Editing Mode 7 (section)

==Overview==
{{Unreferenced section|date=May 2023}}
The Super NES console has eight graphics modes, numbered from 0 to 7, for displaying background layers.  The last one (background mode 7) has a single layer that can be scaled and rotated. Two-dimensional affine transformations can produce any combination of [[translation (geometry)|translation]], [[scaling (geometry)|scaling]], [[reflection (mathematics)|reflection]], [[rotation (mathematics)|rotation]], and [[Shear mapping|shearing]]. However, many games create additional effects by setting a different transformation matrix for each scanline.  In this way, pseudo-perspective, curved surface, and distortion effects can be achieved.

Mode 7 graphics are generated for each pixel by mapping screen coordinates to background coordinates using an [[affine transformation]] and sampling the corresponding background color. The 2D affine transformation is specified for each scanline by 6 parameters: <math>
a
</math>, <math>
b
</math>, <math>
c
</math>, and <math>
d
</math> ( which together define the matrix <math>
\mathbf{M}
</math>), and <math>
x_0
</math> and <math>
y_0
</math> (which define the vector <math>
\mathbf{r}_0
</math>, the origin). Specifically, screen coordinate <math>
\mathbf{r}
</math> is translated to the origin coordinate system, the matrix is applied, and the result is translated back to the original coordinate system to obtain <math>
\mathbf{r}^\prime
</math>.

In 2D matrix notation:

<math>
\mathbf{r}^\prime = \mathbf{M} (\mathbf{r} - \mathbf{r}_0) + \mathbf{r}_0
</math>

<math>
\begin{bmatrix} x' \\ y' \end{bmatrix}
 =
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
\left(
\begin{bmatrix} x \\ y \end{bmatrix}
- \begin{bmatrix} x_0 \\ y_0 \end{bmatrix}
\right)
+ \begin{bmatrix} x_0 \\ y_0 \end{bmatrix}
</math>.

All arithmetic is carried out on 16-bit signed fixed point numbers, while all offsets are limited to 13 bits.  The [[radix point]] is between bits 7 and 8.