bezier Curve
Bezier Curve Topics
Polynomial Curve Constructions
B-Spline Curves
Constructing Spline Curves
Differential Geometry I
Geometric Continuity
Conic Sections
Rational Bezier and B-Spline Curves
Tensor Product Patches
Constructing Polynomial Patches
Composite Surfaces
Bezier Triangles
Practical Aspects of Bezier Triangles
Differential Geometry II
Geometric Continuity for Surfaces
Surfaces with Arbitrary Topology
Coons Patches
Shape
Evaluation of Some Methods
Computer aided geometric design (CAGD) is a discipline dealing with computational aspects of geometric objects. It is best explained by a brief historical sketch.^
Renaissance naval architects in Italy were the first to use drafting techniques that involved conic sections. Prior to that, ships were built "hands on" without any mathematics being involved. These design techniques were refined through the centuries, culminating in the use of splines—wooden beams that were bent into optimal shapes. In the beginning of the twentieth century, airplanes made their first appearance. Their design (or rather, the design of the outside fuselage) was streamlined by the use of conic sections, as pioneered by R. Liming [390]. He devised methods that went beyond traditional drafting with conies—for the first time, certain conic coefficients could be used to define a shape—thus numbers could be used to replace blueprints!
The automobile, one of the defining cultural icons of the twentieth century, also needed new design approaches as mass production started. In the late 1950s, hardware became available that allowed the machining of 3D shapes out of blocks of wood or steel.^ These shapes could then be used as stamps and dies for products such as the hood of a car. The bottleneck in this production method was soon found to be the lack of adequate software. In order to machine a shape using a computer, it became necessary to produce a computer-compatible description of that shape. The most promising description method was soon identified to be in terms of parametric surfaces. An example of this approach is provided in Color Plates I and III: Color Plate I shows the actual hood of a car; Color Plate III shows how it is represented inteally as a collection of parametric surfaces.
The major breakthroughs in CAGD were the theory of Bezier curves and surfaces, later combined with B-spline methods. Bezier curves and surfaces were independently developed by P. de Casteljau at Citroen and by P. Bezier at Renault. De Casteljau's development, slightly earlier than Bezier's, was never published, and so the whole theory of polynomial curves and surfaces in Bestein form now bears Bezier's name. CAGD became a discipline in its own right after the 1974 conference at the University of Utah (see Bahill and Riesenfeld [34]). Several other disciplines have emerged and interacted with CAGD. Computational geometry is conceed with the analysis of geometric algorithms. An example would be finding a bound on the time it takes to triangulate a set of points. Knowledge of such bounds allows a comparison and evaluation of different algorithms. The literature includes Prepata and Shamos [497] and de Berg et al. [135]. Ironically, another book with the term computational geometry in it is the one by Faux and Pratt [228]. It was a very influential text, but today, it would be classified as a CAGD text.
Another related discipline is solid modeling. It is conceed with the representation of objects that are enclosed by an assembly of surfaces, mostly very elementary ones such as planes, cylinders, or tori. The literature includes Hoffmann [327] and Mantyla [416]. CAGD has also influenced fields such as medical imaging, geographic information systems, computer gaming, and scientific visualization. It should go without saying that computer graphics is one of the earliest and most important applications of CAGD; see [238] or [9].
How a Simple System Was Bo
Introductory Material
Linear Interpolation
The de Casteljau Algorithm
The Bestein Form of a Bezier Curve
Bezier Curve Topics
Polynomial Curve Constructions
B-Spline Curves
Constructing Spline Curves
Differential Geometry I
Geometric Continuity
Conic Sections
Rational Bezier and B-Spline Curves
Tensor Product Patches
Constructing Polynomial Patches
Composite Surfaces
Bezier Triangles
Practical Aspects of Bezier Triangles
Differential Geometry II
Geometric Continuity for Surfaces
Surfaces with Arbitrary Topology
Coons Patches
Shape
Evaluation of Some Methods