We introduce a vector representation called diffusion curve textures for mapping diffusion curve images (DCI) onto arbitrary surfaces. In contrast to the original implicit representation of DCIs [Orzan et al. 2008], where determining a single texture value requires iterative computation of the entire DCI via the Poisson equation, diffusion curve textures provide an explicit representation from which the texture value at any point can be solved directly, while preserving the compactness and resolution independence of diffusion curves. This is achieved through a formulation of the DCI diffusion process in terms of Green's functions. This formulation furthermore allows the texture value of any rectangular region (e.g.