The success of using streamline technique for visualizing a vector field usually depends largely on the choice of adequate seed points. Turk and Banks developed an elegant technique for automatically placing seed points to achieve a uniform distribution of streamlines on a 2D vector field. Their method uses an energy function calculated from the low-pass filtered streamline image to guide the optimization process of the streamline distribution. This paper proposes a new technique for creating evenly distributed streamlines on 3D parametric surfaces found in curvilinear grids. We make use of Turk and Banks's 2D algorithm by first mapping the vectors on a 3D surface into the computational space of the curvilinear grid. To take into the consideration the mapping distortion caused by the uneven grid density in a curvilinear grid, a new energy function is designed and used for guiding the placement of streamlines in the computational space with desired local densities.