Eq scan direction connue

Article/notions.tex

@@ -59,15 +59,15 @@ c = min (c_1, c_2)
 \noindent
 where $c_1 = a\cdot x_A + b\cdot y_A$ and $c_2 = a\cdot x_B + b\cdot y_B$.
 
-The scan line $\cal{L}_i(A,B)$ is then defined by :
+The scan line ${\cal L}_i(A,B)$ is then defined by :
 
 \centerline{
-${\cal L}_i(A,B) = {\cal S}(A,B) \cap {\cal D}(a, b, c, \nu)$, with
+${\cal L}_i(A,B) = {\cal S}(A,B) \cap {\cal D}(a', b', c', \nu')$, with
 $\left\{ \begin{array}{l}
-a = y_B - y_A \\
-b = x_A - x_B \\
-\nu = max (|a|,|b|) \\
-c = a\cdot x_A + b\cdot y_A + i \cdot \nu
+a' = y_B - y_A \\
+b' = x_A - x_B \\
+\nu' = max (|a'|,|b'|) \\
+c' = a'\cdot x_A + b'\cdot y_A + i \cdot \nu'
 \end{array} \right.$
 }
 
@@ -76,8 +76,31 @@ is the chessboard distance ($d_\infty = max (|d_x|,|d_y|)$).
 In practice, this difference of length between scan lines is not a drawback,
 as the image bounds should also be processed anyway.
 
-The directional scan can also be defined by its central point $C$, its
-direction $\vec{D}$ and its width $w$. TO BE DEVELOPED.
+The directional scan can also be defined by its central point $C(x_C,y_C)$,
+its direction $\vec{D}(x_D,y_D)$ and its width $w$ :
+
+\centerline{
+${\cal S}(C,\vec{D},w) = {\cal D}(a, b, c, \nu)$, with
+$\left\{ \begin{array}{l}
+a = y_D \\
+b = -x_D \\
+\nu = w \\
+c = a\cdot x_C + b\cdot y_C - w / 2
+\end{array} \right.$
+}
+
+and the scan line ${\cal L}_i(A,B)$ by :
+
+\centerline{
+${\cal L}_i(C,\vec{D},w) =
+{\cal S}(C,\vec{D},w) \cap {\cal D}(a', b', c', \nu')$, with
+$\left\{ \begin{array}{l}
+a' = x_D \\
+b' = y_D \\
+\nu' = max (|a'|,|b'|) \\
+c' = a'\cdot x_C + b'\cdot y_C - w / 2 + i\cdot w
+\end{array} \right.$
+}
 
 \subsection{Adaptive directional scan}