1. Computing Vertex Normals from Arbitrary Meshes
Hyoung-Seok Kim and Ho-Sook Kim
Dongeui University and Dongeui Institute of Technology
KOREA

2. Contents 1. Problem Definition 2. Previous Works
3. Mean Value Coordinates
4. Our Method
5. Experimental Results
6. Conclusion

3. Computer Graphics Algorithms for visual simulation
- Modeling : How to represent 3D objects ?
- Rendering : How to simulate the formation of images ?
Light Simulation
Normal Vector
of Vertex
Modeling
Rendering

4. Normal Vector Mathematical definition in continuous space
 Unit vector orthogonal to the tangent plane of a point
Tangent Plane : Unique N

5. Normal Vector In discrete space (ex, Mesh)
 How many tangent planes of a vertex are there ?
- Infinitely many tangent planes
- Infinitely many normal vectors

6. Problem Definition What is the normal vector of a vertex ?
 No exact definition in discrete spaces.
 Need a good approximation of normal vector
How to compute the normal vector ?
 Our solution :
conformal mapping and mean value coordinates N N

7. 1. Gouroud’s method [1971] Drawback : Dependency on topology
Previous Works
1. Gouroud’s method [1971]
Drawback : Dependency on topology V 1 2 m - Nf
N_C =

8. Previous Works 2. Thurmer’s [1998] : Angle weighted
3. Max’s method [1999] : Area weighted V 1 2 m - Nf q V V m Nf 1 Nf 1 q 1 q m q m q 1 - m V 1 - m V
N_C = 2 V 2 V 1 V 1 V m V

9. 4. Chen’s [2004] : Area + Length
Previous Works
4. Chen’s [2004] : Area + Length V V 1 V 1 V m V
Drawback : different vectors for the same local shape
N_C =

10. Mean Value Coordinates [Float, 97]
How to represent a point with vertices of a polygon? P 1 U 2 m - 2 a 1 m - U
N_C =

11. Mean Value Coordinates
Advantage : Smoothness
Mean Value Coordinates
Barycentric Coordinates
N_C =
Convex
Concave

12. Conformal Mapping + Mean Value Coordinates
Our Method
Conformal Mapping + Mean Value Coordinates
N_C =

13. Conformal Mapping : relatively local shape preserving
Our Method
Conformal Mapping
: relatively local shape preserving

14. Mean Value Coordinates
Our Method
Mean Value Coordinates
New Normal Vector

15. Experiments
Approximated Spheres
Error Measure

16. Experimental Results Mean error for the recursively generated spheres
N_C =
Mean error for the recursively generated spheres

17. Experimental Results Mean error for the recursively generated spheres
N_C =
Mean error for the recursively generated spheres

18. Experimental Results
N_C =
Mean error for the parametric spheres

19. Experimental Results (Rendering)
N_C =

20. Conclusion New Normal Vector Computation
- Based on Conformal Mapping + Mean Value Coordinates
- Efficient for approximated spheres
- Efficient for Rendering
- Available for meshes of arbitrary topology
Future Works
- More tests for a variety of meshes
- Apply this method to Phong Shading
N_C =