Surface Representation as a Boundaryvalued

Transcription

1 Surface Representation as a Boundaryvalued PDE Problem Lihua You Jian J Zhang National Centre for Computer Animation, Bournemouth University Poole, Dorset BH 5BB, United Kingdom jzhang@bournemouth.ac.uk Surface representation as a boundary-valued problem of partial differential equations (PDE) is an important topic of computer graphics and computer aided design. In eisting references, various free-from surfaces were created with a fourth order PDE which is only able to meet the tangential conditions at the surface boundaries. The need for a sith order PDE in surface modelling arises in two situations: one is to generate surfaces with curvature continuity; and the other is to use curvature values as a user handle for surface shape manipulation. In this chapter, we introduce such a sith order PDE for free-form surface generation and develop a finite difference method to solve this PDE. We also investigate the effects of boundary curvature and the vector-valued shape parameters on the surface shape. Introduction Traditional parametric surface representations, such as Bézier, B- spline and NURBS [7] are common methods of surface modelling. Surfaces and curves are manipulated through the use of the control points. In the real world, all objects have certain physical properties and will change shape when they are subject to internal and eternal actions. But the traditional surface modelling methods do not attempt to describe the effect of physics laws. Physically based surface modelling methods have attracted more and more attention in the communities of computer graphics and computer-aided design to complement the traditional geometrically based modelling methods. Terzopolous et al. were among the first researchers to study physically based surface modelling approaches for computer graphics. In their initial work, an elastic deformation

2 model was proposed by Terzopolous et al. [8]. Based on the theories of elasticity and the finite element technique, deformable curve and surface finite element formulae were implemented by Celniker and Gossard [4]. The elastic deformation model was later etended to include viscoelasticity, plasticity and fracture by Terzopolous, et al. [9, ]. In order to describe dynamic surface deformation, various physically based dynamic surface modelling methods were developed by Terzopolous, Qin, Mandal and Vemuri [,, 7, ]. These static and dynamic surface modelling methods were developed from the theories of elasticity, plasticity and fracture mechanics and implemented using the finite element method and finite difference method. Applying other mechanics approaches to generate surfaces has also been studied by some researchers. For eample, Léon and Verson [9], and Guillet and Léon [8] applied a bar network mechanics method to deform free-form surfaces. Partial differential equation based surface modelling method should also be classified as physically based modelling. For cloth and other fleible fabrics, their governing equations that represent the deformed surfaces can be derived from the physical laws of elastic cloth deformation and motion which are the fourth order dynamic PDEs involving the mechanical properties [, 4]. PDE based surface modelling was first proposed by Bloor et al. []. In order to etend the PDE based methods to solve more complicated surface modelling problems, some numerical methods, such as the finite element method [,, ], finite difference method [5, ] and collocation method [] have been developed. Zhang and You [5] also discussed the effectiveness and efficiency of surface modelling using the second, fourth and mied order PDEs. In the above work, only the second and fourth order PDEs were applied. For a fourth order PDE, tangent continuity at the boundary curves of the surfaces to be created can be taken into account. Therefore, such an equation is effective in generating surfaces requiring tangent continuity. However, in many engineering design tasks, curvature continuity must be considered to satisfy the functional or visual requirements. For eample, the streamlined surfaces of an aircraft, ship and submarine with curvature continuity do not only look pleasant, but more importantly reduce the risk of flow separation and turbulence. In contrast, a cam designed without curvature continuity between two connected surfaces will cause abrupt changes in acceleration resulting in harmful impact.

