Research On Spline Curves With Parameters In The Euclidean Space R~3 And S~3 The Unit Sphere

Geometric modeling with parametric curves and surfaces is one of the important research problems in computer-aided geometric design (CAGD). In this field there is a wide range of development space. Design method based on the parametric curves and surfaces has been widely used in aerospace product development,3D animation, video effects processing and so on.In order to improve the flexibility of modeling methods, requirements in case of no changing the control points to modify the curve should be met. A representative method is to use the Non-Uniform Rational B-Spline (NURBS), by modifying the weight factors to change the shape of the curve. But there are some problems in the NURBS method such as the calculation is complicated, the relationship is not very clear between changing the weight factors and modifying the shape. Another approach is to introduce shape parameters into the basis functions and the shape of a curve is adjusted by changing the shape parameters. This method has the advantages of flexible operations, small calculated amount, predictable change in shape etc, and thus become one of the hot research field in CAGD. Firstly, the G1 and G2 smooth connection problems are studied by using a class of generalized Bernstein basis function in this paper. Then a Gamma spline with parameters and a Beta spline with shape parameters in the Euclidean space R3 are constructed respectively. Secondly, taking into account the practical application, there may be some special additional constraints in constructing curves. For example, in spherical objects CNC machining, the toolpath is located in a sphere; In computer animation, it is required to be smooth interpolation of three-dimensional object orientation keyframes; In Robot path planning, robot requires smooth movement, etc. Because with the constraints, handling these problems are more complex with the conventional Bézier curve or spline curve in R3.Therefore, quaternion method is introduced to construct spherical spline with shape parameters on the unit sphere S3. The resulting curve with parameters not only has many excellent properties of the original curve but also has a more flexible adjustability. Finally, the impact of different values of shape parameters is discussed and the feasibility and effectiveness of the algorithm are verified by numerical experiments.