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

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

Divergence Theorem Divergence of a vector Flux Standard Divergence Theorem

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

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

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

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

and

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

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

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

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

Limiting flux The limiting flux goes to zero everywhere.

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

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

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

Medial density example

Medial Densities 1/π 1/4

Medial density