Liming-like curve constructions

Abstract

R.A. Liming developed a method based on classical conic section theory to design cross-sections of aircraft using implicit rather than explicit forms of the quadratic equations. In its basic form the method is used to construct a one-parameter family of conics through four points or, equivalently, a family tangential to two given lines at two points. The parameter, known as the Liming multiplier, is then adjusted to ensure that the conic passes through a fifth chosen point. This method is more stable than the more common approach of two-dimensional interpolation with a general quadratic equation in two variables. The method can be extended to the construction of higher order curves, with applications in computer-aided design and computer graphics. The method results in the efficient use of a hierarchy of algorithms that can be combined to construct curves of increasing complexity. Since all the curves are in implicit form, this also necessitates the development of algorithms for the efficient and aesthetically pleasing graphics representation of higher order curves.