It is probably easiest to put these shadings on a background layer. The following shades the complements of the nodes. Below that I'll show how shade the segments of the nodes:
\documentclass{article}
\usepackage{tikz,etoolbox}
\usetikzlibrary{matrix,calc,shapes,decorations}
\pgfdeclarelayer{background}
\pgfdeclarelayer{main}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\begin{document}
\begin{tikzpicture}[scale=2]
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-1.5,0);
\coordinate (XAxisMax) at (4,0);
\coordinate (YAxisMin) at (0,-0.5);
\coordinate (YAxisMax) at (0,3.3);
\begin{pgfonlayer}{foreground}
\draw [thin] (XAxisMin) -- (XAxisMax);% Draw x axis
\draw [thin] (YAxisMin) -- (YAxisMax);% Draw y axis
\end{pgfonlayer}
\clip (-0.7,-0.7) rectangle (4.18cm,3.3cm); % Clips the picture...
\begin{pgfonlayer}{background}
\clip (-0.7,-0.7) rectangle (4.18cm,3.3cm); % Clips the picture...
\end{pgfonlayer}
\begin{pgfonlayer}{foreground}
\clip (-0.7,-0.7) rectangle (4.18cm,3.3cm); % Clips the picture...
\draw[blue] (-0.25,0.25) node {$\mathbf{A}$};
\draw[blue] (0.35,0.25) node {$\mathbf{B}$};
\draw[blue] (-0.3,-0.25) node {$\mathbf{C}$};
\draw[blue] (0.25,-0.25) node {$\mathbf{D}$};
\end{pgfonlayer}
\pgftransformcm{1}{0.3}{0.7}{1}{\pgfpoint{0cm}{0cm}}
\coordinate (Bone) at (0,2);
\coordinate (Btwo) at (2,-2);
\coordinate (negBonePlusBtwo) at (-2,0);
\begin{pgfonlayer}{foreground}
\draw[style=help lines,dashed] (-14,-14) grid[step=2cm] (14,14);
\end{pgfonlayer}
\begin{pgfonlayer}{background}
\foreach \x in {-2,-1,...,3}{
\foreach \y in {-2,-1,...,4}{
\pgfmathparse{int(3*(\x+\y+4))}
\fill[red!\pgfmathresult] (2*\x,2*\y) -- +(2,0) -- +(2,2) -- +(0,2) --cycle;
}
}
\fill[red!40] (0,0) -- +(2,0) -- +(2,2) -- +(0,2) --cycle;
\end{pgfonlayer}
\foreach \x in {-1,0,...,4}{
\foreach \y in {-1,0,...,4}{
\node[star,scale=5,draw=black,fill=white] at (2 * \x, 2 * \y) {};
% make a lattice of stars
}
}
\begin{pgfonlayer}{foreground}
\draw[very thick,draw=black] (Origin) rectangle ($2*(Bone)+(Btwo)$);
\draw [ultra thick,-latex,red] (Origin) -- (Bone) node [above left]
{$b_1$};
\draw[blue] (Bone) node [below right] {$\mathbf{D} '$}; \draw[blue]
($(Bone)+(Btwo)$) node [above left] {$\mathbf{A} '$}; \draw[blue]
($1.95*(Bone)+0.96*(Btwo)$) node [below left] {$\mathbf{C} '$};
\draw [ultra thick,-latex,red] (Origin) -- ($(Bone)+(Btwo)$) node
[below right] {$b_2$};
\draw [thick,dashed,-latex,red,scale=3] (Origin) -- ($-0.25*(Bone)$)
; \draw [thick,dashed,-latex,red] (Origin) --
($0.35*(negBonePlusBtwo)$) ;
\end{pgfonlayer}
\end{tikzpicture}
\end{document}
The layers need to be declared outside of the picture and need to include a layer main. Also you have to provide the clip specification for each layer. In your case it is easiest before you transform the coordinates, hence the multiple clip commands early in the code. Placing the nodes on the main layer, allows one to fill them white covering the background, but still lying behind the foreground.
Now for shading the segments of the nodes instead, we can just use the anchors provided by the shape star to fill individual regions:
\documentclass{article}
\usepackage{tikz,etoolbox}
\usetikzlibrary{matrix,calc,shapes,decorations}
\pgfdeclarelayer{background}
\pgfdeclarelayer{main}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}
\begin{document}
\begin{tikzpicture}[scale=2]
\coordinate (Origin) at (0,0);
\coordinate (XAxisMin) at (-1.5,0);
\coordinate (XAxisMax) at (4,0);
\coordinate (YAxisMin) at (0,-0.5);
\coordinate (YAxisMax) at (0,3.3);
\begin{pgfonlayer}{foreground}
\draw [thin] (XAxisMin) -- (XAxisMax);% Draw x axis
\draw [thin] (YAxisMin) -- (YAxisMax);% Draw y axis
\end{pgfonlayer}
\clip (-0.7,-0.7) rectangle (4.18cm,3.3cm); % Clips the picture...