3 Pegna [4] proposed a method with which curvature continuous fairing surfaces can be interactively generated. For designing curvature continuous blending surfaces, Pegna and Wolter [5] developed a linkage curve theorem. Curvature continuity between two rectangular or triangular patches was eamined by Zheng et al. [] with the two patches being represented with rational Bézier surfaces. Aumann [] proposed the so-called normal ringed surfaces to form curvature continuous connections of cones and/or cylinders. In this chapter, we will use a sith order PDE for free-from surface representation to account for the effect of boundary curvature. Since finding the closed form solution of a sith order PDE is more difficult than that of a fourth order PDE, we also develop a finite difference technique to solve the proposed sith order partial differential equation. How boundary curvature and the shape parameters affect the surface shape will be investigated. Surface representation eamples will be given to demonstrate the applications of the developed method. Governing equations and boundary conditions Boundary curvature continuity can be taken into account with a sith order PDE. Since the vector-valued shape parameters in the equation eert a strong effect on surface shape, they act as user handles for surface manipulation. Therefore, instead of using one shape parameter proposed by Bloor et al. [], we introduce four shape parameters and propose to use the following sith order PDE to represent freefrom surfaces considering the curvature boundary conditions ( a + b + c + d ) = () 4 4 v v v where a = [ a a a ] T, b = [ b b b ] T, c = [ c c c ] T, [ d d d ] T y z y z d = are vector-valued shape parameters, [ ( u, v), y( u, v), z( u, v) ] T = represents a vector-valued function, and u, v are the parametric variables. Depending on the boundary conditions and the shape parameters, PDE () can represent various free-form surfaces. In order to solve a PDE for the generation of a surface, we must define proper boundary conditions first. Normally, the curvature of a surface = ( u, v) can be described with its second partial deriva- y z y z

4 tives. In this chapter, we only consider the curvature crossing the boundary curves, i. e.,. In this way, the boundary conditions which include the effects of boundary tangent and curvature can be written as u =, = G( v) = G ( v) = G ( v) () u =, = G 4 ( v) = G 5 ( v) = G ( v) The solution of the sith order partial differential equation under boundary conditions () will result in the representation of a surface. Finite difference technique The analytical solution of PDE () subject to boundary conditions () is usually very difficult to obtain. Only for some simple boundary conditions and special combinations of the shape parameters, does the closed form solution of Eq. () eist. Generally, Eq. () can only be solved with numerical methods such as the above-mentioned finite element method, finite difference method and weighted residual method []. Among them, the finite difference method transforms a PDE to a system of algebraic equations by replacing all the partial derivatives in the differential equation with their discretized approimations. In the following, we will develop such a finite difference technique to solve PDE () subject to boundary conditions (). To facilitate the description, we define a new vector product operator whose operands are two vectors of the same dimension and each element of the resultant vector is the product of the corresponding elements of the two vectors, i. e., = [ p ] T q p yq y pzq z where p = [ p p p ] T and = [ q q q ] T pq () y z q y z are two column vectors. Using the Taylor series epansion of a continuous function f ( u, v), we can derive its central finite difference approimations of the first and second partial derivatives at a typical node shown in

5 Figure. Then using the basic finite difference approimations of the first and second derivatives to formulate the fourth and sith partial derivatives, we can obtain the finite difference approimations of all the partial derivatives required. Substituting these finite difference approimations into PDE () and using the notation of Eq. (), we obtain its finite difference formula at the typical node as follows u 8 v δ δ Fig.. Typical finite difference nodes 4(5a + b + c + 4d) + 8c + 5d) ( b + c)( + d( ) ( c + d)( ) + c( (5a + 8b + c) ) = (5a + 8b + c) 7 + ) (a + b)( ) + b( (a + 8c + 5d) 8 + ) + a( (b ) 9 (4)

6 where i ( i = ~,7 ~ 8) represents the co-ordinate value of function at node i. Substituting the basic finite difference approimations of the first and second derivatives into boundary conditions (), the finite difference formulae at the typical node can be written as u = = G ( v ) u = + + = G = δg 4 ( v ) = δg 5 ( v ( v ) = δ G ( v ) ) = δ G where v is the value of the parametric variable v at node. Dividing the resolution domain into m by n discrete nodes, we can obtain the same finite difference formula (4) for all the nodes at the inner resolution domain and finite difference formula (5) for all the nodes at the boundaries. Writing these finite difference formulae into a matri form, we obtain a set of linear algebraic equations which has the form of KX = F () The resolution of equations () will lead to the finite difference solution of partial differential equation () under boundary conditions (). ( v ) (5) 4 Effect of boundary curvature In our previous discussions, we have mentioned that the boundary curvature has a great influence on the shape of surfaces to be created. In order to eplore this effect and develop it into a user handle for shape manipulation, in this section, we will investigate the effect of boundary curvature on the surface shape by a concrete problem of vase surface representation. The vase surface to be generated is defined by two boundary curves represented with two concentric circles of different radii. The boundary conditions considering tangent and curvature effects can be written as

