Abstract: We introduce a novel technique for the construction of smooth stream surfaces of 4th order precision. While common stream surface techniques use linear interpolation for generating seed points for new streamlines in the refinement phase, we use Hermite interpolation. The derivatives needed for Hermite interpolation are obtained by integration along the streamlines. This yields stream surfaces of4th order precision. Additionally, we analyse the accuracy ofthe well known Hultquist approach and our new algorithm and proof that Hultquist's method is exact for linear vector fields. We compare both methods using the well known distance based and a novel error based refinement strategy. Our resulting surface is C1 -continuous, enabling improved rendering among other benefits.