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How FreeType's rasterizer work by David Turner Revised 2007-Feb-01 This file is an attempt to explain the internals of the FreeType rasterizer. The rasterizer is of quite general purpose and could easily be integrated into other programs. I. Introduction II. Rendering Technology 1. Requirements 2. Profiles and Spans a. Sweeping the Shape b. Decomposing Outlines into Profiles c. The Render Pool d. Computing Profiles Extents e. Computing Profiles Coordinates f. Sweeping and Sorting the Spans I. Introduction =============== A rasterizer is a library in charge of converting a vectorial representation of a shape into a bitmap. The FreeType rasterizer has been originally developed to render the glyphs found in TrueType files, made up of segments and second-order Béziers. Meanwhile it has been extended to render third-order Bézier curves also. This document is an explanation of its design and implementation. While these explanations start from the basics, a knowledge of common rasterization techniques is assumed. II. Rendering Technology ======================== 1. Requirements --------------- We assume that all scaling, rotating, hinting, etc., has been already done. The glyph is thus described by a list of points in the device space. - All point coordinates are in the 26.6 fixed float format. The used orientation is: ^ y | reference orientation | *----> x 0 `26.6' means that 26 bits are used for the integer part of a value and 6 bits are used for the fractional part. Consequently, the `distance' between two neighbouring pixels is 64 `units' (1 unit = 1/64th of a pixel). Note that, for the rasterizer, pixel centers are located at integer coordinates. The TrueType bytecode interpreter, however, assumes that the lower left edge of a pixel (which is taken to be a square with a length of 1 unit) has integer coordinates. ^ y ^ y | | | (1,1) | (0.5,0.5) +-----------+ +-----+-----+ | | | | | | | | | | | | | o-----+-----> x | | | (0,0) | | | | | o-----------+-----> x +-----------+ (0,0) (-0.5,-0.5) TrueType bytecode interpreter FreeType rasterizer A pixel line in the target bitmap is called a `scanline'. - A glyph is usually made of several contours, also called `outlines'. A contour is simply a closed curve that delimits an outer or inner region of the glyph. It is described by a series of successive points of the points table. Each point of the glyph has an associated flag that indicates whether it is `on' or `off' the curve. Two successive `on' points indicate a line segment joining the two points. One `off' point amidst two `on' points indicates a second-degree (conic) Bézier parametric arc, defined by these three points (the `off' point being the control point, and the `on' ones the start and end points). Similarly, a third-degree (cubic) Bézier curve is described by four points (two `off' control points between two `on' points). Finally, for second-order curves only, two successive `off' points forces the rasterizer to create, during rendering, an `on' point amidst them, at their exact middle. This greatly facilitates the definition of successive Bézier arcs. The parametric form of a second-order Bézier curve is: P(t) = (1-t)^2*P1 + 2*t*(1-t)*P2 + t^2*P3 (P1 and P3 are the end points, P2 the control point.) The parametric form of a third-order Bézier curve is: P(t) = (1-t)^3*P1 + 3*t*(1-t)^2*P2 + 3*t^2*(1-t)*P3 + t^3*P4 (P1 and P4 are the end points, P2 and P3 the control points.) For both formulae, t is a real number in the range [0..1]. Note that the rasterizer does not use these formulae directly. They exhibit, however, one very useful property of Bézier arcs: Each point of the curve is a weighted average of the control points. As all weights are positive and always sum up to 1, whatever the value of t, each arc point lies within the triangle (polygon) defined by the arc's three (four) control points. In the following, only second-order curves are discussed since rasterization of third-order curves is completely identical. Here some samples for second-order curves. * # on curve * off curve __---__ #-__ _-- -_ --__ _- - --__ # \ --__ # -# Two `on' points Two `on' points and one `off' point between them * # __ Two `on' points with two `off' \ - - points between them. The point \ / \ marked `0' is the middle of the - 0 \ `off' points, and is a `virtual -_ _- # on' point where the curve passes. -- It does not appear in the point * list. 2. Profiles and Spans --------------------- The following is a basic explanation of the _kind_ of computations made by the rasterizer to build a bitmap from a vector representation. Note that the actual implementation is slightly different, due to performance tuning and other factors. However, the following ideas remain in the same category, and are more convenient to understand. a. Sweeping the Shape The best way to fill a shape is to decompose it into a number of simple horizontal segments, then turn them on in the target bitmap. These segments are called `spans'. __---__ _-- -_ _- - - \ / \ / \ | \ __---__ Example: filling a shape _----------_ with spans. _-------------- ----------------\ /-----------------\ This is typically done from the top / \ to the bottom of the shape, in a | | \ movement called a `sweep'. V __---__ _----------_ _-------------- ----------------\ /-----------------\ /-------------------\ |---------------------\ In order to draw a span, the rasterizer must compute its coordinates, which are simply the x coordinates of the shape's contours, taken on the y scanlines. /---/ |---| Note that there are usually /---/ |---| several spans per scanline. | /---/ |---| | /---/_______|---| When rendering this shape to the V /----------------| current scanline y, we must /-----------------| compute the x values of the