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The plotting of streamlines is an effective way of visualizing fluid motion in steady flows. Additional information about the flowfield such as local rotation and expansion can be shown by drawing in the form of a ribbon or tube. In this paper we present efficient algorithms for the construction of streamlines streamribbons and streamtubes on unstructured grids. A specialized version of the Runge-Kutta method has been developed to speed up the integration of particle paths. We have also derived closed-form solutions for calculating angular rotation rate and radius to construct streamribbons and streamtubes respectively. According to our analysis and test results these formulations are two to four times better in performance than previous numerical methods. As a large number of traces are calculated the improved performance could be significant.