free-form design using axial curve-pairs
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
Free-form design using axial curve-pairs
K.C. HuiCUHK
Computer-Aided Design 34(2002)583-595
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 Com
puter-Aided Engineering, CUHK
http://www2.acae.cuhk.edu.hk/~kchui/
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)=[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.
4. Conversion of a point P in A(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 by PN
where PN is closest to P,
lz(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 curve1. 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 PN
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.
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
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 CD(t)
where
Simple approach to adjusting CD(t) when moving C(t)
→
→
Problem of the simple approach
(a).
(b). Overlapping BACK
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
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
Specify the new position of qi after moving Pi
where
Comparing effect
Twisting the curve-pair
Rotation of qi about ti
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!Thank you!
Supplementary1.Burtnyk N, Wein M. Interactive skeleton
techniques for enhancing motion dynamics in key frame animation. CACM 1976; Oct:546-69.
2. Singh K, Fiume E. Wires: a geometric deformation technique. Proc.SIGGRAPH 98 1998:405-14.
3.Lazarus F, Coquillart S, Jancene P. Axial deformations: an intuitive deformation technique. CAD 1994:26(8):607-13.
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