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 fluxZero at non-medial pointsNon-zero at medial points and

computed as shown later.

Modified Divergence Theorem

F is a smooth function, discontinuous at the medial surface.

Define cF : projTM(F) = cF U

Grassfire Flow

F = G = -U projTM(G) = -U

cG = -1

Limiting Flux

Region Γt(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 : voln (Γt(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 points1 dimensional medial axis2 dimensional medial surface

voln-1

Medial density

example

Medial Densities

1/π 1/4

Medial density