X-Splines
A Spline Model Designed for the End-User

Copyright 1995 by the Association for Computing Machinery Inc.
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Authors

C. Blanc, C. Schlick

Keywords

Computer Aided Geometric Design, Curves and Surfaces, Splines, Intuitive Modelling.

Abstract

This paper presents a new model of spline curves and surfaces. The main characteristic of this model is that it has been created from scratch by using a kind of mathematical engineering process. In a first step, a list of specifications was established. This list groups all the properties that a spline model should contain in order to appear intuitive to a non-mathematician end-user. In a second step, a new family of blending functions was derived, trying to fulfill as many items as possible of the previous list. Finally, the degrees of freedom offered by the model have been reduced to provide only shape parameters that have a visual interpretation on the screen. The resulting model includes many classical properties such as affine and perspective invariance, convex hull, variation diminution, local control and C2/G2 or C2/G0 continuity. But it also includes original features such as a continuum between B-splines and Catmull-Rom splines, or the ability to define approximation zones and interpolation zones in the same curve or surface.

Document

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Extended Field Functions for Soft Objects

Authors

C. Blanc, C. Schlick

Keywords

Implicit Surfaces, Soft Objects, Blobs, Metaballs, Convolution Surfaces, Field Functions.

Abstract

In the field of geometric design, the generic term "soft object" embeds several implicit models (blobs, metaballs, distance surfaces, convolution surfaces) proposed over the years for modelling and animating free-form 3D objects. All these models share the property that curved surfaces are defined by computing isosurfaces of a set of potential fields. The topic of this paper is to presents some innovative ways for defining these potential fields. First, it proposes a set of ready-to-use functions and second describes an environment which allows the user to design interactively his own functions.

Document

Part 1 (6.7 MB)
Part 2 (5.8 MB)



Generic Implementation of Axial Deformation Techniques

Author

C. Blanc

Keywords

Geometric Modeling, Deformation Techniques, Implementation

Abstract

Global deformation techniques were first introduced to extend the set of primitives that may be used in constructive solid modeling. In fact, these techniques are more general and have been consequently adapted to tessellated surfaces or parametric patches as well. This paper proposes a generic implementation of several of these global deformation techniques. The term "generic" focuses on the fact that the implementation depends neither on a given geometric model for surfaces nor on a specific data structures for internal representation.

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More Accurate Representation of Conics by NURBS

Authors

C. Blanc, C Schlick

Keywords

Conic Sections, NURBS, Reparametrization, Rational Polynomials, Continuity.

Abstract

One of the argument usually given to explain the popularity of NURBS is the fact that they allow to define free-form curves and surfaces (as almost every spline model) and provide also an exact representation of conic sections and thus of a large set of curves and surfaces that are intensively used in CAD : circular arcs, circles, cylinders, cones, spheres, surfaces of revolution, etc. This paper deals with the two following problems :
  1. All the known representations by NURBS of curves and surfaces based on conics have only a C1 continuity. Moreover, there does not exist any technique which would eventually allow to find a parametrization with a higher level of continuity.
  2. The parametrization resulting from the representation of conics by NURBS can deviate significantly from the ideal arc length parametrization. The only known solution to reduce this deviation is to increase the number of control points of the spline (using refinement algorithms, for instance), but such a process converges quite slowly to the ideal parametrization.

The solution that we propose in this paper uses an original reparametrization process, that we have called zigzag reparametrization, which is based on a particular family of rational polynomials. This techniques allows to get a higher order continuity as well as a more uniform parametrization.

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Ratioquadrics: An Alternative Model for Superquadrics

Authors

C. Blanc, C Schlick

Keywords

Superconics, Superquadrics, Ratioconics, Ratioquadrics, Rational Polynomials.

