Mode 7 transform

Background mode 7 has the ability to perform an affine transformation on its tilemap.

Instead of rendering normally, this tilemap (effectively a 1024x1024 pixel square) can be given an arbitrary 2D affine transformation, which means this is a square that can:
 * Translate or slide its position up, down, left or right.
 * Rotate at any angle.
 * Zoom in and out or be squashed, but the scaling must be uniform along any axis (i.e. it can stretch along a straight line, but it cannot "bend").
 * Shear or skew.

It is conceptually similar to texture mapping a single quad on a modern GPU. Conversely, it is also like selecting a parallelogram region from the tilemap, and stretching its four corners to the rectangle of the screen.

The affine transformation can be changed every scanline via HDMA, allowing versatile perspective and distortion effects.
 * See: Mode 7 perspective effects

The usual BG1 scrolling registers are replaced with M7HOFS, M7VOFS, which scroll the tilemap after transformation.

The affine transformation is applied by M7A, M7B, M7C, M7D, with an additional pivot-point center offset via M7X, M7Y.

ABCD defines a transformation matrix, which combined with the offset and pivot maps screen pixel coordinats (Sx,Sy) to texel coordinates (Tx,Ty): +-      -+   +-                 -+   +-   -+   +-  -+ | M7A M7B |   | Sx + M7HOFS - M7X |   | M7X |   | Tx | |        | * |                   | + |     | = |    | | M7C M7D |   | Sy + M7VOFS - M7Y |   | M7Y |   | Ty | +-      -+   +-                 -+   +-   -+   +-  -+

Affine Matrix
M7A, M7B, M7C, M7D together define how to map the tilemap "texture" to the screen, as pixels are rasterized left to right, top to bottom. In this explanation a pixel is an output pixel on the screen, and a texel is the color fetched from the 1024x1024 background tilemap. Each of these is an 8.8 fixed point value.

When you move one pixel to the right on the screen:
 * M7A is how many texels to move to the right on the background.
 * M7C is how many texels to move down.

When you move one pixel down on the screen:
 * M7B is how many texels to move right.
 * M7D is how many texels to move down.

In modern computer graphics terms: (M7A,M7C) and (M7B,M7D) are 2D vectors defining Δu and Δv for texture mapping.

This is why the recommended default values of (1,0) (0,1) makes mode 7 behave like a normal background. 1 pixel to the right = 1 texel to the right. 1 pixel down = 1 texel down.

Scaling can be accomplished by changing the length of these vectors.
 * (2,0) (0,1) will move 2 texels right for every 1 pixel, shrinking by 1/2 in the horizontal.
 * (1,0) (0,-0.1) will move 0.1 texels up for every 1 pixel, stretching by 10 in the vertical and flipping upside down.

Rotation can be accomplished by rotating these vectors.
 * (cos ϴ, sin ϴ) (-sin ϴ, cos ϴ) form a standard rotation matrix that will rotate the map by ϴ degrees.
 * On screen: it will rotate the map counter-clockwise relative to the pivot-point.
 * On map: it will rotate the view parallelogram clockwise.


 * (0.866, 0.500) (-0.500, 0.866) will rotate by 30 degrees (CCW on screen, CW on map).

Shearing creates a slanted mapping by adding to one coordinate unevenly.
 * (1,0) (0.2,1) causes the background to gradually slide to the left as it proceeds down the screen.

Scaling, rotation, and shearing together can be combined into a single transformation matrix in A/B/C/D. This can be computed by matrix multiplication.

Center Adjustment
M7X and M7Y define a texel coordinate that becomes the center (a.k.a. pivot-point) of the scaling/rotation applied by the affine transformation matrix ABCD. This allows you to rotate around some other point besides the top left of the map.

M7HOFS and M7VOFS define a starting point for rasterization. After the transformation ABCD/XY is applied, this is a pixel coordinate that shifts the top-left of the screen. E.g. increasing M7HOFS by 1 will have the effect of moving whatever is in view 1 pixel to the left. With the default matrix this is exactly the same as just scrolling the map in other modes.

Summary
On screen:
 * Start with a view of the top left 256x224 pixels of the tilemap.
 * From that view, pick a pivot-point on the map (M7X,M7Y) and rotate and scale around that point (ABCD).
 * Now (M7HOFS,M7VOFS) will move (right,down) in screen-space, scrolling over the transformed map.

On the map:
 * (A,C) and (B,D) are vectors that define the angle and size of a parallelogram that will be the screen's view.
 * The top-left corner of the parallelogram is a more complicated computation involving all 8 register values.
 * Increasing M7HOFS or M7VOFS by 1 will move the top-left corner along the direction of the parallelogram sides equivalent to 1 screen pixel.