7 u = ui = ri cos πv = r i cos πv = r cos i πv y y y = ri sin πv = r i sin πv = r sin πv i (7) z z z = hi = h i = h i ( i =, ) where u =, u =, h, = and h = h. For the surface to be generated, the resolution domain is a unit square in a two-dimensional space of parametric variables u and v. Uniformly dividing this square resolution domain into nodes and solving the above-developed finite difference equation (), we obtain the co-ordinate values of the surface at these nodes. In this section, the basic geometric parameters are taken to be: r =., r = r = r =. 8, h = h = h == r =, r =, h = and h =.. In order to investigate the effect of boundary curvature on surface shape, we fi the shape parameters in a vector-valued form a = c = d = [ ] T and b = [ ] T for all the case studies in this section and only change the boundary curvature. Let us first consider the effect of the boundary curvature whose value is determined by r on the vase surface shape. The initial value of r given above produces the vase surface in Figure a. Then it is set to r = and the image in Figure a is changed to that in Figure b. Further setting it to r = 5 gives the surface shape in the Figure c. Net, we investigate the effect of the boundary curvature whose value is determined by r on vase surface shape. With the above basic geometric parameters but changing r to, the vase shape is changed to that in Figure d. The image in Figure e is created with r =. Similarly, the boundary curvature determined by h and h also have an obvious effect on surface shape. Firstly, taking h = 8, we obtained the vase shape in Figure f. It is changed to that in Figure g when h = 5. For the boundary curvature determined by h, a value of gives the shape in Figure h and the value of -9 creates that in Figure i.

8 a b c d e f g h i Fig.. Effect of boundary curvature on surface shape These images clearly show that the boundary curvature can effectively affect the shape of the surfaces. By adjusting the values of the coefficients of the boundary curvature functions in Eq. (7), we can obtain different surface shapes.

9 5 Effect of vector-valued shape parameters Similar to the effect of the vector-valued shape parameters in a fourth order PDE, the shape parameters in our proposed sith order PDE should also have a strong influence on the surface shape. In order to study this influence, in this section, we eamine how the shape parameters in PDE () affect the surface shape. a b c d Fig.. Effect of vector-valued parameters on surface shape For simplicity, we demonstrate this with the same vase surface representation eample and the same node collocation within the resolution domain as those in the previous section. All the geometric parameters are also taken to be those given in the previous section.

10 Only the shape parameters are changed. Firstly, we set the values of the shape parameters to a =, b = [ ] T, [ ] T c = d = resulting in the vase surface in Figure a. Then fiing parameters b and d, and changing parameters a to [ ] T and c to [ 5 5 5] T, the surface shape in Figure b is obtained. Keeping a and b unchanged, and setting c and d to [ ] T and [ 7 7 7] T, respectively, the vase surface shape in Figure c is generated. Finally, by only changing b to [ ] T, the shape in Figure c is changed to that in d. From these images, it is clear that all the shape parameters in the sith order partial differential equation () have a strong effect on the shape of the surfaces to be generated. Therefore, these parameters can be effectively applied as a surface shape manipulation tool. Summary Free-form surfaces can be treated as a boundary-valued problem of partial differential equations. The ability of incorporating boundary curvature is a valuable merit in both engineering design and computer graphics. Eisting surface representation methods based on the solution of PDEs did not tackle this problem. In this chapter, we have proposed a sith order PDE which is capable of coping with such boundary conditions for surface generation. In addition, the proposed PDE introduces four vector-valued shape parameters, which can be effectively used as surface shaping tools. Since the analytical solution of a sith order PDE is far more difficult to achieve than that of a fourth or lower order PDE, a numerical resolution method is usually sought as an alternative. In order to solve the proposed sith order PDE effectively, we have derived the basic finite difference approimation formulae taking up to the sith partial derivatives. We have also transformed the proposed sith order PDE and the boundary conditions into a resultant finite difference equation represented in a matri form. We have made some investigations to understand the effects of the boundary curvature and the shape parameters in shape manipulation. It was found that all boundary curvature conditions and the shape parameters have a great influence on the surface. Therefore,

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