Normal based subdivision scheme for curve and surface design
description
Transcript of Normal based subdivision scheme for curve and surface design
![Page 1: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/1.jpg)
Normal based subdivision scheme for curve and surface design
杨勋年2004.12
http://www.math.zju.edu.cn/yxn
![Page 2: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/2.jpg)
What is CAGD
Computer science
CAGD
Engineering
mathematics
![Page 3: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/3.jpg)
Content
• What is subdivision?- corner cutting algorithms
- interpolating subdivision• Normal based subd. Scheme
- the scheme- for curve design- for surface design
• Summary
![Page 4: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/4.jpg)
What is subdivision
• Recursive refinementfor the generation of- functions (approx. theory, wavelet)- curves and surfaces (CAGD)
• Classification- Steady vs nonsteady- rational vs nonrational- Linear vs nonlinear
![Page 5: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/5.jpg)
Corner cutting algorithms
• Corner cutting: Chaikin, B-spline
• Convergence: de Boor, Riesenfeld, Gregory, et al
![Page 6: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/6.jpg)
Subdivision of B-spline
• Uniform cubic B-spline
• Derive the rule by knots insertion
![Page 7: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/7.jpg)
Arbitrary control mesh
• The topological rule
• The geometric rule
Catmull-clark scheme
![Page 8: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/8.jpg)
Catmull-clark subdivision surface
![Page 9: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/9.jpg)
Interpolating subdivision
• Edge split
• Vertex refinement
![Page 10: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/10.jpg)
Four-point scheme
• Cubic precision (Dyn, et al 1987)
• Linear subdivision
Add a point by local cubic curve interpolation
A geometric look at four point scheme
![Page 11: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/11.jpg)
Butterfly scheme
• Extension of 4-point scheme (Dyn, et al 1990)• Triangular control mesh (1 to 4)• Local bicubic surface interpolation
Control meshParametric domain
![Page 12: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/12.jpg)
Limitations
• Interpolating or fitting- efficient representation- scanning data processing
• By CC scheme- solve inverse problem
• By butterfly scheme- not fair- not easy for normal control
![Page 13: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/13.jpg)
Content
• What is subdivision?- corner cutting algorithms
- interpolating subdivision• Normal based subd. Scheme
- the scheme- for curve design- for surface design
• Summary
![Page 14: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/14.jpg)
Our approach
• Normal refinement
- for each vertex for each level
• Vertex refinement
- subdivide each edge
- project sub-edges onto normals
- compute displacement vector
- compute new vertex
![Page 15: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/15.jpg)
The basic scheme
![Page 16: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/16.jpg)
Normal refinement
• Fixed normal at selected vertexes
- the normal will be interpolated
• Refine other normal for each subdivision
• The rule for normal computation
- chord tangent angles are close
![Page 17: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/17.jpg)
Normal computation
Curve case Surface case
![Page 18: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/18.jpg)
Convergence
• Active chord tangent angles- converge to zero- within fixed scale
• Fixed chord tangent angles- are bounded- convergence
• Polygon series- converge- tangent continuous
![Page 19: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/19.jpg)
For curve design
• The freedoms
- subd. ratio of edges
- scale for displacement vector
• Shape preserving
- same scheme
- explicit choices of freedoms
![Page 20: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/20.jpg)
Shape preserving scheme
12 1kip
ki k
i
1kip
mp
1kin
kip
kin
![Page 21: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/21.jpg)
Freeform curve
![Page 22: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/22.jpg)
Bottle design
Control polygon Subdivision curve
![Page 23: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/23.jpg)
For surface design
• Triangular control mesh
• Topology split
• Vertex refinement
- Normal based scheme
![Page 24: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/24.jpg)
Topology split
![Page 25: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/25.jpg)
Head model
Control mesh Subdivision surface
![Page 26: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/26.jpg)
Solid star
Control mesh Subdivision surface
Butterfly subdivision surface Modified butterfly subd. surface
![Page 27: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/27.jpg)
Knot surface
Control mesh
Butterfly subd. Normal based subd.
![Page 28: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/28.jpg)
Summary
• Normal based subdivision - a geometric scheme
- tangent continuous- natural shape
• Contributions - normal refinement as well as vertex refinement- geometric dependent instead of parametric dependent
![Page 29: Normal based subdivision scheme for curve and surface design](https://reader036.fdocuments.net/reader036/viewer/2022062500/568151dd550346895dc015e6/html5/thumbnails/29.jpg)
Thank you !