The Nature Of The Interval B¨¦zier Curves And Surfaces

As an important research problem, interval B(?)zier curves and surfaces have been investigated widely. This paper mainly discuss their properties. Like general B(?)zier curves and surfaces, interval B(?)zier curves and surfaces have many nice geometric properties ,such as degree elevation ,subdivision,and so on.The number of interval control point will increase if the degree of interval B(?)zier curve is elevated. And so does the number of elements made up of the interval curve. Some cuves degree of n+l can not be obtained by elevating degree of the arbitrary original cuve.So we doubt that the region is changed in the view of cuves.In the other hand , the two region is equal in the view of point collection. We will get the same conclusion when the interval B(?)zier curves are subdivided. So the interval B(?)zier curves have the properties of degree elevation and subdivision.In chapter two, the degree elevation formulae for B(?)zier curves and surfaces are generalized to interval B(?)zier curves and surfaces. We give the definition of interval control polygon and prove that the interval control polygon converges to the original interval B(?)zier curve as the degree elevated incessantly. So do interval B(?)zier surfaces. These enable us to convert an interval B(?)zier curve or surface of lower degree to a higher one which has more flexibility in controlling its shape. Also they provide us an efficient geometric method to generate an interval B(?)zier curve or surface.In chapter three, the subdivision formula for B(?)zier curves is generalized to interval B(?)zier curves . We prove that the interval control polygon converges to the original interval B(?)zier curve as the curve subdivided incessantly. Subdivision enable us to get new control polygon which has more interval control points and more flexibility in controlling its shape. So they provide us another efficient geometric method to generate an interval B(?)zier curve .In chapter four, we generalize the properties of degree elevation and subdivision to rational interval B(?)zier curves and surfaces.Finally, we point out that interval B(?)zier curves have the properties of convexity preserving and variation diminishing .