Effective computational geometry for curves and surfaces (original) (raw)
Implicit Surfaces that Interpolate
Implicit surfaces are often created by summing a collection of radial basis functions. Recently, researchers have begun to create implicit surfaces that exactly interpolate a given set of points by solving a simple linear system to assign weights to each basis function. Due to their ability to interpolate, these implicit surfaces are more easily controllable than traditional "blobby" implicits. There are several additional forms of control over these surfaces that make them attractive for a variety of applications. Surface normals may be directly specified at any location over the surface, and this allows the modeller to pivot the normal while still having the surface pass through the constraints. The degree of smoothness of the surface can be controlled by changing the shape of the basis functions, allowing the surface to be pinched or smooth. On a point-by-point basis the modeller may decide whether a constraint point should be exactly interpolated or approximated. Applications of these implicits include shape transformation, creating surfaces from computer vision data, creation of an implicit surface from a polygonal model, and medical surface reconstruction.
Interpolation of surfaces over scattered data
2001
We investigate the performance of DEI, an approach [2] that computes off-mesh approximations of PDE solutions, and can also be used as a technique for scattered data interpolation and surface representation. For the general case of unstructured meshes, we found it necessary to modify the original DEI. The resulting method, ADEI, adjusts the parameter of the interpolant, obtaining better performance. Finally, we measure ADEI's performance using different sets of scattered data and test functions and compare ADEI against two methods from the collection of ACM algorithms: Algorithms 752 [10] and 790 . The results show that ADEI is better than, if not comparable to, the best of the compared scattered data interpolation techniques.
Chapter 2: Surface Fitting Using Implicit Algebraic Surface Patches
Society for Industrial and Applied Mathematics eBooks, 1992
Interpolation and least-squares approximation provide! efficient ways of generating Ck.continuous meshes of surface patches, necessary for the consLrucLion of accurate computer geometric models of solid physical objects [see for e.g. [8.7]. Two surfaces J(:I;, Y,z) = 0 and g(x, y, z) = 0 meet with Ck-continuity along a curve C if and only if there exists functions a:(x, Y, z) and J3(x, v, z) such that all derivatives upta order ,;; of Ct.! -tJg equals zero [~ce for e.g.[G2]]. Ck-conl.inuity of two surface patches follows if the above conditioll is true along the common boundary curves between the two patches. This paper surveys the use of low dCF;L'Ce. implicitly defined, algebraic surfaces and surface patches in three dimensional real space m. 3 for various scat.tered data Ck-fitting problems. The use of low degree algebraic surface patches to constrHct models of physical ol>jects stems from the advantage of faster computations in sul>sequent geometric model manipulation operations such as computer graphics display, animation. and physica.l ol>ject simulations, see for e.g. (10]. Why algebraic surfaces? A real algebraic surface S in m. 3 is implicitly defined by a single polynomial equation F: !(x,y, z) = 0, where coeIfici<lllts of! arc over the real numlJers m... 1fanipulating polynomials, as opposed to arlJiLrary anaJytic fUllctions. is computationally more efficient. Furthermore algebraic surfaces provide enough gencl'illi\.y lo accurately model almost all complicated rigid objects. IThc oUlpul of olle opera.tion acts as [he inpul. 10 another operaLioll
A Strategy for the Interpolation of Surfaces through the Use of Basis Functions
2003
For the construction of digital terrain models based on surface interpolation, it is defined a bivariate function ¡ £ ¢ ¥ ¤ § ¦ © that interpolates a finite set of sample points, § ¢ ¤ ¦ © ¦ , such that, ¡ £ ¢ ¤ ¦. In this work, it is presented a strategy for the generation of interpolation surfaces through the use of basis functions. This methodology is based on a work by Chaturvedi and Piegl, where improvements related to the construction of the basis functions were made. The proposed strategy allows a larger expansion of the basis function's support region, represented by the interior of a trajectory curve, composed of quadratic rational Bézier segments and reduces the approximation error between the reference surface and the interpolation surface.
