Accurately and efficiently approximating complex geometries is crucial for the success of numerical simulations, especially when dealing with poorly defined geometries encountered in engineering and physics. These geometries may include non-watertight surfaces, intricate details, or be represented as point clouds or triangle soups, presenting significant challenges for constructing reliable representations. In this work we introduce the concept of intrinsic triangulation into an adaptive meshing process, offering a novel form of intrinsic mesh generation for arbitrary input geometries. The method begins with a high-resolution triangulation of the surface, which serves as the ground truth and can be reliably constructed from any input geometry. Starting from this surface, the approach adapts the principles of the MeshAdapt algorithm, replacing local optimization techniques with intrinsic operations such as edge swap, edge collapse, etc. These operations involve the computation of the shortest geodesic paths between points in the surface triangulation, which can be achieved through the MMP algorithm. The resulting intrinsic mesh is a segmentation of the initial triangulated surface where each edge in the mesh represents the shortest path between two vertices on the surface. Importantly, the segmentation process does not involve approximation; the vertices, edges, and faces are composed of or lie on parts of the original surface. An additional approximation must be introduced to take advantage of this intrinsic mesh for numerical simulations. Several approaches are available, with the simplest being the approximation of the intrinsic mesh using high-order polynomial elements. Alternatively, the segmentation can be used to construct a NURBS representation for isogeometric methods, or directly applied in hybrid high-order methods, exploiting the multiscale nature of the mesh. In this study, we present the method as the generation of an intrinsic triangulation of a closed surface with the primary goal of approximating the surface using high-order polynomial elements. This objective imposes specific requirements and constraints on the computed intrinsic mesh, which significantly affect the final outcome. We will also discuss the limitations of the method and propose improvements. In addition, the approach will be extended to open surfaces, exploring how the methodology can be adapted to address the challenges associated with boundary management in these cases.