EUROGRAPHICS 2001 / Jonathan C. Roberts Short Presentations A Simple Validity Condition for B-Spline Hyperpatches F.A. Conde and J.C. Torres Department of Lenguajes y Sistemas Informáticos, The University of Granada, Granada, Spain Abstract The use of hyperpatches as a method for solid modelling has a problem: the validity of the model is not guaranteed. The problem of ensuring the validity of hyperpatch representations of solids is discussed in this work, and a validity condition for cubic uniform b-spline hyperpatches is presented. Our validity condition is based on comparisons among points, and it is robust and easy to implement. 1. Introduction A volumetric dataset is a special kind of non-homogeneous solid in which each point of its interior has a different prop- erty value varying continuously. Volumetric datasets can be used to model volumetric solids with smooth shapes. Hyperpatches can be used as a modelling tool with the advantage of being able to model the interior as well as the boundary surfaces. Also, hyperpatches can be edited easily by moving control points that make them suitable for inter- active design. In any representation scheme, validity is a very important property 123 , as it ensures that we can not obtain represen- tations of meaningless objects. To apply algorithms to an in- valid representation of a solid may produce bad results. It is not a good idea to rely on human operators to ensure the va- lidity of the model, because the complexity of the model can make it impossible to check its validity without the help of a computer. On the other hand, models are not always created by humans and, in this case, visual checking of validity is impossible. As validity is a very important property of a representation scheme, it is essential to develop an automatic condition to ensure the validity of our hyperpatches. This paper explains a simple condition of validity for uniform b-spline tri-cubic hyperpatches. 2. Previous work There are several references dealing with the problem of checking if a hyperpatch is invalid. In this section we out- line them. The early work of Sederberg and Parry 4 focused on free- form deformation states that the jacobian is a very interesting tool to determine the inner distribution of points in the inte- rior of a hyperpatch. They don’t use the jacobian for check- ing validity, but to ensure that a deformation is volume pre- serving. Joy and Duchaineau 5 , in the context of finding the sur- faces that represent the boundary of a trivariate tensor- product b-spline solid, propose a method based on the Ja- cobian that can be seen as a validity test. They are not interested in checking if a solid is or is not valid, but in finding the correct boundary surfaces of that solid. Using the implicit function theorem, they state that the real boundary of the solid is a subset of the union of a set of parametric b-spline patches and the isosurface where the determinant of the jacobian is zero. They approximate the ja- cobian by using interval techniques and adaptive subdivision of the domain space. Gain 6 , in the context of free-form deformation derivates a validity test for hyperpatches that is very similar to the work of Joy and Duchaineau. Gain, like Joy, realizes that the jacobian is zero if and only if the three partial derivatives of the solid are linearly dependent, and they use a conic-hull hodograph to bound these derivatives. The proposed method works by comparing relative posi- tions of the control points of the hyperpatch. His method is a sufficient condition and so, it restricts the range of allow- able free-form deformations, but it can be used repeatedly in short steps to achieve the same result without obtaining an invalid deformation. Another slightly different approach is provided by Choi c The Eurographics Association 2001.