TABLE OF CONTENTS Algebraic geometry deals with algebraic manifolds in projective spaces. In particular, algebraic geometry is concerned with invariants of curves and surfaces under projective and birational transformations. Isaac Newton (1643 1729) and Leonhard Euler (1707 1783) were the first to investigate algebraic properties of curves, but the actual founder of algebraic geometry is Max Noether (1844 1921). In the 20th century, algebraic geometry grew quickly and became more abstract. The more geometric part of algebraic geometry generalizes the methods developed above for quadrics to curves and surfaces of higher degree.
In 1984 Tom Sederberg introduced methods of algebraic geometry into geometric design when he considered calculating the intersection of algebraic curves and surfaces. Nearly at the same time an interesting relation between polynomial splines and polar forms was discovered by de Casteljau. This discovery lead to new insight into the properties of splines and their algorithmic construction.
Furthermore, the implicitization of parametric curves and surfaces and the so-called inverse problem can be solved by methods of classical algebraic geometry.
A curve in the plane or a surface in space may be given either parametrically, i.e., explicitly, or by an equation for the coordinates of its points, i.e., implicitly. If the curve or surface is given by one or more polynomials it is referred to as an algebraic curve or surface.
Literature: Brieskorn Kn rrer, Griffiths Harris,...
