1. Divergence Theorem, flux and applications Chapter 3, section 4.5, 4.6 Chapter 4, part of section 2.3 Rohit Saboo

2. Distance Fields Distance Field (left) Gradient of distance field (bottom)

3. Divergence Theorem Divergence of a vector Flux Standard Divergence Theorem

4. Flux of a vector field defined on the medial axis Flux with discontinuities along the medial axis

5. Average outward flux Grey: (near) zero flux Black: large negative flux

6. Modified Divergence Theorem Γ a region in Ω Γ has regular piecewise smooth boundaries

7. Modified Divergence Theorem Divergence of G Average outward flux Medial Volume Grassfire Flow G

8. and

9. Average Outward Flux Average outward flux Zero at non-medial points Non-zero at medial points and computed as shown later.

10. Modified Divergence Theorem F is a smooth function, discontinuous at the medial surface. Define c F : proj TM (F) = c F U

11. Grassfire Flow F = G = -U proj TM (G) = -U c G = -1

12. Limiting Flux Region Γ t (x) as t -> 0 x is a point on M x

13. Limiting flux The limiting flux goes to zero everywhere.

14. Average flux Limiting value of the average flux N-dimensional volume : vol n (Γ t (x))

15. Invariants at a medial point is the minimum non-zero values for the different values of U and N at x.

16. Medial density Different types of medial points 1 dimensional medial axis 2 dimensional medial surface vol n-1

17. Medial density example

18. Medial Densities 1/π 1/4

19. Medial density