ACM SIGGRAPH 2005 Courses on - SIGGRAPH '05, 2005
We describe algebraic methods for creating implicit surfaces using linear combinations of radial basis interpolants to form complex models from scattered surface points. Shapes with arbitrary topology are easily represented without the usual interpolation or aliasing errors arising from discrete sampling. These methods were first applied to implicit surfaces by Savchenko, et al. and later developed independently by Turk and O'Brien as a means of performing shape interpolation. Earlier approaches were limited as a modeling mechanism because of the order of the computational complexity involved. We explore and extend these implicit interpolating methods to make them suitable for systems of large numbers of scattered surface points by using compactly supported radial basis interpolants. The use of compactly supported elements generates a sparse solution space, reducing the computational complexity and making the technique practical for large models. The local nature of compactly supported radial basis functions permits the use of computational techniques and data structures such as k-d trees for spatial subdivision, promoting fast solvers and methods to divide and conquer many of the subproblems associated with these methods. Moreover, the representation of complex models permits the exploration of diverse surface geometry. This reduction in computational complexity enables the application of these methods to the study of shape properties of large complex shapes.
Defining, contouring, and visualizing scalar functions on point-sampled surfaces
Computer-Aided Design, 2011
This paper addresses the definition, contouring, and visualization of scalar functions on unorganized point sets, which are sampled from a surface in 3D space; the proposed framework builds on moving leastsquares techniques and implicit modeling. Given a scalar function f : P → R, defined on a point set P , the idea behind our approach is to exploit the local connectivity structure of the k-nearest neighbor graph of P and mimic the contouring of scalar functions defined on triangle meshes. Moving least-squares and implicit modeling techniques are used to extend f from P to the surface M underlying P . To this end, we compute an analytical approximationf of f that allows us to provide an exact differential analysis off , draw its iso-contours, visualize its behavior on and around M, and approximate its critical points. We also compare moving least-squares and implicit techniques for the definition of the scalar function underlying f and discuss their numerical stability and approximation accuracy. Finally, the proposed framework is a starting point to extend those processing techniques that build on the analysis of scalar functions on 2-manifold surfaces to point sets.
Function Representation of Solids Reconstructed from Scattered Surface Points and Contours
Computer Graphics Forum, 1995
This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality The volume spline is based on use of the Green's function for interpolation of scalar function values of a chosen "carrier" solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross-sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD f o r bodies with complex surfaces.
A C2 triangular patch for the interpolation of functional scattered data
Computer-Aided Design, 1997
Given a set of data points with their positional. first-and second-order partial derivative values, we wish to construct a smooth surface which interpolates these values. The method requires triangulation of the data, and a rationally corrected quintic Bezier triangular patch scheme is then employed on each triangle. Each patch is rational of degree 9 over degree 4 but can be controlled by only 27 Bezier points. The data given enables us to determine appropriate Bezier control points so that adjacent patches meet with C' continuity.
Symmetry
Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve C 1 continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered...
Range-Restricted Surface Interpolation Using Rational Bi-Cubic Spline Functions With 12 Parameters
This paper discusses the constraint data interpolation or range restricted interpolation for surface data arranges on rectangular meshes that lie above or below an arbitrary plane and between two arbitrary planes by using partially blended rational bi-cubic spline function with 12 parameters. Common research in range restricted surface interpolation is to construct the constrained surface lie above linear plane. However, in this paper, we consider the constraint surfaces up to degree three (cubic). To construct the constrained surface with shape preserving properties, i.e., the resulting surface will lie below or above single planes or between two respective planes, the data dependent sufficient conditions are derived on four parameters; meanwhile, the remaining eight parameters are free parameters to change the shape of the interpolating surface locally. The proposed scheme is tested with various types of data test, including some well-known functions. From the numerical results, we found that the proposed scheme is easy to use, locally control via free parameters, and require less computation compared with some existing schemes as well as visually pleasant for visualization. Furthermore, based on root mean square error (RMSE) and coefficient of determination (R 2), the proposed scheme is better than existing scheme with the value of R 2 achieved is in between 0.9701 (97.01%) and 0.9954 (99.54%). This is quite good for range restricted surface data interpolation since we can explain at least 97.01% of the variance in the interpolation by using the proposed scheme. Furthermore, the proposed scheme requires less CPU time (in seconds) compared with the existing scheme. INDEX TERMS Constrained surface, shape preserving, interpolation, rational bi-cubic spline, local control, constraint planes.
