A rational spline with tension: some CAGD perspectives

It is often necessary to represent a hand-drawn shape accurately. Modeling such shapes manually is both cumbersome and commercially expensive. The user's concern for curves that are easy to manipulate has been a major influence on the development of free-form curves. Rational parametric curves have been receiving considerable attention in the area of geometric modeling because any parametric polynomial curve can be expressed as a rational curve and most polynomial splines have rational extensions. Rational parametric curves can be used to model any object, like ships and airplanes, and even in the medical field for modeling the heart or other parts of the body. In object modeling, the rational cubic spline is most popular because it is the lowest degree that can define space curves and curves with points of zero curvature. The foremost objective of this work is to show how conic sections can adequately be used to represent curves and objects that were previously thought to require rational cubic splines. A single rational cubic curve is represented by two conics by splitting the rational cubic at its mid-point. The degree of smoothness has been taken into account to have a visually pleasing effect. It has been realized that the conic representation is advantageous over the rational cubic one in terms of computational requirements and shape control. All of this work is carried out on both 2D curves and 3D objects.