# Geometry Matrix

## Code

To execute this program, you need the image file `gopher.png`

at the same directory:

The result will be like this:

## How the code works

The program is almost same as render-an-image example. The difference is whether `DrawImageOptions`

is specified or not at the `DrawImage`

call. In this example, a `GeoM`

is adjusted. `GeoM`

represents an affine matrix to translate, scale or rotate the image.

In this example, the image is translated by $(50, 50)$ first, and scaled by $(1.5, 1)$. In Ebiten, Y axis is not upward but downward. Then, tralslating by $(50, 50)$ means that the image is moved by 50 pixels rightward, and by 50 pixels downward. For scaling, the origin point is upper-left and the directions are also downward and rightward.

\begin{equation} \begin{bmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1 \\ \end{bmatrix} \end{equation}

While `GeoM`

is an matrix for 2D dimension, `GeoM`

's dimension is 3. It is because `GeoM`

is an affine matrix, which can represent not only scaling or rorating, but also translating. While scaling and rotating don't change the origin point (upper-left), translating does, and such change cannot be achieved with a non-affine 2-dimension matrix. The last row is always $(0, 0, 1)$.

The geometry matrix is used to determine how to convert the position in the source image to the position in the destination image. In other words, the geometry matrix represents a converting rule of positions. By adjusting the geometry matrix, we can adjust how to render the source image on the destination image.

\begin{equation} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \end{equation}

The initial value of `GeoM`

represents an identity matrix. This means that nothing was changed. In Ebiten, the default location is the upper-left corner.

\begin{equation} \begin{bmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} g \end{equation}

`(g *GeoM).Scale`

left-multiplies a scaling geometry matrix. The origin position is the upper-left corner of the destination image.

\begin{equation} \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \\ \end{bmatrix} g \end{equation}

`(g *GeoM).Translate`

left-multiplies a translating geometry matrix.

\begin{equation} \begin{bmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} g \end{equation}

`(g *GeoM).Rotate`

left-multiplies a rotating geometry matrix. The origin position is the upper-left corner of the destination image.