A simple method for interpolating meshes of arbitrary topology by Catmull-Clark surfaces
The Visual Computer, 2010
Interpolating an arbitrary topology mesh by a smooth surface plays important role in geometric modeling and computer graphics. In this paper we present an efficient new algorithm for constructing Catmull–Clark surface that interpolates a given mesh. The control mesh of the interpolating surface is obtained by one Catmull–Clark subdivision of the given mesh with modified geometric rule. Two methods—push-back operation based method and normal-based method—are presented for the new geometric rule. The interpolation method has the following features: (1) Efficiency: we obtain a generalized cubic B-spline surface to interpolate any given mesh in a robust and simple manner. (2) Simplicity: we use only simple geometric rule to construct control mesh for the interpolating subdivision surface. (3) Locality: the perturbation of a given vertex only influences the surface shape near this vertex. (4) Freedom: for each edge and face, there is one degree of freedom to adjust the shape of the limit surface. These features make interpolation using Catmull–Clark surfaces very simple and thus make the method itself suitable for interactive free-form shape design.
Technical section Subdivision interpolating implicit surfaces
Interpolating implicit surfaces using radial basis functions can directly specify surface points and surface normals with closed form solutions, so they are elegantly used in surface reconstruction and shape morphing. This paper presents subdivision interpolating implicit surfaces, a new progressive subdivision tessellation scheme for interpolating implicit surfaces controlled by a triangular mesh with arbitrary topology. We use a recursive polyhedral subdivision scheme to subdivide the control triangular mesh, and the new generated vertices are mapped to the implicit surface using Newton iteration. A multiresolution surface representation is automatically built with the proposed approach. Based on this approach, a newly surface modeling tool with more flexible control is developed by blending the interpolating subdivision surfaces with the subdivision interpolating implicit surfaces. r 2003 Elsevier Ltd. All rights reserved.
Surface construction from planar contours
Computers & Graphics, 1987
calculated of the interactive ve was generated (1 S) (this implies r n) and then inin monotonicity re the conditions urv, 1 a spline 317 (l':lo6). sciola, Analysis of of"Basic L-splines." sciola, Using intercattered data. IEEE 4(7), 43-45 (1984). \onotone piecewise • Anal. 17(2), 238n bv some Hermite 1ppi. Math. 4, 7-9 ,ch verkoppelte un-1e interpolation. El.
Modelling with implicit surfaces that interpolate
ACM Transactions on Graphics, 2002
We introduce new techniques for modelling with interpolating implicit surfaces . This form of implicit surface was first used for problems of surface reconstruction and shape transformation, but the emphasis of our work is on model creation. These implicit surfaces are described by specifying locations in 3D through which the surface should pass, and also identifying locations that are interior or exterior to the surface. A 3D implicit function is created from these constraints using a variational scattered data interpolation approach, and the iso-surface of this function describes a surface. Like other implicit surface descriptions, these surfaces can be used for CSG and interference detection, may be interactively manipulated, are readily approximated by polygonal tilings, and are easy to ray trace. A key strength for model creation is that interpolating implicit surfaces allow the direct specification of both the location of points on the surface and the surface normals. These ar...
Local surface interpolation with B'ezier patches
Cagd, 1988
A surface interpolation method for meshes of cubic curves is described. A mesh of cubic curves is constructed between the given vertices. This mesh is filled with B6zier patches, so that the surface is represented as a union of geometrically continuous bicubic quadrilateral and/or quartic triangular B~zier patches. The method is local and uses Farin's [Farin '83l conditions of G 1 continuity between patches. The procedure for finding the needed control points of the B6zier patches is simple and efficient.
Modeling scattered function data on curved surfaces
1994
We present efficient algorithms to model a collection of scattered function data defined on a given smooth domain surface D in three dimensional real space (lR 3 ), by a C 1 cubic or a C2 quintic piecewise trivariate polynomial approximation F (a mapping from D into lR 4 ). The smooth polynomial pieces or finite elements of F are defined on a three dimensional triangulation called the simplicial hull and defined over the domain surface D. Our smooth polynomial approximations allows one to additionally control the local geometry of the modeled function F. We also present two different techniques for visualizing the graph of the function F.
ACM Transactions on Graphics, 1990
In industrial design, the tool of choice for constructing surfaces that interpolate curves is the Boolean sum surface technique. However, if curves do not lie on constant parameter lines, reparametrizations will be needed, and this may introduce derivative discontinuities. A new technique which shows promise in overcoming this problem is described here. The method is based on describing the interpolation problem directly as a system of linear equations rather than as a curve-blending problem. The resulting system of equations is usually underdetermined and can be solved using numerical linear algebra methods without the a priori determination of certain parameters. The “free” parameters can be used to control the shape of the resulting surface. Two examples of the procedure are given.