Abstract: After constructing the three-parameter basis function with multiple shape parameters, this paper analyzed the excellent properties of the three-parameter basis function, defined the three-parameter curve, and presented the properties of the three-parameter curve. Next, based on the definition of continuity, the condition for smooth blending of three-parameter curves was obtained, and the segmented three-parameter curves were combed ensure the continuity of the combined curves. Then, by comparing the properties of three-parameter curve with the original basis function curve and the cubic B-spline curve, it can be seen that the three-parameter curve is closer to the control polygon, easier to meet the continuous condition, and also capable of adjusting the curve shape. Different models of curve rotating surface can be constructed by changing the values of different parameters. In addition, the energy minimization model was used to determine the single parameter value of the three-parameter curve, for the purpose that a most smooth transition of the curve can be accomplished. On this basis, the parameter optimization of the combined three-parameter curve under the constraint of bending energy was also studied and proved by the curvature vector. Finally, a unique combined three-parameter curve with energy constraints obtained.

Key words: three-parameter curve, blending, continuity, energy optimization, parameter selection