基于NURBS权因子调整的FFD直接操作

FFD（Free-Form Deformation）是计算机动画和变形中最重要的方法之一。FFD的直接操作为变形提供了更直观更方便的控制手段。该文用约束优化方法，通过调整张量积网格控制点对应的权因子，解决基于NURBS的FFD直接操作问题，给出了变形物体在单点约束下FFD直接操作的解，使变形后的物体通过给定的目标点。该方法计算简便，易于实现。文中给出的变形实例也表明，变形结果比较自然。     

Hermite approximation for free-form deformation of curves and surfaces

Free-Form Deformation Techniques (FFD) are commonly used to generate animations, where a polygonal approximation of the final object suffices for visualization purposes. However, for some CAD/CAM applications, we need an explicit expression of the object, rather than a collection of sampled points. If both object and deformation are polynomial, their composition yields a result that is also polynomial, albeit very high degree, something undesirable in real applications. To solve this problem, we transform each curve or surface composing the object, usually expressed in the Bernstein basis, to a modified Newton form. In this representation, the two-point analogue of Taylor expansions, the composition admits a simple expression in terms of discrete convolutions, and degree reduction corresponding to Hermite approximation is trivial by dropping high-degree coefficients. Furthermore, degree-reduction can be incorporated into the composition. Finally, the deformed curve or surface is converted back to the Bernstein form. This method extends to general non-polynomial deformation, such as bending and twisting, by computing a polynomial approximant of the deformation.     

Free-Form Deformation with Rational DMS-Spline Volumes

In this paper, we propose a novel free-form deformation （FFD） technique, RDMS-FFD （Rational DMS-FFD）, based on rational DMS-spline volumes. RDMS-FFD inherits some good properties of rational DMS-spline volumes and combines more deformation techniques than previous FFD methods in a consistent framework, such as local deformation, control lattice of arbitrary topology, smooth deformation, multiresolution deformation and direct manipulation of deformation. We first introduce the rational DMS-spline volume by directly generalizing the previous results related to DMS-splines. How to generate a tetrahedral domain that approximates the shape of the object to be deformed is also introduced in this paper. Unlike the traditional FFD techniques, we manipulate the vertices of the tetrahedral domain to achieve deformation results. Our system demonstrates that RDMS-FFD is powerful and intuitive in geometric modeling.     

Analysis of 3-D human foot forms using the Free Form Deformation method and its application in grading shoe lasts

An effective way to design well-fitting products is to analyse human body forms and to classify them into several groups. In the present study, a new method is proposed to analyse human body forms using the FFD (Free Form Deformation) technique. The FFD method is a way to deform the shapes of object smoothly by moving control lattice points set around the object. The reference body form is automatically deformed to coincide with the other body forms using the FFD method. The dissimilarity is defined by the movements of the control lattice points. The foot forms of 56 Japanese adult females were analysed with this method, and distributions for them were calculated using multi-dimensional scaling. The first axis contrasts feet with high dorsal arches and low dorsal arches, and the second axis is related to the antero-posterior proportion of the foot. As an application of the present method, a last of width EEEE was designed from an existing last of width E by applying the control lattice points that converted a representative foot of width E into a foot of width EEEE. The new EEEE width last reflected the allometric differences between narrow and wide feet better than one obtained by a conventional method. It was found that the present method with FFD is not only useful for classifying 3-D human body forms, but also has potential as applications for designing well-fitting products.     

Analysis of 3-D human foot forms using the Free Form Deformation method and its application in grading shoe lasts

An effective way to design well-fitting products is to analyse human body forms and to classify them into several groups. In the present study, a new method is proposed to analyse human body forms using the FFD (Free Form Deformation) technique. The FFD method is a way to deform the shapes of object smoothly by moving control lattice points set around the object. The reference body form is automatically deformed to coincide with the other body forms using the FFD method. The dissimilarity is defined by the movements of the control lattice points. The foot forms of 56 Japanese adult females were analysed with this method, and distributions for them were calculated using multi-dimensional scaling. The first axis contrasts feet with high dorsal arches and low dorsal arches, and the second axis is related to the antero-posterior proportion of the foot. As an application of the present method, a last of width EEEE was designed from an existing last of width E by applying the control lattice points that converted a representative foot of width E into a foot of width EEEE. The new EEEE width last reflected the allometric differences between narrow and wide feet better than one obtained by a conventional method. It was found that the present method with FFD is not only useful for classifying 3-D human body forms, but also has potential as applications for designing well-fitting products.     

