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Free-form design using axial curve-pairs

Free-form design using axial curve-pairs. K.C. Hui CUHK Computer-Aided Design 34(2002)583-595. OUTLINE. About Author Overall View of The paper Previous Work Axial Curve-pairs Implementation and Results Conclusions. Kin-chuen Hui 许健泉. Professor

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Free-form design using axial curve-pairs

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  1. Free-form design using axial curve-pairs K.C. Hui CUHK Computer-Aided Design 34(2002)583-595

  2. OUTLINE • About Author • Overall View of The paper • Previous Work • Axial Curve-pairs • Implementation and Results • Conclusions

  3. Kin-chuen Hui许健泉 Professor Department of Automation and Computer-Aided Engineering, CUHK http://www2.acae.cuhk.edu.hk/~kchui/

  4. Overall View of the Paper • What problem does the paper solve? • Freeform deformation of 3D shapes. • The essence of the paper: • Construct a local coordinate frame by a curve-pair.

  5. Previous Work • Free-form deformation(FFD), Sederberg and Parry • Initially propose • Skeleton-based technique,Burtnyk • Paper Link • Using wires for deformation, Singh and Fiume • Paper Link • Axial deformation techinque, Lazarus • Paper Link

  6. Axial deformation technique • Basic idea of the technique

  7. 2. Axial Space—— A(C,l) • Defined by a curveC(t), and a localcoordinate systeml(t)=[lx(t),ly(t),lz(t)] on the curve. • P = (t, u, v, w) • 3. Instance of an axial space • t = t0,the local coordinate frame.

  8. 4. Conversion of a pointPinA(C,l)to 3D f:R4→R3 P= f(t,u,v,w)=C(t)+ulx(t)+vly(t)+wlz(t) 5. Reverse conversion:f-1 f-1: R3 → R4 The value of t is generally decided byPN

  9. where • PNis closest toP, • lz(t) is the direction of the tangent atC(t), • hence:

  10. The point P in A(C,l) is expressed as • The major problem of the axial curve deforamtion: • Lack of control on the local coordinate frame of the axial curve • Cannot be twisted by manipulating the axial curve.

  11. Axial curve-pair technique Framing a curve • Frenet Frame • No user control of the orientation of the • C’’(t) vanishes. • 2.Direction curve approach, Lossing and Eshleman

  12. Cannot be control intuitively • 3. Local coordinate frame of a curve-pair • the coordinate frame atPN

  13. C(t): Primary curve • CD(s): Orientation curve • PD: the intersection of CD(s) with a plane passing through PN and having a normal direction C’(t). • Problem of the Coordinate frame: • Considerable amount of computation for getting PN.

  14. Improvement: • PD is obtained by projecting the point CD(t) to the plane • Local coordinate frame of a curve-pair • Axial curve-pair • An ordered pair (C, CD), | C(t) - CD(t)|≤ r

  15. The construction of orientation curve • The orientation curve lies within a circular tube • Similar to construct an offset of the primary curve

  16. Primary curve C(t) is a B-Spline curve The process of construction is below: (a).

  17. (b). (c). • The detailed process is the same to the process of • adjusting the local coordinate frame.

  18. Manipulating axial curve-pairs • Primary curve C(t) • Orientation curve CD(t) where

  19. Simple approach to adjusting CD(t) when moving C(t) • → • →

  20. Problem of the simple approach • (a). (b). Overlapping BACK

  21. New approach • the local coordinate of the vector relative to Pi keep constant while relocating Pi . • The local coordinate frame at Pi is specified with • a polygon tangent at Pi • a vector normal to the polygon tangent. • Polygon tangent • Give a polygon with vertices Pi , 0<i<n, the polygon tangent ti at Pi is

  22. Local coordinate frame at a control point • The frame at Pi is given by the unit vectors • Where ti is the polygon tangent at Pi , • Configuration of a curve-pair • The set of all the tuples • where

  23. Specify the new position of qi after moving Pi where

  24. Comparing effect

  25. Twisting the curve-pair • Rotation of qi about ti • Keep the configuration

  26. The axial skeletal representation • The hierarchy of axially represented shapes. • Axial Skeletal Representation(ASR) of the object.

  27. Implementation and results ASR Single axial

  28. The deformed dolphin model

  29. A vase with the dolphin as decorative component

  30. Construction of a ribbon knot

  31. Construction of a leave pattern

  32. Deformation of a squirrel shaped brooch

  33. Conclusions (a). Propose a new method to construct the local coordinate frame. (b). Using a hierarchy of axial curve-pairs to constitute a complex object.

  34. Thank you!

  35. Supplementary • Burtnyk N, Wein M. Interactive skeleton techniques for enhancing motion dynamics in key frame animation. CACM 1976; Oct:546-69. • Singh K, Fiume E. Wires: a geometric deformation technique. Proc.SIGGRAPH 98 1998:405-14. • Lazarus F, Coquillart S, Jancene P. Axial deformations: an intuitive deformation technique. CAD 1994:26(8):607-13. • BACK

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