Nine-point stencil

<h2 id="motivation">Motivation</h2>
If we discretize the 2D <a href="/facts/Laplace_operator/kov9u3PT">Laplacian</a> by using <a href="/facts/Finite_difference_method/EoG81Qn9">central-difference methods</a>, we obtain the commonly used <a href="/facts/Five-point_stencil/9PHAJh2B">five-point stencil</a>, represented by the following <a href="/facts/Convolution/PVPrdz9J">convolution</a> kernel:

 
 
 
 
 D
 
 C
 D
 
 
 =
 
 
 [
 
 
 
 0
 
 
 1
 
 
 0
 
 
 
 
 1
 
 
 −
 4
 
 
 1
 
 
 
 
 0
 
 
 1
 
 
 0
 
 
 
 ]
 
 
 
 
 {\displaystyle D_{CD}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}}

Even though it is simple to obtain and computationally lighter, the central <a href="/facts/Difference_kernel/4eD7pB8y">difference kernel</a> possess an undesired intrinsic <a href="/facts/Anisotropy/YMqTQWVB">anisotropic</a> property, since it doesn't take into account the diagonal neighbours. This intrinsic anisotropy poses a problem when applied on certain numerical simulations or when more accuracy is required, by propagating the Laplacian effect faster in the <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">coordinate axes</a> directions and slower in the other directions, thus distorting the final result.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
This drawback calls for finding better methods for discretizing the Laplacian, reducing or eliminating the anisotropy.

<h2 id="implementation">Implementation</h2>
The two most commonly used <a href="/facts/Isotropy/mc3QEY2x">isotropic</a> nine-point stencils are displayed below, in their convolution kernel forms. They can be obtained by the following formula:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

 
 
 
 D
 =
 (
 1
 −
 γ
 )
 
 
 [
 
 
 
 0
 
 
 1
 
 
 0
 
 
 
 
 1
 
 
 −
 4
 
 
 1
 
 
 
 
 0
 
 
 1
 
 
 0
 
 
 
 ]
 
 
 +
 γ
 
 
 [
 
 
 
 1
 
 /
 
 2
 
 
 0
 
 
 1
 
 /
 
 2
 
 
 
 
 0
 
 
 −
 2
 
 
 0
 
 
 
 
 1
 
 /
 
 2
 
 
 0
 
 
 1
 
 /
 
 2
 
 
 
 ]
 
 
 
 
 {\displaystyle D=(1-\gamma ){\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}+\gamma {\begin{bmatrix}1/2&0&1/2\\0&-2&0\\1/2&0&1/2\end{bmatrix}}}

The first one is known by Oono-Puri,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> and it is obtained when γ=1/2.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

 
 
 
 
 D
 
 O
 P
 
 
 =
 
 
 [
 
 
 
 1
 
 /
 
 4
 
 
 2
 
 /
 
 4
 
 
 1
 
 /
 
 4
 
 
 
 
 2
 
 /
 
 4
 
 
 −
 12
 
 /
 
 4
 
 
 2
 
 /
 
 4
 
 
 
 
 1
 
 /
 
 4
 
 
 2
 
 /
 
 4
 
 
 1
 
 /
 
 4
 
 
 
 ]
 
 
 =
 
 
 [
 
 
 
 0.25
 
 
 0.5
 
 
 0.25
 
 
 
 
 0.5
 
 
 −
 3
 
 
 0.5
 
 
 
 
 0.25
 
 
 0.5
 
 
 0.25
 
 
 
 ]
 
 
 
 
 {\displaystyle D_{OP}={\begin{bmatrix}1/4&2/4&1/4\\2/4&-12/4&2/4\\1/4&2/4&1/4\end{bmatrix}}={\begin{bmatrix}0.25&0.5&0.25\\0.5&-3&0.5\\0.25&0.5&0.25\end{bmatrix}}}

The second one is known by Patra-Karttunen or Mehrstellen,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> and it is obtained when γ=1/3.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

 
 
 
 
 D
 
 P
 K
 
 
 =
 
 
 [
 
 
 
 1
 
 /
 
 6
 
 
 4
 
 /
 
 6
 
 
 1
 
 /
 
 6
 
 
 
 
 4
 
 /
 
 6
 
 
 −
 20
 
 /
 
 6
 
 
 4
 
 /
 
 6
 
 
 
 
 1
 
 /
 
 6
 
 
 4
 
 /
 
 6
 
 
 1
 
 /
 
 6
 
 
 
