In this paper we consider a new approach for single object segmentation in 3D images. Our method improves the classical geodesic active surface model. It greatly simplifies the model initialization and naturally avoids local minima by incorporating user extra information into the segmentation process. The initialization procedure is reduced to introducing 3D curves into the image. These curves are supposed to belong to the surface to extract and thus, also constitute user given information. Hence, our model finds a surface that has these curves as boundary conditions and that minimizes the integral of a potential function that corresponds to the image features. Our goal is achieved by using globally minimal paths. We approximate the surface to extract by a discrete network of paths. Furthermore, an interpolation method is used to build a mesh or an implicit representation based on the information retrieved from the network of paths. Our paper describes a fast construction obtained by exploiting the Fast Marching algorithm and a fast analytical interpolation method. Moreover, a Level set method can be used to refine the segmentation when higher accuracy is required. The algorithm has been successfully applied to 3D medical images and synthetic images.