1 INTRODUCTION Parametric surfaces are the basis of the geometric models used in several domains of engineering. For some specific applications, such surfaces can be re- placed by linear approximations assuming that every point from the resulting mesh will be within a given tolerance. The main motivation for this work is the application to the simplified representation of the ship’s hull form. In some engineering applications such as the fast prototyping of the hull structures, it is a relevant factor to speed-up the process the re- placement of the surface by a mesh of planar ele- ments. The tessellation of parametric surfaces can be car- ried out in the R 2 parametric domain or on simplified surface approximations, in R 3 . Polygonal meshes can be classified in structured or non-structured. For some engineering applications such as CFD compu- tations or ship dynamics, structured meshes are gen- erally required to obtain results. Non-structured meshes, namely the triangular ones, have typical ap- plication to visualization and finite element compu- tations, because they are able to be better adapted to the hull form. For the intended application, the final number of triangles of the mesh and their topological correctness is more important that any metric used to qualify the triangles shape. Triangular meshes present several advantages: they allow a simple topological structure, more compact data storage and are more easily adaptable to more complex shapes without the use of subdivi- sion or of block generation methods, which generally require an interactive definition of the boundaries. Subdivision methods generate the mesh by subdi- viding the surface recursively in the parametric do- main into quadrangular faces whose size depends of a pre-defined level of approximation. This subdivi- sion generates a topologically incorrect grid in the parametric domain, namely with vertices over the edges of adjacent faces (the normally called T con- nections). As a consequence, the triangular mesh generated from the grid is also topologically incorrect and may have undesired gaps between triangles. In order to avoid this incorrectness, the proposed method identi- fies and stores adjacent faces and common vertices in a Binary Space Partitioning (BSP) tree data struc- ture during the process of subdivision. With this ad- ditional data, the triangulation of the grid in the par- ametric domain is executed without Ts or holes and final result is an adaptive and topologically correct triangular on a parametric surface. This work is organized as follows: first it is dis- cussed the state of the art in mesh generation from parametric surfaces; next, it is presented the adaptive method proposed. Finally, some results of the appli- cation of the method to a sample free surface with different curvature distribution regions are presented and discussed. Generation of an Adaptive Triangular Mesh from a Parametric Surface J.M. Varela 1 & Manuel Ventura 2 Centre for Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 1 varela@mar.ist.utl.pt 2 mventura@mar.ist.utl.pt ABSTRACT: This paper presents a new method for the generation of an adaptive topologically correct trian- gular mesh on parametric surfaces. The level of approximation is controlled through a given tolerance result- ing into a density of triangles that increases with the local curvature of the surface. The surface is subdivided in the parametric domain by a recursive algorithm which simultaneously stores the sequence of operations into a Binary Space Partitioning tree data structure. To each interior edge defined by two or more vertices corre- spond always only two faces. After the subdivision, the resulting polygonal faces are triangulated using all the points generated. Finally the mesh obtained in the parametric domain is mapped back into the Cartesian 3D space. The generated mesh is topologically correct and provides a basis for a wide range of engineering appli- cations.