 ]
 
 
 =
 
 
 [
 
 
 
 0.16
 
 
 0.66
 
 
 0.16
 
 
 
 
 0.66
 
 
 −
 3.33
 
 
 0.66
 
 
 
 
 0.16
 
 
 0.66
 
 
 0.16
 
 
 
 ]
 
 
 
 
 {\displaystyle D_{PK}={\begin{bmatrix}1/6&4/6&1/6\\4/6&-20/6&4/6\\1/6&4/6&1/6\end{bmatrix}}={\begin{bmatrix}0.16&0.66&0.16\\0.66&-3.33&0.66\\0.16&0.66&0.16\end{bmatrix}}}

Both are isotropic forms of <a href="/facts/Discrete_Laplace_operator/w3fdFNC4">discrete Laplacian</a>,<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> and in the limit of small Δx, they all become equivalent,<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> as Oono-Puri being described as the optimally isotropic form of discretization,<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> displaying reduced overall error,<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> and Patra-Karttunen having been systematically derived by imposing conditions of rotational invariance,<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> displaying smallest error around the origin.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

<h2 id="desired-anisotropy">Desired anisotropy</h2>
On the other hand, if controlled <a href="/facts/Anisotropy/YMqTQWVB">anisotropic</a> effects are a desired feature, when solving <a href="/facts/Anisotropic_diffusion/HK9nIVKt">anisotropic diffusion</a> problems for example, it is also possible to use the 9-point stencil combined with <a href="/facts/Tensor/pNjFo5tV">tensors</a> to generate them.
Consider the laplacian in the following form:

 
 
 
 c
 
 ∇
 
 2
 
 
 A
 =
 c
 
 D
 
 O
 P
 
 
 ∗
 A
 
 
 {\displaystyle c\nabla ^{2}A=cD_{OP}*A}

Where c is just a constant coefficient. Now if we replace c by the <a href="/facts/Cauchy_stress_tensor/5MsXPQVH">2nd rank tensor</a> C:

 
 
 
 C
 =
 
 
 [
 
 
 
 
 c
 
 1
 
 
 
 
 0
 
 
 
 
 0
 
 
 
 c
 
 2
 
 
 
 
 
 ]
 
 
 
 
 {\displaystyle C={\begin{bmatrix}c_{1}&0\\0&c_{2}\end{bmatrix}}}

Where c1 is the constant coefficient for the principal direction in x axis, and c2 is the constant coefficient for the secondary direction in y axis. In order to generate anisotropic effects, c1 and c2 must be different.
By multiplying it by the <a href="/facts/Rotation_matrix/c2K2Ftl8">rotation matrix</a> Q, we obtain C', allowing anisotropic propagations in arbitrary directions other than the <a href="/facts/Coordinate_axes/lkTqvMmB">coordinate axes</a>.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a><a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>

 
 
 
 Q
 =
 
 
 [
 
 
 
 cos
 ⁡
 θ
 
 
 sin
 ⁡
 θ
 
 
 
 
 −
 sin
 ⁡
 θ
 
 
 cos
 ⁡
 θ
 
 
 
 ]
 
 
 
 
 {\displaystyle Q={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}}

C
 ′
 
 =
 Q
 C
 
 Q
 
 T
 
 
 
 
 {\displaystyle C'=QCQ^{\operatorname {T} }}

C
 ′
 
 =
 
 
 [
 
 
 
 
 c
 
 x
 x
 
 
 
 
 
 c
 
 x
 y
 
 
 
 
 
 
 
 c
 
 x
 y
 
 
 
 
 
 c
 
 y
 y
 
 
 
 
 
 ]
 
 
 =
 
 
 [
 
 
 
 
 c
 
 1
 
 
 
 cos
 
 2
 
 
 ⁡
 θ
 +
 
 c
 
 2
 
 
 
 sin
 
 2
 
 
 ⁡
 θ
 
 
 (
 
 c
 
 2
 
 
 −
 
 c
 
 1
 
 
 )
 cos
 ⁡
 θ
 sin
 ⁡
 θ
 
 
 
 
 (
 
 c
 
 2
 
 
 −
 
 c
 
 1
 
 
 )
 cos
 ⁡
 θ
 sin
 ⁡
 θ
 
 
 
 c
 
 2
 
 
 
 cos
 
 2
 
 
 ⁡
 θ
 +
 
 c
 
 1
 
 
 
 sin
 
 2
 
 
 ⁡
 θ
 
 
 
