Interactive modeling of implicit surfaces using a direct visualization approach with signed distance functions

Modeling appealing virtual scenes is an elaborate and time-consuming task, requiring not only training and experience, but also powerful modeling tools providing the desired functionality to the user. In this paper, we describe a modeling approach using signed distance functions as an underlying representation for objects, handling both conventional and complex surface manipulations. Scenes defined by signed distance functions can be stored compactly and rendered directly in real-time using sphere tracing. Hence, we are capable of providing an interactive application with immediate visual feedback for the artist, which is a crucial factor for the modeling process. Moreover, dealing with underlying mathematical operations is not necessary on the user level. We show that fundamental aspects of traditional modeling can be directly transferred to this novel kind of environment, resulting in an intuitive application behavior, and describe modeling operations which naturally benefit from implicit representations. We show modeling examples where signed distance functions are superior to explicit representations, but discuss the limitations of this approach as well.     

隐式磨光曲面的保凸性研究

对隐式代数磨光曲面的保凸性进行了研究。证明了如果两个隐式代数曲面是凸的,则它们光滑拼接后,得到的磨光曲面在一定条件下仍然是凸的。以凸三面角为例,提出了利用代数样条磨光的方法并证明了利用这种磨光方法得到的磨光曲面具有保凸性。     

High-order implicit blending surfaces of low degree

The applicability of the so-called potential method for blending implicitly defined surfaces is extended. The extended method is able to produce blendings which are C^k-continuous, where k may be chosen arbitrarily large. The blendings consist of piecewise algebraic surfaces of low degree. Specifically, the degree is k + 1 for so-called convex corners, and 2k for an important class of non-convex corners. In general the degree is linear in k, the constant of proportionality depending on the geometry of the corner. The method is illustrated by a concrete example, where a C² blending surface of degree 4 is constructed for a certain non-convex corner.     

基于结式的曲面拼接与三维造型

针对很多几何造型是带有约束条件的曲面拼接问题,在线性连续同伦的基础上提出了利用非线性同伦连续计算拼接曲面以进行三维造型的方法。首先,根据得到的截面(切片)的位置及其曲线方程确定插值点并得到插值多项式;其次,将此插值多项式作为非线性连续同伦映射函数并分别代入主曲面和辅助曲面的多项式方程得到过渡曲面的方程;然后,仅将插值变元作为变元而主、辅助曲面方程的变元作为参数,利用Sylvester结式消去过渡方程中的变元得到关于主曲面的拼接方程即造型曲面。利用该方法能实现带有控制点的曲面造型以及多曲面约束的几何造型,而且它可以确定造型过程中的中间形状及中间形状的位置,从而更加具有实用性。     

Hyperfinite Interpolation, Wu s Method and Blendingof Implicit Algebraic Surfaces

One of the central questions in CAGE^[1] is blending surfaces which provides the theoretical basis for the design technology of space surfaces.We will discuss the general theories and algorithms for multivariate hyperfinite interpolation and their application to the blending of implicit algebraic surfaces,and investigate the existence conditions of hyperfinite interpolation.Based on Wu‘s theory on blending implicit algebraic surfaces,the problem of blending two quadric surfaces is studied.The conditions for the coefficient of gi under which there exists the cubic blending surface S(f) (the lowest degree)are obtained and the concrete expressions of f are presented is they exist.These results can be applied directly to CAGD.     

A new implicit blending technique for volumetric modelling

Current implicit blending techniques are mostly designed for use in surface modelling, where only boundaries of the object defined by the implicit primitives are important. In contrast, in volumetric implicit modelling the interior of the object is also significant, which requires different and more suitable techniques for combining implicit primitives. In this paper, we first discuss irregularities that occur using the current techniques. Then, a new technique for blending implicit primitives, especially appropriate in volumetric modelling (e.g., cloud modelling), is introduced. It overcomes these abnormalities and gives us better results than current techniques.     

Computing minimum distance between two implicit algebraic surfaces

The minimum distance computation problem between two surfaces is very important in many applications such as robotics, CAD/CAM and computer graphics. Given two implicit algebraic surfaces, a new method based on the offset technique is presented to compute the minimum distance and a pair of points where the minimum distance occurs. The new method also works where there are an implicit algebraic surface and a parametric surface. Quadric surfaces, tori and canal surfaces are used to demonstrate our new method. When the two surfaces are a general quadric surface and a surface which is a cylinder, a cone or an elliptic paraboloid, the new method can produce two bivariate equations where the degrees are lower than those of any existing method.     

基于动态偏微分方程构造过渡面

为了实现交互式的偏微分方程曲面造型,针对传统静态偏微分方程构造过渡面存在的不足,提出了基于动态偏微分方程构造C1连续的过渡面,并引入迭代有限差分法求解偏微分方程的数值解,在此基础上构造了光滑过渡曲面.讨论了形状控制因子、密度、阻尼系数等物理参数的变化对曲面形状的影响,其中形状控制因子对生成曲面的形状影响最为明显.造型实例表明,利用动态偏微分方程构造过渡面具有更高的灵活性,大大提高了工业几何设计的交互性,在CAD/CAM中具有重要的应用价值.     

