CORDIC (COordinate Rotation Digital Computer) is an iterative algorithm for the calculation of the rotation of a two-dimensional vector, in linear, circular or hyperbolic coordinate systems, using only add and shift operations. This paper presents a novel algorithm and architecture for the rotation-mode in circular and hyperbolic coordinate systems in which the directions of all micro-rotations are pre-computed while maintaining a constant scale factor. Thus, an examination of the sign of the angle after each iteration is no longer required. By using a redundant adder, the critical path (without scaling and conversion) of the entire CORDIC architecture only requires (1.5n+2) full-Adders (n corresponds to the word-length of the inputs) for rotation mode. This is a speed improvement of about 20% compared to the previously fastest reported rotation mode implementations. Additionally, there is a higher degree of freedom in choosing the pipeline cutsets due to the novel feature of independence of the iterations i and i-1 in the CORDIC rotation. Optional pipelining can lead for example in the rotation mode to an on-line delay of three clock cycles including scaling and conversion, where every clock cycle corresponds to a delay of twelve full-Adders.