Section: Scientific Foundations

Scientific Foundations

In Geometry Modeling, shapes are described in term of Parametric Surfaces. The pioneering work in this domain was the theory of Bézier curves and surfaces (theory of polynomial curves and surfaces in Bernstein form), later combined with B-spline methods. Today, Non-Uniform Rational B-Spline (NURBS) have become the standard curves and surfaces description in the field of CAD. Differential Geometry is also an important scientific foundation for Geometry Modeling. Differential geometry is based largely of the pioneering work of L. Euler (1707-1783), C. Monge (1746-1818) and C.F. Gauss (1777-1855). One of their concerns was the description of local curves and surface properties such as curvature. These concepts are also of interest in modern computer-aided geometry design. The main tool for the development of general results is the use of local coordinate systems, in term of which geometric properties are easily described and studied.