Dyna, Year 78, Nro. 166, pp. 133-141 Medellin, April, 2011. ISSN 0012-7353 AUTOMATIC CONSTRUCTION OF NURBS SURFACES FROM UNORGANIZED POINTS RECONSTRUCCIÓN AUTOMÁTICA DE SUPERFICIES NURBS A PARTIR DE PUNTOS NO ORDENADOS NALLIG EDUARDO LEAL NARVÁEZ Facultad de Ingeniería, Universidad Autónoma del Caribe, Barranquilla, nleal@uac.edu.co ESMEIDE ALBERTO LEAL NARVÁEZ Facultad de Ingeniería, Universidad Autónoma del Caribe, Barranquilla, esleal@uac.edu.co JOHN WILLIAN BRANCH Facultad de Minas, Universidad Nacional de Colombia, Medellín, jwbranch@unal.edu.co Received for review May 25 th, 2010, accepted November 23th, 2010, final version November, 25 th, 2010 ABSTRACT: Modeling with Non Uniform Rational B-Splines (NURBS) surfaces has become a standard in CAD/CAM systems due to its stability, flexibility, and local modification properties. The advantage of fitting with NURBS surfaces is well known, but it is also known that NURBS surfaces have several deficiencies. A NURBS surface cannot be fitted over an unorganized and scattered set of points and the representation of sharp features like edges, corners, and high curvatures is poor. This paper presents a new method for fitting a NURBS surface over an unorganized and scattered cloud of points, preserving its sharp features. In contrast with other methods, ours does not need either to construct a network of NURBS patches or polygon meshes. By reducing the dimensionality of the point cloud using ISOMAP algorithms, our method detects both regions with lacking points, and regions where the cloud is too dense. Then, the cloud is regularized by inserting and removing points, and it is approximated by a NURBS surface. An evolutionary strategy obtains the weights of the NURBS surface in order to improve the representation of sharp features. KEYWORDS: NURBS, ISOMAP, evolutionary strategies. RESUMEN: El modelamiento con Superficies B-Splines Racionales no Uniformes (NURBS) se ha convertido en un estándar en los sistemas CAD/CAM debido a su estabilidad, flexibilidad y propiedades de modificación local. Las ventajas de ajustar con superficies NURBS son bien conocidas, aunque también son conocidas las limitaciones que éstas presentan. Una superficie NURBS no puede ser ajustada sobre un conjunto de puntos dispersos no ordenados. Adicionalmente, la representación de detalles finos como aristas, esquinas y altas curvaturas, es pobre. Este artículo presenta un nuevo método para ajustar superficies NURBS sobre conjuntos de puntos dispersos no ordenados, preservando los detalles finos. A diferencia de otros métodos, el nuestro no necesita construir una red de parches NURBS ni mayas poligonales. Para reducir la dimensionalidad de la nube de puntos usando el algoritmo ISOMAP, nuestro método detecta regiones con carencia de puntos y regiones donde la nube es muy densa; luego, la nube es regularizada por inserción y remoción de puntos, para finalmente ser ajustada por una única superficie NURBS. Para mejorar la representación de los detalles finos, una estrategia evolutiva obtiene los pesos de la superficie NURBS. PALABRAS CLAVE: NURBS, ISOMAP, estrategias evolutivas. 1. INTRODUCTION NURBS is one of the most employed surface fitting models, provided that it is a standard representation of curves and surfaces [1]; and is widely supported by modern standards like OpenGL and IGES, which are used for graphics and geometric data exchange [2]. In addition, the NURBS surface model has stability, flexibility, local modification properties; and is robust to noise. However, NURBS surfaces models has some disadvantages: the input data points should be mapped on a regular grid structure [3] and the representation of sharp features is poor [14]. In the 3D reconstruction process, the registration and integration stages produce massive scattered and unorganized point clouds that cannot be mapped on a regular grid structure. Such point clouds cannot be fitted by a NURBS surface and are not suitable for usage in computer-aided design (CAD) systems [4]. In order to fit a NURBS surface to an unorganized and scattered point cloud, several approaches have been presented [3, 4, 5, 6, 7, 8]. Such approaches fit to the cloud a network of NURBS patches with some degree of continuity between them. The construction of the network requires constructing