footile

A 2D vector graphics library written in Rust. It uses pix for the underlying raster images.

See the documentation for examples and API usage.

Rasterizing: Bird's Eye View

There is nothing novel here — this is merely a guide to the code.

We have a 2D path made up of lines, bézier splines, arcs, etc., and we want to make a high-quality raster image out of it. But how?

Modules

• path: Defines path operations and Path2Ds
• plotter: Defines Plotter struct and flattens curves
• stroker: Creates stroked paths for plotter
• fixed: Defines Fixed struct used by fig module
• fig: Rasterizes paths

Curve Flattening

Flattening refers to approximating a curve with a series of line segments.

We use the recursive algorithm described by De Casteljau. There might be opportunities for optimization here if we ever determine this is a bottleneck. One other thing to note: this method could cause a stack overflow with the wrong input data.

Once complete, we have a series of line segments forming one or more closed polygons.

Sorting Vertices

For the next step, we create a sorted list of the vertices in (Y, X) order. This is needed because we will scan the polygon onto a grid one row at a time.

Every path has a winding order: either clockwise or the other direction, typically called counter- or anti-clockwise. Let's avoid that debate by calling it widdershins, since clocks rarely go backwards.

The first vertex must be on the outside of the path, so we can check the angle between its neighbors to determine the winding order.

Active Edges

As rows are scanned from top to bottom, we keep track of a list of active edges. If an edge crosses the current row, it is added to the list, otherwise, it is removed. Since horizontal edges cannot cross a row, they can safely be ignored.

For each row, vertices from the list are compared to its top and bottom. If the new vertex is above the bottom, one or more edges are added. When the new vertex is above the top, existing edges are removed.

           v0
           /\
          /  \ (a)
         /    \      v2
    (b) /      +-----+
       /      v1      \
      /                \ (c)
     /                  \
    +--------------------+
     v3                  v4

Example:

• Starting with v0, add edges (a) and (b)
• Scan until the bottom of the current row is below v1 / v2
• Add edge (c)
• Scan until the row top is below v1
• Remove edge (a)
• Scan until the row top is below v3 / v4
• Remove edges (b) and (c)

Signed Area

The active edges are used to find the signed area at any point. Count the number of edge crossings needed to reach the exterior of the path. For example:

    +------+
    |      |
    |  +1  |
    |      |
    +------+

A self-intersecting polygon looks like this:

    +-----------------+
    |                 |
    |      +------+   |
    |      |       \ /
    |  +1  |  +2    X
    |      |       / \
    |      +------+   |
    |                 |
    +-----------------+

What about sub-paths with opposite winding order? In that case, subtract instead of adding:

    +----------------+
    |      +-----+   |
    |      |     |   |
    |  +1  |  0  |   |
    |      |     |   |
    |      +-----+   |
    |        <--     |
    |                |
    +----------------+
          -->

Scanning Rows

When scanning a row, the signed area is sampled for each pixel. The direction of each edge from top to bottom determines whether it adds to or subtracts from the area. In the normal winding order, it adds to the area; otherwise, it subtracts.

This row is 4 pixels wide:

      -  -  - | -  -  -   -  -  - | -  -  -
    |         |         |         |         |
              | +1                | -1
    |         |         |         |         |
              |                   |      
    | -  -  - | -  -  - | -  -  - | -  -  - |
          0         1         1         0

The cumulative sum of these values is the signed area of each pixel.

Anti-Aliasing

Sometimes edges don't fall on pixel boundaries. In this case, the trick is to use fractional numbers for anti-aliasing.

      -  -  - | -  |  -   -  -  -   -  -  -
    |         |    |    |         |         |
              | +1 | -½   -½
    |         |    |    |         |         |
              |    |                     
    | -  -  - | -  |  - | -  -  - | -  -  - |
          0         ½         0         0

Notice how the remainder of the second edge coverage is added to the pixel to the right (third pixel). This is necessary to keep the cumulative sum correct.

      -  -  - | -  \  -   -  -  -   -  -  -
    |         |     \   |         |         |
              | +1   \-¼  -¾
    |         |       \ |         |         |
              |        \
    | -  -  - | -  -  - \ -  -  - | -  -  - |
          0         ¾         0         0

Compositing

The signed area buffer can be composited with a raster, using a source color.