Topological correction of hypertextured implicit surfaces for ray casting

Hypertextures are a useful modelling tool in that they can add three-dimensional detail to the surface of otherwise smooth objects. Hypertextures can be rendered as implicit surfaces, resulting in objects with a complex but well defined boundary. However, representing a hypertexture as an implicit surface often results in many small parts being detached from the main surface, turning an object into a disconnected set. Depending on the context, this can detract from the realism in a scene, where one usually does not expect a solid object to have clouds of smaller objects floating around it. We present a topology correction technique, integrated in a ray casting algorithm for hypertextured implicit surfaces, that detects and removes all the surface components that have become disconnected from the main surface. Our method works with implicit surfaces that are C² continuous and uses Morse theory to find the critical points of the surface. The method follows the separatrix lines joining the critical points to isolate disconnected components.     

Iso-planar interpolation for the machining of implicit surfaces

The paper presents an approach for on-line path generation and interpolation for the machining of implicit surfaces. For a given implicit surface, once the cutting plane direction and cut-in points have been selected, iso-planar tool paths and interpolated points can be calculated on-line according to the feedrate and scallop height requirements. The approach enables the tool position and orientation to be correctly calculated at each interpolated point. Validation examples are provided for the interpolation of cyclide surfaces with planar and curved boundaries.     

Trimming implicit surfaces

Algorithms for trimming implicit surfaces yielding surface sheets and stripes are presented. These two-dimensional manifolds with boundaries result from set-theoretic operations on an implicit surface and a solid or another implicit surface. The algorithms generate adaptive polygonal approximation of the trimmed surfaces by extending our original implicit surface polygonization algorithm. The presented applications include modeling several spiral shaped surface sheets and stripes (based on M. Eschers artworks) and extraction of ridges on implicit surfaces. Another promising application of the presented algorithms is modeling heterogeneous objects as implicit complexes.     

Isotopic meshing of implicit surfaces

Implicit surfaces are given as the zero set of a function F:ℝ³→ℝ. Although several algorithms exist for generating piecewise linear approximations, most of these are based on a user-defined stepsize or bounds to indicate the precision, and therefore cannot guarantee topological correctness. Interval arithmetic provides a mechanism to determine global properties of the implicit function. In this paper we present an algorithm that uses these properties to generate a piecewise linear approximation of implicit curves and surfaces, that is isotopic to the curve or surface itself. The algorithm is simple and fast, and is among the first to guarantee isotopy for implicit surface meshing.     

Ray casting implicit fractal surfaces with reduced affine arithmetic

A method is presented for ray casting implicit surfaces defined by fractal combinations of procedural noise functions. The method is robust and uses affine arithmetic to bound the variation of the implicit function along a ray. The method is also efficient due to a modification in the affine arithmetic representation that introduces a condensation step at the end of every non-affine operation. We show that our method is able to retain the tight estimation capabilities of affine arithmetic for ray casting implicit surfaces made from procedural noise functions while being faster to compute and more efficient to store.     

Dynamic mesh optimization for polygonized implicit surfaces with sharp features

Published online: 14 February 2003     

PDE blending surfaces with C continuity

In this paper, we propose to use a general sixth-order partial differential equation (PDE) to solve the problem of C² continuous surface blending. Good accuracy and high efficiency are obtained by constructing a compound solution function, which is able to both satisfy the boundary conditions exactly and minimise the error of the PDE. This method can cope with much more complex surface-blending problems than other published analytical PDE methods. Comparison with the existing methods indicates that our method is capable of generating blending surfaces almost as fast and accurately as the closed-form method and it is more efficient and accurate than other extant PDE-based methods.     

一种参数化可调的NURBS曲面G~2光滑拼接算法

提出一种用于NURBS曲面G2光滑拼接算法。在创建拼接曲面时,采用"参数可调"的思想,用拼接函数和指重参数先统一两基曲面的参数,计算出拼接曲面上的插值点,并以这些插值点为参考点根据G2连续的几何性质对拼接曲面的内部控制点进行修正。此算法适用于各类曲面的拼接,通过调整平衡因子和指重参数可以得到在满足G2连续的前提下各种曲率的拼接曲面,简化曲面拼接的计算过程。     

Parametric G n blending of curves and surfaces

We introduce a simple blending method for parametric curves and surfaces that produces families of parametrically defined, G ⁿ –continuous blending curves and surfaces. The method depends essentially on the parameterizations of the curves/surfaces to be blended. Hence, the flexibility of the method relies on the existence of suitable parameter transformations of the given curves/surfaces. The feasibility of the blending method is shown by several examples. The shape of the blend curve/surface can be changed in a predictable way with the aid of two design parameters (thumb weight and balance).     

具有变动边界的过渡面的热传导方程构造法

提出过渡面的热传导方程构造方法。使用热传导方程、初值条件和变动的边界条件,得到过渡面的偏微分方程模型,再利用分离变量法或者数值方法得到模型的解,则方程的像空间就是满足连续性要求的过渡面。数值实验也表明这种方法是可行的、有效的。