Modeling Mean Curvature Flow using Cellular Automata Dr. Jeremy Scott LeCrone, Barbara Joy Smith, and Samantha Carol Zerger Kansas State University 22 July 2014 Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 1 / 19
Mean Curvature � � dT � � H ( � x 0 ) = ± � � � ds � Given a curve Γ with a parametrization γ : R → R 2 , the curvature of Γ at a point x 0 = γ ( t 0 ) is given by the formula x 0 ) = ±|| γ ′ ( t 0 ) × γ ′′ ( t 0 ) || H ( � , || γ ′ ( t 0 ) || 3 where the sign is determined if one fixes an orientation on the curve. Meanwhile, if f is known, the curvature takes the form − f ′′ ( x ) H ( � x 0 ) = . 3 (1 + f ′ ( x ) 2 ) 2 Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 2 / 19
Mean Curvature Flow V Γ ( t ) = −H Γ ( t ) Evolves based on Geometric Evolution Equations Surface area is always going to be decreasing Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 3 / 19
Cellular Automata Finite dimensional grid of cells – such as an integer lattice. Each cell stores a finite number of states. Set of rules governing the evolution of these states. Rules are based on local conditions of neighboring cells. Goes through several generations. Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 4 / 19
Theoretical Justification for Local Geometric Approximations Theorem The curvature at a point on a closed curve Γ ∈ C 2 can be calculated using x 0 ∈ Γ rotations and translations so that the mean curvature at a point � follows from A f ,δ lim = H ( � x 0 ) . δ 3 δ → 0 + 3 Γ n γ ′ ( t ) f ′ (0) = 0 γ ( t ) f (0) = 0 n Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 5 / 19
Definitions B δ (0) ∩ A q Γ = graph ( f ) B δ (0) ∩ A f q ( x ) δ A q ,δ A f ,δ δ A f := { ( x , y ) | y > f ( x ) } and A q := { ( x , y ) | y > q ( x ) } . A f ,δ := | B δ (0) ∩ A f | − 1 2 | B δ (0) | and A q ,δ := | B δ (0) ∩ A q | − 1 2 | B δ (0) | . Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 6 / 19
Claim 1 Claim 1: A q ,δ lim = H Γ ( � x 0 ) . δ 3 δ → 0 + 3 Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 7 / 19
Claim 1 Claim 1: A q ,δ lim = H Γ ( � x 0 ) . δ 3 δ → 0 + 3 √ �� � 3 3 f ′′ ( x ) 2 2 + 3 π A q ,δ • = � δ 3 f ′′ ( x ) 2 f ′′ ( x ) 2 δ 2 + 1 + 1 2 δ 3 √ �� � 3 2 f ′′ ( x ) − 3 2 δ sin − 1 + � � f ′′ ( x ) 2 δ 2 + 1 + 1 3 f ′′ ( x ) 2 δ 2 + 1 + 1) ( 2 A q ,δ = − f ′′ (0) = H Γ ( � • lim x 0 ) δ 3 δ → 0 3 Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 7 / 19
Claim 2 Claim 2: A f ,δ − A q ,δ lim = 0 . δ 3 δ → 0 + Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 8 / 19
Claim 2 Claim 2: A f ,δ − A q ,δ lim = 0 . δ 3 δ → 0 + � δ � δ | A f ,δ − A q ,δ | | f ( x ) − q ( x ) | dx | g ( x ) | dx • ≤ = δ 3 δ 3 δ 3 − δ − δ � δ � δ M | x 3 | dx = M | x 3 | dx = M δ ≤ δ 3 δ 3 2 − δ − δ Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 8 / 19
Proof of Claim 2 Since f is twice differentiable, we can write f ( x ) = f (0) + f ′ (0) x + 1 2 f ′′ (0) x 2 + g ( x ) , � �� � q ( x ) where g ( x ) = O ( x 3 ) as x → 0. That is, there exists some positive constant M and ǫ > 0 such that | g ( x ) | ≤ M | x 3 | for | x | < ǫ . Therefore, | f ( x ) − q ( x ) | = | g ( x ) | ≤ M for | x | < ǫ. x 3 x 3 Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 9 / 19
Discrete Setting using Cellular Automata Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 10 / 19
Defining the Interface Neighbors The 8 adjacent cells surrounding a specific cell If at most 7 neighbors are alive around a particular cell it is declared an interface point. Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 11 / 19
Template Circle Size π *radius 2 Box Radius Box Count Percent Error 3.5 37 38.4845 -3.8574 6 113 113.0973 -.0861 7 149 153.9380 -3.2100 7.5 177 176.7146 .1615 9 253 254.4690 -.5773 10.5 349 346.3606 -0.7620 radius = 6 α radius = 7.5 α radius = 10.5 α Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 12 / 19
Burning Algorithm This important algorithm provides the most information about our curve because it relates to the curvature at a point. Extra live cells that we don’t Cells that are not alive that need want to consider to be considered Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 13 / 19
Add/Remove Algorithm Thought at first to make a “mean curvature function.” Decided to directly use ranges of burncounts and apply the add/remove algorithm (determined with previous knowledge) Modified to use two template circles -1 +1 -3 0 Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 14 / 19
Outline of Entire Algorithm 1 Initialize some surface and obstacles. 2 Identify interface points. 3 Place a template circle around an interface point. 4 Run through the burning algorithm. 5 Use this count to determine how many cells should be added or removed. 6 Repeat 3-6 for every interface point. 7 Designate which cells are to be added or removed. 8 Change the cell values so the “to be removed” cells are now dead and those “to be added” are now alive. 9 Repeat steps 2-8 (one generation) as long as desired. Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 15 / 19
Batman! Because everyone wants to see how Batman evolves... Initial Object After 1 step After 2 steps After 10 steps After 20 steps After 45 steps Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 16 / 19
Embedding Obstacles An obstacle matrix is overlayed on the larger cell matrix. Fix the obstacle on the lattice. The obstacle remains fixed, “immortal” in a way. Our algorithm removes these cells from the list of potential cells when determining which ones to add or remove. The evolution of the closed curve with one obstacle is expected to evolve toward and then wrap around the obstacle. Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 17 / 19
Embedded Obstacles Two obstacles After 30 steps After 50 steps After 80 steps After 100 steps After 122 steps Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 18 / 19
Potential Research Other methods to determine the curvature discretely looking at only the interface points. Determining the normal direction and velocity at an interface point. Prioritizing which cells to add or remove based on the normal direction. Relationship between interface velocity and concentration of add/remove values. Analysis in higher dimensions. Dr. LeCrone, B. Smith, and S. Zerger Discrete Mean Curvature Flow 22 July 2014 19 / 19
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