 ]
 
 
 
 
 {\displaystyle C'={\begin{bmatrix}c_{xx}&c_{xy}\\c_{xy}&c_{yy}\end{bmatrix}}={\begin{bmatrix}c_{1}\cos ^{2}\theta +c_{2}\sin ^{2}\theta &(c_{2}-c_{1})\cos \theta \sin \theta \\(c_{2}-c_{1})\cos \theta \sin \theta &c_{2}\cos ^{2}\theta +c_{1}\sin ^{2}\theta \end{bmatrix}}}

Which is very similar to the <a href="/facts/Cauchy_stress_tensor/5MsXPQVH">Cauchy stress tensor</a> in 2 dimensions. The angle 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 can be obtained by generating a <a href="/facts/Vector_field/ccFNuPPp">vector field</a> 
 
 
 
 
 V
 
 =
 
 V
 
 x
 
 
 
 
 i
 
 
 +
 
 V
 
 y
 
 
 
 
 j
 
 
 
 
 {\displaystyle \mathbf {V} =V_{x}{\mathbf {i} }+V_{y}{\mathbf {j} }}
 
 in order to orientate the pattern as desired.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> Then:

 
 
 
 θ
 =
 arctan
 ⁡
 (
 
 V
 
 y
 
 
 
 /
 
 
 V
 
 x
 
 
 )
 
 
 {\displaystyle \theta =\arctan(V_{y}/V_{x})}

Or, for different anisotropic effects using the same vector field<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

 
 
 
 θ
 =
 arctan
 ⁡
 (
 
 V
 
 y
 
 
 
 /
 
 −
 
 V
 
 x
 
 
 )
 
 
 {\displaystyle \theta =\arctan(V_{y}/-V_{x})}

It is important to note that, regardless of the values of 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
, the anisotropic propagation will occur parallel to the secondary direction c2 and perpendicular to the principal direction c1:.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> The resulting convolution kernel is as follows<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>

 
 
 
 
 D
 
 A
 n
 i
 s
 o
 
 
 =
 
 
 [
 
 
 
 
 
 
 −
 
 c
 
 x
 y
 
 
 
 2
 
 
 
 
 
 c
 
 y
 y
 
 
 
 
 
 
 
 c
 
 x
 y
 
 
 2
 
 
 
 
 
 
 
 c
 
 x
 x
 
 
 
 
 −
 2
 (
 
 c
 
 x
 x
 
 
 +
 
 c
 
 y
 y
 
 
 )
 
 
 
 c
 
 x
 x
 
 
 
 
 
 
 
 
 
 c
 
 x
 y
 
 
 2
 
 
 
 
 
 c
 
 y
 y
 
 
 
 
 
 
 
 −
 
 c
 
 x
 y
 
 
 
 2
 
 
 
 
 
 ]
 
 
 
 
 {\displaystyle D_{Aniso}={\begin{bmatrix}{\frac {-c_{xy}}{2}}&c_{yy}&{\frac {c_{xy}}{2}}\\c_{xx}&-2(c_{xx}+c_{yy})&c_{xx}\\{\frac {c_{xy}}{2}}&c_{yy}&{\frac {-c_{xy}}{2}}\end{bmatrix}}}

