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

OUTLINE 1.About Author 2.Overall View of The paper 3.Previous Work 4.Axial Curve-pairs 5.Implementation and Results 6.Conclusions

Kin-chuen Hui 许健泉 Professor Department of Automation and Computer-Aided Engineering, CUHK

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.

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

Axial deformation technique 1.Basic idea of the technique

2. Axial Space—— A(C,l) Defined by a curve C(t), and a local coordinate system l(t)=[l x (t), l y (t), l z (t)] on the curve. P = (t, u, v, w) 3. Instance of an axial space t = t 0,the local coordinate frame.

4. Conversion of a point P in A(C,l) to 3D f: R 4 → R 3 P = f(t,u,v,w)= C(t)+ul x (t)+vl y (t)+wl z (t) 5. Reverse conversion: f -1 f -1 : R 3 → R 4 The value of t is generally decided by P N

where P N is closest to P, l z (t) is the direction of the tangent at C(t), hence:

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.

Framing a curve 1.Frenet Frame No user control of the orientation of the C ’’ (t) vanishes. 2.Direction curve approach, Lossing and Eshleman Axial curve-pair technique

Cannot be control intuitively 3. Local coordinate frame of a curve-pair the coordinate frame at P N

C(t): Primary curve C D (s): Orientation curve P D : the intersection of C D (s) with a plane passing through P N and having a normal direction C ’ (t). Problem of the Coordinate frame: Considerable amount of computation for getting P N.

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

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

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

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

Manipulating axial curve-pairs Primary curve C(t) Orientation curve C D (t) where

Simple approach to adjusting C D (t) when moving C(t) →

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

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

Local coordinate frame at a control point The frame at P i is given by the unit vectors Where t i is the polygon tangent at P i, Configuration of a curve-pair The set of all the tuples where

Specify the new position of q i after moving P i where

Comparing effect

Twisting the curve-pair Rotation of q i about t i Keep the configuration

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

Implementation and results Single axial ASR

The deformed dolphin model

A vase with the dolphin as decorative component

Construction of a ribbon knot

Construction of a leave pattern

Deformation of a squirrel shaped brooch

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.

Thank you!

Supplementary 1.Burtnyk N, Wein M. Interactive skeleton techniques for enhancing motion dynamics in key frame animation. CACM 1976; Oct: Singh K, Fiume E. Wires: a geometric deformation technique. Proc.SIGGRAPH : Lazarus F, Coquillart S, Jancene P. Axial deformations: an intuitive deformation technique. CAD 1994:26(8): BACK