Research On Surface Developability On C-Bézier Curve In CAGD

This paper profoundly studies surface developablity on new a set of curves in the field of CAGD and pays attention to the methods of developabe surface modeling and surface development on C-Bézier curve .The research achievements and main contents are as follows:1. The history and representative work of methods of the developable surface modeling and surface development in the field of CAGD are briefly reviewed.2. A succinct formula for calculating the derivative of the degree 3 C-Bézier curve is given. Then an important property is obtained : when a - 0, the first order derivative of the degree 3 C-Bézier curve is that of the degree 3 Bezier curve, aber the second order derivative of the degree 3 C-Bézier curve has no the property.3. By using the dual concept between points and planes in 3D projective space, two classes of developable surfaces on C-Bézier curve are constructed. The former is as a envelope of one-parameter plane family. The latter is a tangent surface with C-Bézier curve as spine curve.4. The methods of development for cones and tangent surface on C-Bézier curve and the complex surfaces are obtained. First, a map between the cones and tangent surface on C-Bézier curve und two-dimensional plane is found respectively and then a ruled surface between two Bezier curves is constructed and a sufficient condition on its developablity is given. Finally, the complex surfaces are developed by partition of the surfaces into small surface segment ,and approximation of the small segment with the developable surface between two Bezier curves. The number of partition is decided by the area accuracy a.5. Two classes of developable surfaces with any C-Bézier curve as curvature curve and geodesic curve respectively are constructed.Finally, when a 00, the limt situations for the developable modeling and surface development above are all disscussed and Compared with those on Bezier curve .The programs of the above methods und algorithms are developed, which shall be used in the system of developable surface modeling for CAD/CAM.