\begin{pgfonlayer}{background}
\clip (-0.7,-0.7) rectangle (4.18cm,3.3cm);
\end{pgfonlayer}
\begin{pgfonlayer}{foreground}
\clip (-0.7,-0.7) rectangle (4.18cm,3.3cm);
\draw[blue] (-0.2,0.2) node {$\mathbf{A}$};
\draw[blue] (0.28,0.2) node {$\mathbf{B}$};
\draw[blue] (-0.28,-0.21) node {$\mathbf{C}$};
\draw[blue] (0.2,-0.2) node {$\mathbf{D}$};
\end{pgfonlayer}
\pgftransformcm{1}{0.3}{0.7}{1}{\pgfpoint{0cm}{0cm}}
\coordinate (Bone) at (0,2);
\coordinate (Btwo) at (2,-2);
\coordinate (negBonePlusBtwo) at (-2,0);
\begin{pgfonlayer}{foreground}
\draw[style=help lines,dashed] (-14,-14) grid[step=2cm] (14,14);
\end{pgfonlayer}
\foreach \x in {-1,0,...,4}{
\foreach \y in {-1,0,...,4}{
\node[star,scale=5,draw=black] (x) at (2 * \x, 2 * \y) {};
\begin{pgfonlayer}{background}
\fill[green] (x.outer point 3) -- (x.center) -- (x.outer
point 5) -- (x.inner point 4) -- (x.outer point 4) --
(x.inner point 3) -- cycle;
\fill[yellow] (x.inner point 2) -- (x.center) -- (x.outer
point 3);
\fill[green!30] (x.inner point 5) -- (x.center) -- (x.inner
point 2) -- (x.outer point 2) -- (x.inner point 1) --
(x.outer point 1) -- cycle;
\fill[yellow!30] (x.outer point 5) -- (x.center) -- (x.inner
point 5) -- cycle;
\end{pgfonlayer}
}
}
\begin{pgfonlayer}{foreground}
\draw[very thick,draw=black] (Origin) rectangle ($2*(Bone)+(Btwo)$);
\draw [ultra thick,-latex,red] (Origin) -- (Bone) node [above left]
{$b_1$};
\draw[blue] (Bone) node [below right] {$\mathbf{D} '$}; \draw[blue]
($(Bone)+(Btwo)$) node [above left] {$\mathbf{A} '$}; \draw[blue]
($1.95*(Bone)+0.96*(Btwo)$) node [below left] {$\mathbf{C} '$};
\draw [ultra thick,-latex,red] (Origin) -- ($(Bone)+(Btwo)$) node
[below right] {$b_2$};
\draw [thick,dashed,-latex,red,scale=3] (Origin) -- ($-0.25*(Bone)$)
; \draw [thick,dashed,-latex,red] (Origin) --
($0.35*(negBonePlusBtwo)$) ;
\end{pgfonlayer}
\end{tikzpicture}
\end{document}
The anchors are labelled (x.inner point n) and (x.outer point n) with n running from 1 to 5.
In general, you lattice dirrections may not coincide nicely with anchors in your shape. In such cases, you can recreate the boundary path of the shape and use that to clip. In your case the node loop would be:
\foreach \x in {-1,0,...,4}{
\foreach \y in {-1,0,...,4}{
\coordinate (x) at (2 * \x, 2 * \y);
\node[star,scale=5,draw=black] (n) at (x) {};
\begin{pgfonlayer}{background}
\begin{scope}
\path[clip] (n.outer point 1) -- (n.inner point 1) --
(n.outer point 2) -- (n.inner point 2) --
(n.outer point 3) -- (n.inner point 3) --
(n.outer point 4) -- (n.inner point 4) --
(n.outer point 5) -- (n.inner point 5) -- cycle;
\fill[green] (x) -- +(1,0) -- +(1,-1) -- +(0,-1) -- cycle;
\fill[yellow] (x) -- +(0,-1) -- +(-1,-1) -- +(-1,0) -- cycle;
\fill[green!30] (x) -- +(-1,0) -- +(-1,1) -- +(0,1) -- cycle;
\fill[yellow!30] (x) -- +(0,1) -- +(1,1) -- +(1,0) -- cycle;
\end{scope}
\end{pgfonlayer}
}
}
which produces the same result as the previous picture.
The techniques at Clipping a path using a node might enable you to just use the node to do the clipping...
\clip (Origin) rectangle (Bone -| Btwo);\node[star,scale=5,draw=black,fill=yellow] at (0,2) {};you can get an idea how to proceed I hope. So you can first fill the stars and then place the vectors. Also in question http://tex.stackexchange.com/questions/42564/ we didn't have anything else to draw but here you might constrain the effect of the transformation matrix for other additions – percusse May 29 '13 at 09:06