Fig. 2 Gaussian surfaces with different standard deviations

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1 Free-Form Deformations Based on Gaussian Functions - Fundamental Theory for Interactive Modeling- Norimasa YOSHIDA, Ken'ya KANOU and Katsuhiro KITAJIMA Interactive and intuitive modeling is one of the most important issues in geometric modeling. However, modeling a complex object interactively and intuitively is still a difficult problem. In this paper, new Free-Form Deformations (FFDs) based on Gaussian functions are introduced. Different from previous FFDs that are based on Bernstein or B-spline functions, FFDs based on Gaussian functions allow an arbitrary number of control points to be placed in arbitrary positions. Moreover, the standard deviations of Gaussian functions can control the localness of the effects of control points. These facts provide designers with more interactivity and more intuitiveness. The designer can place a manipulation point in the region to be modified, and by changing the standard deviations, the designers can choose the effect of localness of the movement of the manipulation point. The definition of FFDs based on Gaussian functions, as well as several examples of the FFDs, is presented. Key words: geometric modeling, free-form deformation, direct manipulation, Gaussian functions Fig. 1 A parallelpipedical lattice

2 Fig. 2 Gaussian surfaces with different standard deviations

3 Fig. 3 Flow of FFDs based on Gaussian functions Fig. 4 FFDs of a cube with different standard deviations

4 (b) 吉田加納北鳴:がウス関数に基づくFree-FOrmDeformation (a)original cow model (a)σs=σt=σu=10.0 (b)ƒð9 = 0.03 σs=10.0σt=0.05σn=0.01 (c)ƒð9 = 0.1 (c)shade image of(a)(d)shadedimageof(b) Fig.5FFDsofacowmodel(1) 標準偏差の変更も可能であり,図3に破線で示すように全く異なる操作点を配置しなおして変形することも可能である. 5.ガ図4は,立ウス関数に基づくFFDを方体のコーナの8か所に操作点を置き,すの操作点の標準偏差を変えてFFDを変形する様子を示している.立用いた変形べて行ったときの立方体の方体は,各間隔にパラメータ化されている.図4(a)に (d)ƒð9 = 0.2 軸方向に関して等示されるように, 標準偏差が大きい場合にはパラメータの等間隔性が保たれ, 標準偏差が小さくなると,各パラメータ点は操作点に近づいていく(図4(b),(c)).図4(a),(b),(c)にの標準偏差は,そおけるすべての操作点れぞれ,10.0,0.5,035である(標準偏差はモデルの大きさに依存するので,各 0 x,y,z 1とモデルの座標はなるようにスケーリングされている). 図5(a)は,cowモデルを囲む立方体の頂点上に8個の操作点を配置した図である.図5(b)は,この状態から操作点の位置を変えずに,すよびu方べての操作点のtおだけを小さくしてFFDを 974精向の標準偏差行ったものである.図5(c),(d)は, 密工学会誌Vol,65,No.7,1999 (e)shaded image of (d) Fig. 6 FFDs of a cow model (2)

5 1) S. Coquillart: Extended Free-Form Deformation: A Sculpting Tool for 3D Geometric Modeling, Computer Graphics (SIGGRAPH'90), 24, 4, (1990) ) O.R.Davis and R. P. Burton: Free-Form Deformation as an Interactive Modeling Tool, J. Imaging Technology, 17, 4, (1991), ) A. Goshtasby: Design and Recovery of 2-D and 3-D Shapes Using Rational Gaussian Curves and Surfaces, Int. J. Computer Vision, 10, 3, (1993) ) A. Goshtasby: Geometric Modelling Using Rational Gauss-Ian Curves and Surfaces, Computer-Aided Design, 27, 5, May (1995), ) J. Griessmair and W. Purgathofer: Deformation of Solids with Trivariate B-splines, Computer Graphics Forum (EUROGRAPHICS'89), (1989) ) W. M. Hsu, J. F. Hughes and H. Kaufman: Direct Manipulation of Free-Form Deformation, Computer Graphics (SIGGRAPH), 26, 2, (1992) ) H. J. Lamousin and W. N. Waggenspack, Jr.: NURBS- Based Free-Form Deformations, IEEE CG&A, 14, 6, (1994) 59. 8) T. W. Sederberg and S. R. Parry: Free-Form Deformation of Solid Geometric Models: Computer Graphics (SIGGRAPH'86), 20, 4, (1986), ) D. Zorin, P. Schroder and W. Sweldens: Interactive Multiresolution Editing, Computer Graphics (SIGGRAPH'97), (1997) 259.

Developement of Plastic Collocation Method Extension of Plastic Node Method by Yukio Ueda, Member Masahiko Fujikubo, Member Masahiro Miura, Member Sum

Developement of Plastic Collocation Method Extension of Plastic Node Method by Yukio Ueda, Member Masahiko Fujikubo, Member Masahiro Miura, Member Summary Previously, the authors developed the plastic