If, for example, c1=c2=1, the cxy component will vanish, resulting in the simple <a href="/facts/Five-point_stencil/9PHAJh2B">five-point stencil</a>, rendering no controlled anisotropy.
If c2>c1 and 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
=0, the anisotropic effects will be more pronounced in the vertical axis.
If c2>c1 and 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
=45 degrees, the anisotropic effects will be more pronounced in the upper-right / lower-left diagonal.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Patra, Michael; Karttunen, Mikko (2006). "Stencils with isotropic discretization error for differential operators". Numerical Methods for Partial Differential Equations. 22 (4): 3–7. doi:10.1002/num.20129. S2CID 123145969. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Fomel, Sergey; Claerbout, Jon F. (9 October 1997). "Constructing an isotropic Laplacian operator". Exploring three-dimensional implicit wavefield extrapolation with the helix transform. Stanford Exploration Project. <a href="http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html" target="_blank">http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Lindeberg, Tony. "Scale-Space for Discrete Signals" (PDF). Stockholm: Royal Institute of Technology. pp. 22–23. <a href="http://kth.diva-portal.org/smash/get/diva2:472968/FULLTEXT01.pdf" target="_blank">http://kth.diva-portal.org/smash/get/diva2:472968/FULLTEXT01.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Oono, Y.; Puri, S. (1987). "Computationally efficient modeling of ordering of quenched phases" (PDF). Physical Review Letters. 58 (8): 837. Bibcode:1987PhRvL..58..836O. doi:10.1103/PhysRevLett.58.836. PMID 10035049.
 <a href="http://repository.ias.ac.in/31904/1/31904.pdf" target="_blank">http://repository.ias.ac.in/31904/1/31904.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Oono, Y.; Puri, S. (1988). "Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling" (PDF). Physical Review A. 38 (1): 436. Bibcode:1988PhRvA..38..434O. doi:10.1103/PhysRevA.38.434. PMID 9900182.
 <a href="http://repository.ias.ac.in/32406/1/32406.pdf" target="_blank">http://repository.ias.ac.in/32406/1/32406.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Sevink, G. J. A. (2015). "Rigorous embedding of cell dynamics simulations in the Cahn-Hilliard-Cook framework: Imposing stability and isotropy". Physical Review E. 91 (5): 4. Bibcode:2015PhRvE..91e3309S. doi:10.1103/PhysRevE.91.053309. hdl:1887/3194035. PMID 26066281.
 <a href="https://scholarlypublications.universiteitleiden.nl/access/item%3A3194036/download" target="_blank">https://scholarlypublications.universiteitleiden.nl/access/item%3A3194036/download</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Rotation-invariant Laplacian for 2D grids". 21 March 2021. <a href="https://eng.aurelienpierre.com/2021/03/rotation-invariant-laplacian-for-2d-grids/" target="_blank">https://eng.aurelienpierre.com/2021/03/rotation-invariant-laplacian-for-2d-grids/</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"Investigation of Isotropic Laplacian Operators by Computer Simulation for A-B Diblock Copolymers" (PDF). IJCSNS International Journal of Computer Science and Network Security. 18 (5): 102–103. May 2018. <a href="http://paper.ijcsns.org/07_book/201805/20180514.pdf" target="_blank">http://paper.ijcsns.org/07_book/201805/20180514.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Fomel, Sergey; Claerbout, Jon F. (9 October 1997). "Constructing an isotropic Laplacian operator". Exploring three-dimensional implicit wavefield extrapolation with the helix transform. Stanford Exploration Project. <a href="http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html" target="_blank">http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Patra, Michael; Karttunen, Mikko (2006). "Stencils with isotropic discretization error for differential operators". Numerical Methods for Partial Differential Equations. 22 (4): 3–7. doi:10.1002/num.20129. S2CID 123145969. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">"Rotation-invariant Laplacian for 2D grids". 21 March 2021. <a href="https://eng.aurelienpierre.com/2021/03/rotation-invariant-laplacian-for-2d-grids/" target="_blank">https://eng.aurelienpierre.com/2021/03/rotation-invariant-laplacian-for-2d-grids/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">"Investigation of Isotropic Laplacian Operators by Computer Simulation for A-B Diblock Copolymers" (PDF). IJCSNS International Journal of Computer Science and Network Security. 18 (5): 102–103. May 2018. <a href="http://paper.ijcsns.org/07_book/201805/20180514.pdf" target="_blank">http://paper.ijcsns.org/07_book/201805/20180514.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Thampi, Sumesh P.; Ansumali, Santosh; Adhikari, R.; Succi, Sauro (2013), "Isotropic discrete Laplacian operators from lattice hydrodynamics", Journal of Computational Physics, 234: 3–7, arXiv:1202.3299, Bibcode:2013JCoPh.234....1T, doi:10.1016/j.jcp.2012.07.037, S2CID 14633171 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Lynch, Robert (7 January 1992), "Fundamental Solutions of 9-point Discrete Laplacians; Derivation and Tables", Department of Computer Science Technical Reports: 2
