Characterization of Spherical and Plane Curves Using Rotation Minimizing Frames

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies, up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves as curves whose so-called natural mates are spherical.Comment: 8 pages. This version is an improvement of the previous one. In addition to a study of some properties of plane and spherical curves, it contains a characterization of Bertrand curves in terms of the so-called natural mate

Axial deformation with controllable local coordinate frames.

Chow, Yuk Pui.Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (leaves 83-87).Abstracts in English and Chinese.Chapter 1. --- Introduction --- p.13-16Chapter 1.1. --- Motivation --- p.13Chapter 1.2 --- Objectives --- p.14-15Chapter 1.3 --- Thesis Organization --- p.16Chapter 2. --- Related Works --- p.17-24Chapter 2.1 --- Axial and the Free Form Deformation --- p.17Chapter 2.1.1 --- The Free-Form Deformation --- p.18Chapter 2.1.2 --- The Lattice-based Representation --- p.18Chapter 2.1.3 --- The Axial Deformation --- p.19-20Chapter 2.1.4 --- Curve Pair-based Representation --- p.21-22Chapter 2.2 --- Self Intersection Detection --- p.23-24Chapter 3. --- Axial Deformation with Controllable LCFs --- p.25-46Chapter 3.1 --- Related Methods --- p.25Chapter 3.2 --- Axial Space --- p.26-27Chapter 3.3 --- Definition of Local Coordinate Frame --- p.28-29Chapter 3.4 --- Constructing Axial Curve with LCFs --- p.30Chapter 3.5 --- Point Projection Method --- p.31-32Chapter 3.5.1 --- Optimum Reference Axial Curve Point --- p.33Chapter 3.6 --- Advantages using LCFs in Axial Deformation --- p.34Chapter 3.6.1 --- Deformation with Smooth Interpolated LCFs --- p.34-37Chapter 3.6.2 --- Used in Closed-curve Deformation --- p.38-39Chapter 3.6.3 --- Hierarchy of Axial Curve --- p.40Chapter 3.6.4 --- Applications in Soft Object Deformation --- p.41Chapter 3.7 --- Experiments and Results --- p.42-46Chapter 4. --- Self Intersection Detection of Axial Curve with LCFs --- p.47-76Chapter 4.1 --- Related Works --- p.48-49Chapter 4.2 --- Algorithms for Solving Self-intersection Problem with a set of LCFs --- p.50-51Chapter 4.2.1 --- The Intersection of Two Plane --- p.52Chapter 4.2.1.1 --- Constructing the Normal Plane --- p.53-54Chapter 4.2.1.2 --- A Line Formed by Two Planes Intersection --- p.55-57Chapter 4.2.1.3 --- Problems --- p.58Chapter 4.2.1.4 --- Sphere as Constraint --- p.59-60Chapter 4.2.1.5 --- Intersecting Line between Two Circular Discs --- p.61Chapter 4.2.2 --- Distance between a Mesh Vertex and a Curve Point --- p.62-63Chapter 4.2.2.1 --- Possible Cases of a Line and a Circle --- p.64-66Chapter 4.3 --- Definition Proof --- p.67Chapter 4.3.1 --- Define the Meaning of Self-intersection --- p.67Chapter 4.3.2 --- Cross Product of Two Vectors --- p.68Chapter 4.4 --- Factors Affecting the Accuracy of the Algorithm --- p.69Chapter 4.3.1 --- High Curvature of the Axial Curve --- p.69-70Chapter 4.3.2 --- Mesh Density of an Object. --- p.71-73Chapter 4.5 --- Architecture of the Self Intersection Algorithm --- p.74Chapter 4.6 --- Experimental Results --- p.75- 79Chapter 5. --- Conclusions and Future Development --- p.80-82Chapter 5.1 --- Contribution and Conclusions --- p.80-81Chapter 5.2 --- Limitations and Future Developments --- p.82References --- p.83-8

