A new series of the generalized Ball basises of degree-n with parameter k(2≤k≤「n/2」+1) are constructed as the transition from Wang-Ball basis (k=2) to Said-Ball basis (k=n/2+n), and some properties of them are given. Then, the new generalized Ball curves are based on these new basises and the algorithms for recursive evaluation, degree-elevation, and degree-reduction are introduced. Finally, the corresponding trigonal Ball basises are shown and the algorithms for the recursive evaluation and degree-elevation of trigonal Ball surfaces are given.