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Cubic spline curves

Tài liệu Cubic spline curves provides bezier curves with zero second derivative at one end; gluing two bezier curves; B-spline curves; interpolation by relaxed cubic splines; higher dimesions ans applications to animation; anonparametic version; basis fuctions; other possible end conditions.  | Math 149 W02 DD. Cubic spline curves 1. Overview Polynomial parametric curves of high degree have a disadvantage Requirements placed on one stretch of such a curve can have a very strong effect some distance way. In Figure 1 the jump in the height of the data points near the middle has a strong effect on the interpolating polynomial curve near the ends. In contrast Figure 2 shows an example of a cubic spline curve through the same data points. Notice how it follows them much more closely. The spline curve was constructed by using a different cubic polynomial curve between each two data points. In other words it is a piecewise cubic curve made of pieces of different cubic curves glued together. The pieces are so well matched where they are glued that the gluing is not obvious. In fact if the whole curve shown is described with a single function p i then p i is so smooth that it has a second derivative everywhere and this derivative is continuous. DD 1 Definition. A cubic spline curve is a piecewise cubic curve with continuous second derivative. The word spline actually refers to a thin strip of wood or metal. At one time curves were designed for ships and planes by mounting actual strips of wood or metal so that they went through the desired data points but were otherwise free to move. For reasons of physics such curves are approximately piecewise cubic with continuous second derivative if they are suitably parameterized. You may recall from calculus that the curvature of a curve at. each point depends on the second derivative there. At. the end points an actual wood or metal strip has no reason to bend and the second derivative of its curve is zero. Definition. A cubic spline curve is relaxed if its second derivative is zero at each endpoint. We shall concentrate OU relaxed cubic spline curves. As you will see they can be used either for controlled design B-splines or for interpolation. To describe the cubic pieces simply and conveniently we shall use cubic Bezier