A Real-Time Deformable Clay Model for Rotationally Symmetric Objects

2. VIRTUAL CLAY MODELER We have developed the interactive defonable clay modeler to provide a human designer with a rnethd to interact with the virtual objects. I n our model. the smoothness of the surface and the volume constancy are gu.mnteed even i f any dcSormation is applied, Our method consists of two fundamental rndcling concepts. One IS the representation of the objects by the sliced cylinder mndel(SCM) and the other is smm~hed contour model bascd on the Snzke. The SCM determines the rough shape and the Snake makes the surface xmooth. 2.1 Representation m&t of the clay In our model, we assume the clay object has the following design consmintx. (1) A clay object has a symmemcd form which is given by rotating the contour line mund the axis. (2) The volume of the clay object does not change, even if its shap is deformed. ( 3 ) The 2-D contour is con~inuous and differentiable (smooth). The represenration of the d a y is shown in Fig. I . Let R be a rotationally s y m m e ~ c object. R can he ohrained by rotating 11s contour C. P. INTRODUCTION C : 1 7 1 ~ ) = (X(t i ) .Y(n)) (1 ) Recently. the rndcling methods for representing the natural and non-functional shapes as well as the deformation of their shapes, called the dcfonnabIe mdels, have been proposed. These methods are applied to several fields such ns CG, computer usion and palrtrn recognition. Our research aims at the development nF thc computer supported conceptual design system for the 3D objects, using these deformable modeling methods. Pentland and WiEliamsl 1 ] proposed h e usage of the s ~ p e ~ q ~ a d r i c ~ to represen1 the elastlc deformable ohjects and applicd i t for the 3D object recognition. Me also applied the modal dynamics to reduce the computational cost for the determination of the shapes. Tertopoulos and Eleischer[2t represented both the mot~on and the form of the objects hy the new expressing method based on the theory of eIasticity. He applicd thc methcd lor the represenration of the plastic clay as well. These methods, however, need morc effon to rcalizc real-time deformanion. A rndel proposed by Sato and Nurnazaki(31 can provide the ~nteractive deformation of the plastic rnzatinnally symmezric clay. I n this model. ro reduce computational iinic. the shape is represented by its 2dimensions! contour described by s small number of points. The modification operations can be applied just on one of those points at one: time. Furthermore. the deformed shape i s determined by only a few points near the actually modified point. The disadvantages nf this methd are that thc number and [he positions of those points affect the deformed shape. and that the smoothness of the cnntour(riurface} is not considered at all. We have developed the inferactive defonnabte clay modeler which guarantees the smoo~hnese of the surface and the volume constancy. Our method consists of two fundamental modeling concepts. One i~ the representation of the objects hy the sliced cylinder mdeE(SCM) and the other is the srnmthed contour model based on the Snake. Thc f m e t detem~ncs the mugh shape and the latter makes the surface smooth. In both methods, the energy minimi7ation techniques are employed. around y-axis, where O<u<l. X(u) 2 0 and Y(u) 2 [I. For simplicity, hereinafter, we assume an abject has no hoEes Inside, that is , the inside of R is full with the clay. We also introduce the following constraints. in order to give fine operational environment to the uqers. (4) The area to be m d i f i e d by one action should not be limited to the namw mpion. ( 5 ) The rigidity of the clay can be varied by tuning some parameters. (6) Deformations are perfwed in real-time. 2.2 Sliced cylinder model (SCM) In our model, the object R is constructed by a collection of thin cylinders (F1g.2). Each of these cylinders can bc represented by its radius (x value of rhe contour C) and rhickness (difference aF y value: dY). The s h a ~ of the object R is modified by changing the radius i1nt1 the thickness of each cylinder under volume constancy. We define the energy functions corresponding to the changes of the shape and the psition of the cylinders. Fig.1 Representation of R by the contour C. Fig.2 Sliced cylinder model. Suppose that R is deformed fmm a cylinder S with a unit height and with radius ro. Under volume constancy. Therefore, dYldu decides the new radius X. The shape of R i s presented by the Y(u). Here, we assume that Y(u) is increasing, a function of class ~ 2 , and Y(O)=O. Next, we define the evaluate function. Let V ( u ) and Yuu) be the shapes of R before and after deformation, respecrive1 y. First. for the change of the thickness of each diced cylinder, dY dY*, we define the following strain I I u En = +(y 1l2 2 k l # dY d Y P (4) = y;=du where k is a constant value. For the shift ol each sIiced cylinder. Y Yo, we define the following energy proportional to the shifr. whcm p is constant. The total energy with respect to the mdification of the sliced cylinders is defined as