An investigation on the framework of dressing virtual humans

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Realistic human models are widely used in variety of applications. Much research has been carried out on improving realism of virtual humans from various aspects, such as body shapes, hair, and facial expressions and so on. In most occasions, these virtual humans need to wear garments. However, it is time-consuming and tedious to dress a human model using current software packages [Maya2004]. Several methods for dressing virtual humans have been proposed recently [Bourguignon2001, Turquin2004, Turquin2007 and Wang2003B]. The method proposed by Bourguignon et al [Bourguignon2001] can only generate 3D garment contour instead of 3D surface. The method presented by Turquin et al. [Turquin2004, Turquin2007] could generate various kinds of garments from sketches but their garments followed the shape of the body and the side of a garment looked not convincing because of using simple linear interpolation. The method proposed by Wang et al. [Wang2003B] lacked interactivity from users, so users had very limited control on the garment shape.This thesis proposes a framework for dressing virtual humans to obtain convincing dressing results, which overcomes problems existing in previous papers mentioned above by using nonlinear interpolation, level set-based shape modification, feature constraints and so on. Human models used in this thesis are reconstructed from real human body data obtained using a body scanning system. Semantic information is then extracted from human models to assist in generation of 3 dimensional (3D) garments. The proposed framework allows users to dress virtual humans using garment patterns and sketches. The proposed dressing method is based on semantic virtual humans. A semantic human model is a human body with semantic information represented by certain of structure and body features. The semantic human body is reconstructed from body scanned data from a real human body. After segmenting the human model into six parts some key features are extracted. These key features are used as constraints for garment construction.Simple 3D garment patterns are generated using the techniques of sweep and offset. To dress a virtual human, users just choose a garment pattern, which is put on the human body at the default position with a default size automatically. Users are allowed to change simple parameters to specify some sizes of a garment by sketching the desired position on the human body.To enable users to dress virtual humans by their own design styles in an intuitive way, this thesis proposes an approach for garment generation from user-drawn sketches. Users can directly draw sketches around reconstructed human bodies and then generates 3D garments based on user-drawn strokes. Some techniques for generating 3D garments and dressing virtual humans are proposed. The specific focus of the research lies in generation of 3D geometric garments, garment shape modification, local shape modification, garment surface processing and decoration creation. A sketch-based interface has been developed allowing users to draw garment contour representing the front-view shape of a garment, and the system can generate a 3D geometric garment surface accordingly. To improve realism of a garment surface, this thesis presents three methods as follows. Firstly, the procedure of garment vertices generation takes key body features as constraints. Secondly, an optimisation algorithm is carried out after generation of garment vertices to optimise positions of garment vertices. Finally, some mesh processing schemes are applied to further process the garment surface. Then, an elaborate 3D geometric garment surface can be obtained through this series of processing. Finally, this thesis proposes some modification and editing methods. The user-drawn sketches are processed into spline curves, which allow users to modify the existing garment shape by dragging the control points into desired positions. This makes it easy for users to obtain a more satisfactory garment shape compared with the existing one. Three decoration tools including a 3D pen, a brush and an embroidery tool, are provided letting users decorate the garment surface by adding some small 3D details such as brand names, symbols and so on. The prototype of the framework is developed using Microsoft Visual Studio C++,OpenGL and GPU programming

Tangent-ball techniques for shape processing

Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes.  Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing.  Many applications of shape processing can be found in the entertainment and medical industries. In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes. We propose a set of ball-based operators and discuss their properties, implementations, and applications.  We divide the group of ball-based operations into unary and binary as follows: Unary operators include: * Identifying details (sharp, salient features, constrictions) * Smoothing shapes by removing such details, replacing them by fillets and roundings * Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures Binary operators include: * Measuring the local discrepancy between two shapes * Computing the average of two shapes * Computing point-to-point correspondence between two shapes * Computing circular trajectories between corresponding points that meet both shapes at right angles * Using these trajectories to support smooth morphing (inbetweening) * Using a curve morph to construct surfaces that interpolate between contours on consecutive slices The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis.  These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon

Geometrical positioning of a flow diverter in case of aneurysm

Aquest projecte de final de carrera s'emmarca dins un estudi de recerca d'un estudiant de doctorat, Julien Egger de la “Swiss Federal Institute of Technology of Zürich (ETH)”. Té com a finalitat el desenvolupament d’un programa per optimitzar el posicionament virtual d'un desviador de flux (FD) dins d'una artèria en el cas d'un aneurisma, definint un aneurisma com la dilatació de l’artèria causada per la degradació de la paret. També servirà de base per després fer diferents simulacions de flux i així tenir una anàlisi de sensibilitat sobre les diferents característiques del FD, una vegada implementat en un pacient. Concretament, saber com el FD té inflüència sobre l'artèria, en el flux sanguini, ja que l'objectiu d'utilitzar un FD és canviar la direcció del flux per tal d’evitar que entri a l'interior del sac, fet que portaria a un augment del mateix podent-ne provocar l’explosió. A més a més, trobar una solució per resoldre els possibles problemes en cas que no funcioni i per donar més informació als cirurgians sobre el procediment de implementació. Per exemple, pot servir alhora de comparar les diferents opcions en cas d'una bifurcació, en quina direcció ha d'anar el FD? Per provar nous enfocaments pràctics (dos FD un dins l'altre ), etc. Per tant, donada les dades d’un volum en tres dimensions obtingut a partir d’una CT d'un pacient, el mètode desenvolupat en aquest treball té per objectiu el posicionament d'un FD virtual dins de l’artèria. El procediment emprat es pot dividir en els següents subpassos: - Obtenir una representació en 3D de la paret de l'artèria per mitjà de la segmentació. - Trobar la representació matemàtica de la línia central de l'artèria per utilitzar-la com a base per a la descripció geomètrica del desviador. - Donades les característiques del FD considerat, adaptar la geometria al llarg de la línea central per tal que encaixi amb l’artèria. - Exportar i guardar la representació volumètrica del FD en un format determinat, compatible amb els productes comercials utilitzats per a mallats volumètrics i simulació de flux. Per acabar dir que comparacions qualtitatives del posicionament virtual de FD obtingut a partir d’aquest mètode i el posicionament de FD obtingut a partir de dades reals de pacients són presentats en aquest projecte

Ship Hull Representation by Non-Uniform Rational B-Spline Surface Patches

The purpose of this work is to propose a new method for representing the ship hull shape with mathematic surfaces so that geometric data can be generated for any point on the hull where required to assist the production process. An extensive survey of previous work is presented covering both the use of parametric curves and surfaces to model the ship hull and also the most relevant software systems developed for that purpose. The main methods and algorithms available for the generation and edition of curves and surfaces are presented and compared taking into consideration the intended application. From the analysis of the formulations available it was concluded that the most adequate one, which however had not yet been extensively used to model ship hulls was the Non-Uniform Rational B-Splines (NURBS), due to the potential of their capability to represent exactly conic curves and surfaces. Therefore these surfaces were selected as the basis of the method developed in this thesis. A procedure is proposed for the representation of a given hull form in a two step approach, creating first a wireframe model over which the surface patches are generated. Both curves and surfaces are based on the NURBS formulation. To create the wireframe model, first a set of longitudinal boundary lines is selected, dividing the surface into areas of similar shape. Then, these lines are fitted by curves and faired to some extent. Next, transverse sections are defined and split by the boundary lines. Surface patches are then generated over the transverse section curves within the limits of each patch. Finally, to obtain the traditional representation of the ship surface by transverse sections, buttocks and waterlines, contour lines are generated for constant values of x, y and z coordinates. A computer system has been developed incorporating an interface that allows the visualization of the curves and surfaces being modeled. The system incorporates several algorithms for generation and edition of curves and surfaces, in addition to the main contribution of this thesis which is the use of NURBS to represent the ship hull surface. The system also incorporates curve and surface analysis tools and some basic fairing algorithms so that during the several steps of the creation of the model, the fairness of the curves and surfaces can be evaluated and improved to some extent. The procedure is tested and compared with an existing commercial system through some application examples, of a complete hull and in more detail in the bow region, showing that good results can be obtained with the system presented here

On the use of the Rotation Minimizing Frame for Variational Systems with Euclidean Symmetry

We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimizing frame, also known as the Normal, Parallel or Bishop frame (see [1], [36]). Such systems have previously been studied using the Frenet–Serret frame. However, the Rotation Minimizing frame has many advantages, and can be used to study a wider class of examples. We achieve our results by extending the powerful symbolic invariant cal- culus for Lie group based moving frames, to the Rotation Minimizing frame case. To date, the invariant calculus has been developed for frames defined by algebraic equations. By contrast, the Rotation Minimizing frame is defined by a differential equation. In this paper, we derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants. We then derive the syzygy operator needed to obtain Noether’s conservation laws as well as the Euler–Lagrange equations directly in terms of the invariants, for variational problems with a Euclidean symmetry. We show how to use the six Noether laws to ease the integration problem for the minimizing curve, once the Euler–Lagrange equations have been solved for the generating differential invariants. Our applications include variational problems used in the study of strands of pro- teins, nucleid acids and polymers