 <a href="https://docs.lib.purdue.edu/cstech/929" target="_blank">https://docs.lib.purdue.edu/cstech/929</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Fomel, Sergey; Claerbout, Jon F. (9 October 1997). "Constructing an isotropic Laplacian operator". Exploring three-dimensional implicit wavefield extrapolation with the helix transform. Stanford Exploration Project. <a href="http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html" target="_blank">http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">"Investigation of Isotropic Laplacian Operators by Computer Simulation for A-B Diblock Copolymers" (PDF). IJCSNS International Journal of Computer Science and Network Security. 18 (5): 102–103. May 2018. <a href="http://paper.ijcsns.org/07_book/201805/20180514.pdf" target="_blank">http://paper.ijcsns.org/07_book/201805/20180514.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Provatas, Nikolas; Elder, Ken (2010), Phase-Field Methods in Materials Science and Engineering (PDF), p. 219, doi:10.1002/9783527631520, ISBN 9783527631520
 <a href="9783527631520" target="_blank">9783527631520</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">"Investigation of Isotropic Laplacian Operators by Computer Simulation for A-B Diblock Copolymers" (PDF). IJCSNS International Journal of Computer Science and Network Security. 18 (5): 102–103. May 2018. <a href="http://paper.ijcsns.org/07_book/201805/20180514.pdf" target="_blank">http://paper.ijcsns.org/07_book/201805/20180514.pdf</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Fomel, Sergey; Claerbout, Jon F. (9 October 1997). "Constructing an isotropic Laplacian operator". Exploring three-dimensional implicit wavefield extrapolation with the helix transform. Stanford Exploration Project. <a href="http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html" target="_blank">http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Thampi, Sumesh P.; Ansumali, Santosh; Adhikari, R.; Succi, Sauro (2013), "Isotropic discrete Laplacian operators from lattice hydrodynamics", Journal of Computational Physics, 234: 3–7, arXiv:1202.3299, Bibcode:2013JCoPh.234....1T, doi:10.1016/j.jcp.2012.07.037, S2CID 14633171 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Fomel, Sergey; Claerbout, Jon F. (9 October 1997). "Constructing an isotropic Laplacian operator". Exploring three-dimensional implicit wavefield extrapolation with the helix transform. Stanford Exploration Project. <a href="http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html" target="_blank">http://sepwww.stanford.edu/public/docs/sep95/sergey2/paper_html/node12.html</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">McGinty, Bob (2012). "Coordinate Transforms". continuummechanics.org. <a href="http://www.continuummechanics.org/coordxforms.html" target="_blank">http://www.continuummechanics.org/coordxforms.html</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Witkin, Andrew; Kass, Michael (1991), "Reaction-diffusion textures" (PDF), ACM SIGGRAPH Computer Graphics, 25 (4): 3–4, doi:10.1145/127719.122750 <a href="http://www.cs.cmu.edu/afs/cs.cmu.edu/user/aw/www/pdf/texture.pdf" target="_blank">http://www.cs.cmu.edu/afs/cs.cmu.edu/user/aw/www/pdf/texture.pdf</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Witkin, Andrew; Kass, Michael (1991), "Reaction-diffusion textures" (PDF), ACM SIGGRAPH Computer Graphics, 25 (4): 3–4, doi:10.1145/127719.122750 <a href="http://www.cs.cmu.edu/afs/cs.cmu.edu/user/aw/www/pdf/texture.pdf" target="_blank">http://www.cs.cmu.edu/afs/cs.cmu.edu/user/aw/www/pdf/texture.pdf</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Sanderson, Allen R.; Kirby, Robert M.; Johnson, Chris R.; Yang, Lingfa (2006), "Advanced Reaction-Diffusion Models for Texture Synthesis" (PDF), Journal of Graphics Tools, 11 (3): 4, doi:10.1080/2151237X.2006.10129222, S2CID 13132043 <a href="http://www.sci.utah.edu/~allen/materials/Sanderson_JGT_2006.pdf" target="_blank">http://www.sci.utah.edu/~allen/materials/Sanderson_JGT_2006.pdf</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Garnier, David-Henri; Schmidt, Martin-Pierre; Rohmer, Damien (2022), "Growth of oriented orthotropic structures with reaction/Diffusion", Structural and Multidisciplinary Optimization, 65 (11): 7–8, doi:10.1007/s00158-022-03395-7, S2CID 253304840 <a href="https://hal.science/hal-03842830v1/document" target="_blank">https://hal.science/hal-03842830v1/document</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Witkin, Andrew; Kass, Michael (1991), "Reaction-diffusion textures" (PDF), ACM SIGGRAPH Computer Graphics, 25 (4): 3–4, doi:10.1145/127719.122750 <a href="http://www.cs.cmu.edu/afs/cs.cmu.edu/user/aw/www/pdf/texture.pdf" target="_blank">http://www.cs.cmu.edu/afs/cs.cmu.edu/user/aw/www/pdf/texture.pdf</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
</ol>

Nine-point stencil open-in-new

Nine-point stencil