The term computer graphics was often associated with creation of realistic scenes and animated images. With the advent of both computers and applied mathematics more ambitious applications of computer graphics were sought in the last few decades. Today, one of the most important areas where computer graphics plays a central role is computer-aided design (CAD) and computer-aided manufacture (CAM). Both CAD and CAM are extensively used in a large number of areas, including aerospace, automotive engineering, marine engineering, civil engineering, and electronic engineering. The successful applications of computer graphics in engineering is largely due to the progress of computer aided geometric design (CAGD) which provides the mathematical basis for describing and processing geometric shapes and data. The term geometric modeling(which includes curve modeling, surface modeling, and solid modeling) is often used as the synonym of computer aided geometric design, although some authors argue that geometric modeling means the building up of computer representations of complex shapes from representations of similar components.
The most promising description method of geometric shape is the parametric curves and surfaces. The theory of parametric curves and surfaces are well understood in algebraic geometry and differential geometry. However, their possibilities and advantages for representing geometric entities in a CAD environment were not known until the late 1950s. The major breakthroughs in CAGD were undoubtedly the theory of Ferguson curves and patches, Coons patches, Bézier curves and surfaces, later combined with B-spline methods. Today, Bézier and B-spline representations of curves and surfaces have been the industrial standard.
In this chapter, we shall briefly review different curve representation methods and tell you why the parametric representation of curves is of mots importance. Then, we shall discuss the mathematics on Bézier and B-spline curves, which is the foundation of surface and solid modeling.