Abstract

This paper presents a new family of 2D curves and its extension to 3D surfaces, respectively called ratioconics and ratioquadrics, that have been designed as alternatives to the well-known superconics and superquadrics. This new model is intended to improve the original one on three main points : first it is several times faster to compute, second it provides higher order continuities (C1/G2 or C2/G2 instead of C0/G0), and third it provides a greater variety of shapes for the resulting curves and surfaces. All these improvements are obtained by replacing the signed power function involved in the formulation of superconics and superquadrics by linear or quadratic rational polynomials.

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Easy Transformations between Cartesian, Cylindrical and Spherical Coordinates

Authors

C. Blanc, C Schlick

Keywords

Geometrical Transformations, Implementation.

Abstract

This paper proposes a set of functions that realize all possible transformations of a point in a threedimensional euclidian space between global and local frames using either cartesian, cylindrical or spherical coordinates.

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coord.ps



A Generic Implementation of Free Form Deformation Techniques

Authors

C. Blanc, C. Schlick

Keywords

Geometric Modeling, Deformation Techniques, FFD, EFFD, Implementation.

Abstract

The present paper proposes a generic implementation of FFD and EFFD techniques. The term "generic" focuses on the fact that the implementation depends neither on a given geometric model for surfaces nor on a specific data structures for internal representation. Each deformation routine acts only on points : it takes the coordinates of a point on the original object, computes its displacement according to the deformed lattice, and finally returns the new coordinates of the point.

Document

gen_ffd.ps



Techniques de Modelisation et de Deformation pour la Synthese d'Images

Author

C. Blanc

Keywords

Computer Graphics, Geometric Modelling, surface, Splines, NURBS, Deformations.

Abstract

In the field of computer graphics, geometric modelling is the step in which all the geometrical characteristics of a threedimensional object (or environment) are defined. This document deals more precisely with the use of spline surfaces for geometric modelling. In a first step, we propose a general form which allows to embed every spline model that has been presented in the literature. Then we study more deeply non-uniform rational B-splines (NURBS) which have become a de-facto standard for surface modelling during the last years. In particular, we present a technique to improve the parametrization of such curves and surfaces in order to fulfill some specific constraints. During this study, we enlight several drawbacks of NURBS which are in contradiction with the characteristics that should include a modelling tool intended for a non-specialist user. Therefore, we propose a new spline model that is expected to make the modelling process more intuitive. In a second step, we focus on several techniques, such as combination or deformation of spline surfaces, which enable a more global approach of the modelling of a threedimensional object. After having shown that combination methods are not well adapted to parametric surfaces, we conclude this document by proposing a set of original deformation algorithms for spline.

Document

these0.ps Part 0
these1.ps Part 1 (3.5 MB)
these2.ps Part 2 (8.5 MB)
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A Methodology for Description of Geometrical Deformations

Authors

C. Blanc, P. Guitton, C. Schlick

Keywords

Geometric Modeling, Deformation Techniques, Normalized Operators, RRIFF.

Abstract

A methodology is presented that enables to express any deformation technique in a unique framework. The driving idea of this paper is that every deformation can be reformulated as combination of three kinds of normalized operators, transformation, modulation and perturbation. Moreover, using some pleasant properties of the methodology, some innovative ways to use, combine and enhance classical techniques are provided.

Document

pac.ps (4.8 MB)


Deformation of Spline Surfaces by Normalized Operators

Authors

C. Blanc, P. Guitton

Keywords

Geometric Modeling, NURBS, Deformation Techniques, Normalized Operators.

Abstract

In order to get an optimal use of all the interesting features for free-form modeling provided by sophisticated geometric models (Bezier, B-Spline, NURBS), we propose a deformation technique based on the concept of normalized operators. Such an operator can be viewed as a black-box which encloses --- and hides to the user --- the high number and the complexity of the parameters involved during a deformation. The normalization property of an operator means that it does not modify the structure of the geometric model on which it is applied. Therefore, sequential applications of such operators can be used in order to create a progressive deformation of the object.

Document

eguk94.ps (8.5 MB)