CEG 777-01: Computer Aided Geometric Design

To cover the fundamentals of geometric modeling, including design of curves and surfaces, composite curves and surfaces, and subdivision techniques for creation of free-form shapes

Synthesis analysis of the free form curves and surfaces parametrical models

Parametrical models have become an important mathematical tool for free form curves and surface description. they allow the use of state-of-the-art computers to do the various processing and analysis with respect to shape (calculation of the volume and surface area, vibration analysis, NC programs preparation, etc.). Without these models, the current product design and manufacturing would be difficult. The paper presents an analysis study for the most important parametrical models of free form curves and surfaces description (Ferguson, Coons, Bezier, B-Spline and rational models). Firstly, it emphasises external and mathematical properties of each model. Secondly, it gives the most interesting interfaces between models. Thus the process of complex surface design and manufacturing would be more efficient in view this analysis. Finally for validation, the study is ended by an application to the design of a car bonnet.Parametrical models have become an important mathematical tool for free form curves and surface description. they allow the use of state-of-the-art computers to do the various processing and analysis with respect to shape (calculation of the volume and surface area, vibration analysis, NC programs preparation, etc.). Without these models, the current product design and manufacturing would be difficult. The paper presents an analysis study for the most important parametrical models of free form curves and surfaces description (Ferguson, Coons, Bezier, B-Spline and rational models). Firstly, it emphasises external and mathematical properties of each model. Secondly, it gives the most interesting interfaces between models. Thus the process of complex surface design and manufacturing would be more efficient in view this analysis. Finally for validation, the study is ended by an application to the design of a car bonnet

CEG 777-01: Computer Aided Geometric Design

This course covers the fundamentals of geometric modeling, including design of curves and surfaces, composite curves and surfaces, and subdivision techniques for creating of free-form shapes. Materials covered in the course will be useful in game design and also in visualization of exploratory data

Query processing of geometric objects with free form boundarie sin spatial databases

The increasing demand for the use of database systems as an integrating factor in CAD/CAM applications has necessitated the development of database systems with appropriate modelling and retrieval capabilities. One essential problem is the treatment of geometric data which has led to the development of spatial databases. Unfortunately, most proposals only deal with simple geometric objects like multidimensional points and rectangles. On the other hand, there has been a rapid development in the field of representing geometric objects with free form curves or surfaces, initiated by engineering applications such as mechanical engineering, aviation or astronautics. Therefore, we propose a concept for the realization of spatial retrieval operations on geometric objects with free form boundaries, such as B-spline or Bezier curves, which can easily be integrated in a database management system. The key concept is the encapsulation of geometric operations in a so-called query processor. First, this enables the definition of an interface allowing the integration into the data model and the definition of the query language of a database system for complex objects. Second, the approach allows the use of an arbitrary representation of the geometric objects. After a short description of the query processor, we propose some representations for free form objects determined by B-spline or Bezier curves. The goal of efficient query processing in a database environment is achieved using a combination of decomposition techniques and spatial access methods. Finally, we present some experimental results indicating that the performance of decomposition techniques is clearly superior to traditional query processing strategies for geometric objects with free form boundaries

3D-2D projective registration of free-form curves and surfaces

International audienceSome medical interventions require knowing the correspondence between an MRI/CT image and the actual position of the patient. Examples occur in neurosurgery and radiotherapy, but also in video surgery (laparoscopy). We present in this paper three new techniques for performing this task without artificial markers. To do this, we find the 3D-2D projective transformation (composition of a rigid displacement and a perspective projection) which maps a 3D object onto a 2D image of this object. Depending on the object model (curve or surface), and on the 2D image acquisition system (X-Ray, video), the techniques are different but the framework is common: Results are presented on a variety of real medical data to demonstrate the validity of our approach

3D-2D projective registration of free-form curves and surfaces

Some medical interventions require knowing the correspondence between an MRI/CT image and the actual position of the patient. Examples occur in neurosurgery and radiotherapy, but also in video surgery (laparoscopy). We present in this paper three new techniques for performing this task without artificial markers. To do this, we find the \bf 3D-2D projective transformation (composition of a rigid displacement and a perspective projection) which maps a 3D object onto a 2D image of this object. Depending on the object model (curve or surface), and on the 2D image acquisition system (X-Ray, video), the techniques are different but \bf the framework is common: \beginitemize \item We first find an estimate of the transformation using bitangent lines or bitangent planes. These are first order semi-differential invariants \citeMundy. \item Then, introducing the normal or tangent, we define a distance between the 3D object and the 2D image, and we minimize it using extensions of the Iterative Closest Point algorithm (\citeBesl,Zhang). \item We deal with the critical problem of outliers by computing Mahalanobis distances and performing generalized $\chi^2$ tests. \enditemize Results are presented on a variety of real medical data to demonstrate the validity of our approach

Aesthetic-oriented classification of 2D free-form curves

Nowadays, it is commonly admitted that the aesthetic appearance of a product has an enhanced role in its commercial success. Therefore, understanding and manipulating the aesthetic properties of shapes in the early design phases has become a very important field of research. There exists an appropriate vocabulary for describing the aesthetic properties of 2D free-form curves that includes terms such as straightness, acceleration, convexity and tension, which are normally used by stylists when describing and modifying shapes. However, the relationships between this vocabulary and the geometric models are not well established. This work investigates the possibility of applying Machine Learning Techniques (MLT) to discover possible classification patterns of 2D free-form curves with respect to the so-called straightness of the curve. First, we verified that MLT can correctly (99.78%) reapply the classification to new curves. In addition, we verified the abilities of the Attribute Selection methods to identify the most important attributes for the considered classification, among a larger set of attributes. As a result, it was possible to recognize as the most characterizing parameters the same curve attributes previously used to compute the measure of straightness (S). Moreover, Linear Regression (LR) was able to extract automatically an exact mathematical model, which can correlate the geometric quantities with the class of the curve, congruent to one we previously specified. This work indeed demonstrates that MLT are very suitable and can be efficiently used in this context. The work is a first step towards the characterization and classification of free form surfaces giving the general directions on how MLT can be exploited to characterize free-form surfaces with respects to the aesthetic properties.This work has been partially supported by the VISIONAIR project funded by the European Commission under grant agreement 262044

An evolutionary algorithm for surface modification

Complex shapes bounded by free form surfaces are designed and modified by specifying the parameters of their mathematical descriptions. Although most graphics and design systems provide high level tools for free form shape modifications, they are not intuitive and suitable for a creative designer or a stylist. In the present paper a technique is proposed for the modification of the shape of a parametric B-spline surface through embedding a set of predefined characteristic curves. Shape modification is performed by an evolutionary algorithm. The evolutionary algorithm minimises the distance between the curve and the surface, while maintaining the global shape and smoothness of the original surface. Several experimental results are presented