Preventing self-intersection under free-form deformation

Free-Form Deformation (FFD) is a versatile and efficient modeling technique which transforms an object by warping the surrounding space. The conventional user-interface is a lattice of movable control points but this tends to be cumbersome and counterintuitive. Directly Manipulated Free-Form Deformation (DMFFD) allows the user to drag object points directly and has proven useful in an interactive sculpting context. A serious shortcoming of both FFD and DMFFD is that some deformations cause self-intersection of the object. This is unrealistic and compromises the object's validity and suitability for later use. An in-built self-intersection test is thus required for FFD and its extensions to be truly robust In this paper, we present the following novel results set of theoretical conditions for preventing self-intersection by ensuring the injectivity (one-to-one mapping) of FFD, an exact. (necessary and sufficient) injectivity test which is accurate but computationally costly, an efficient but approximate injectivity test which is a sufficient condition only, and a new form of DMFFD which acts by composing many small injective deformations. The latter expands the range of possible deformations without sacrificing the speed of the approximate test     

Free-form design using axial curve-pairs

Deformation of 3D shapes usually requires the use of a deformation tool. The freeform deformation technique requires the use of a lattice of control point for deforming an object. This may require a synchronized movement of the lattice control points in order to obtain the desired effects. The axial deformation technique allows an object to be deformed by manipulating an axial curve. However, unexpected twist of the object may be obtained. This is a result of the lack of control on the local coordinate frame of the curve. This paper presents a technique for deforming objects with a set of axial curve-pairs. The use of a curve-pair allows the local coordinate frame to be controlled intuitively. A curve-pair is composed of a primary and an orientation curve. The orientation curve is an approximate offset of the primary curve. A technique is proposed for maintaining the relation between the primary and the orientation curve when the curve-pair is adjusted. By associating a complex 3D object to a curve-pair, the object can be stretched, bended, and twisted intuitively through manipulating the curve-pair. This deformation technique is particularly suitable for manipulating complex shapes (e.g. decorative components) in industrial and aesthetic design, and is also suitable for modelling characters and animals with flexible bodies. Adjusting the curve-pair according to some motion constraints produces different postures of a character or animal model. This in turn can be used as decorative components for aesthetic design.     

A method of transferring polyhedron between the intersection-form and the sum-form

A bounded polyhedron (or a polyhedral cone) can be given by either an intersection of linear inequalities, which is thus called intersection-form, or a convex combination (or nonnegative linear combination) of some given points, which is thus called sum-form. This paper first proposes a simple and effective method to transfer a bounded polyhedron or a polyhedral cone from the intersection-form to the sum-form. A new method is proposed to transfer a bounded polyhedron (or a polyhedral cone) from the sum-form to the intersection form. Both procedures can be completed in finite number of iterations. Numerical examples are given for illustration.     

An algorithm for converting the boundary representation of a CAD model to its octree representation

Present CAD systems store the solid model of an object using a convenient representation. Boundary models and CSG (Constructive Solid Geometry) models are the most frequently used representations. Based on recent research findings, octree representation of an object presents a promising approach in solving problems in the areas of Computer Graphics, Manufacturing and Robotics. The most notable use of octree representations is in CAD-based robotic path planning problems. Octree models have also been used in fast rendering of 3-D solid models using ray tracing methods. This paper presents an algorithm for converting the boundary representation of polyhedral models to its octree representation. Such an algorithm would provide the link between an object generated using a solid modelling system and the application involving an octree representation of an object. The algorithm is demonstrated by converting a polyhedral boundary model of a sample object to its octree representation.     

Real-time resolution of self-intersection in dynamic cylindrical free-form deformation

This paper presents a method of self-intersection detection and resolution for dynamic cylindrical-lattice-based free-form deformation (FFD). The lattice-based approach allows efficient computation of deformation of complex geometries. But excessive deformation can cause visual anomalies such as surface infiltration and distortion. This paper derives a geometrically intuitive sufficient condition to guarantee that the FFD function is a homeomorphism and there is no self-intersection. The FFD function is defined by linear and quadratic B-Spline functions with the control points of the cylindrical lattice cell. The sufficient condition is satisfied if each trilinear function of the nine prism-shaped pentahedrons derived from the cell has a positive Jacobian determinant. The positivity is satisfied if the 12 tetrahedrons derived from the pentahedron have positive volumes. Based on the sufficient condition, the proposed method converts the self-intersection problem into a point-face collision detection and response problem suitable for dynamic simulation. The efficiency and accuracy of the self-intersection detection algorithm is analyzed and compared with a previous method. The results show that the proposed technique allows simulation of excessive deformation of tubular objects in an efficient and realistic manner.     

多边形物体的精确B-样条自由变形*

在计算机动画与几何造型中,自由变形是一种重要的几何形状修改方法.该文从移位算子和函数复合的观点探讨一种方法,即当被变形物体用三角片表示、变形工具为B-样条参数体时,变形后的物体可以精确地表示为一组三角Bzier曲面片,其次数为B-样条参数体3个方向的次数之和.此方法的核心在于自由变形是作用在三角片上,而不是顶点上,所以解决了多边形物体B-样条自由变形的点采样问题.     

人体局部曲面建模与自由变形方法

人体曲面建模与自由变形方法是实现人体数字化的关键技术。给出了提取平面点云数据的方法及平面点云排序算法，介绍了基于平面数据的B样条曲面重构算法，重点叙述了用FFD（Free Form Deformation）技术实现曲面变形的算法及自由变形方法的特点，最后给出了人体胸部曲面建模与自由变形的实例，实例表明使用FFD进行曲面变形的效果比较自然。     

Collision detection for moving polyhedra

We consider the collision-detection problem for a three-dimensional solid object moving among polyhedral obstacles. The configuration space for this problem is six-dimensional, and the traditional representation of the space uses three translational parameters and three angles (typically Euler angles). The constraints between the object and obstacles then involve trigonometric functions. We show that a quaternion representation of rotation yields constraints which are purely algebraic in a seven-dimensional space. By simple manipulation, the constraints may be projected down into a six-dimensional space with no increase in complexity. The algebraic form of the constraints greatly simplifies computation of collision points, and allows us to derive an efficient exact intersection test for an object which is translating and rotating among obstacles.     

Modelling polyhedral solids bounded by multi-curved parametric surfaces

Since the beginning of geometric modelling as a field of CAD a decade ago, the methods for interactive design of solid objects and interactive design of free-formed surfaces (of degree three and higher) were developed along parallel yet disjoint lines. One led to the development of techniques for representing and manipulating the shape of polyhedral solids bounded mostly by planes, while the other led to the development of techniques for the mathematical representation of curved surfaces, without paying attention to their combination into volumetric solids. Though the need for integrating solid object modelling with surface modelling for the design of such artefacts as machine parts, aircraft, cars and ships has been widely recognized, there is so far no single modelling system which provides such capabilities in a general way.An integrated solids modelling system for representing and manipulating polyhedral objects bounded by bicubic parametric surfaces is presented. Its basic capabilities include the representation of solids through a surface-based model, such that the surface underlying any face can be replaced by another surface that has been modelled independently. Other functionalities include scaling, rotation and translation of shapes and their pairwise combination into more complex shapes by means of spatial set operators.     

面向多面体模型的静态控制块识别扩展方法

在编译优化中,多面体模型可以对计算密集型程序中的耗时较多的循环代码进行并行性和数据局部性优化.但是,多面体建模过程中存在诸多限制,程序中只有少量代码可以被识别进而转换为多面体表示进行优化.基于LLVM编译框架提出一种分析方法,对多面体建模中的非规则因素进行了规范化处理,对非仿射因素提出一种定值扩展方法,消除了多面体建模...     

一种应用参数曲面的动态自由变形方法

本文提出了一种动态自由变形方法，被变形物体首先附在两张参数曲面上，当参数曲面的形状发生变化时，物体会随之变形。这两张参数曲面分别被称为形状曲面和高度曲面，其中形状曲面用于控制变形物体的形状，高度曲面用于控制物体沿着形状曲面法向的高度。     

Polyhedral Flows in Hybrid Automata

A hybrid automaton is a mathematical model for hybrid systems, which combines, in a single formalism, automaton transitions for capturing discrete updates with differential constraints for capturing continuous flows. Formal verification of hybrid automata relies on symbolic fixpoint computation procedures that manipulate sets of states. These procedures can be implemented using boolean combinations of linear constraints over system variables, equivalently, using polyhedra, for the subclass of linear hybrid automata. In a linear hybrid automaton, the flow at each control mode is given by a rate polytope that constrains the allowed values of the first derivatives. The key property of such a flow is that, given a state-set described by a polyhedron, the set of states that can be reached as time elapses, is also a polyhedron. We call such a flow a polyhedral flow. In this paper, we study if we can generalize the syntax of linear hybrid automata for describing flows without sacrificing the polyhedral property. In particular, we consider flows described by origin-dependent rate polytopes, in which the allowed rates depend, not only on the current control mode, but also on the specific state at which the mode was entered. We identify necessary and sufficient conditions for a class of flows described by origin-dependent rate polytopes to be polyhedral. We also propose and study additional classes of flows: strongly polyhedral flows, in which the set of states that can be reached up to a given time starting from a polyhedron is guaranteed to be a polyhedron, and polyhedrally sliced flows, in which the set of states that can be reached at a given time starting from a polyhedron is guaranteed to be a polyhedron. Finally, we discuss an application of the above classes of flows to approximate exponential behaviours.     

一种面片约束的变形技术

如何方便地设计和编辑修改实体或曲面模型，一直是CAGD和CG中的一个重要问题．提出一种基于面片(patches)约束的变形技术．用一个参考面片定义要变形的物体的某些局部特征，另一个面片用来定义参考面片将会变形到的位置．通过对变形物体上的各点分别进行比例、旋转和平移操作，得到变形结果．扩展因子的应用可增强变形的可控性。