636 lines
24 KiB
Plaintext
636 lines
24 KiB
Plaintext
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How FreeType's rasterizer work
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by David Turner
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Revised 2007-Feb-01
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This file is an attempt to explain the internals of the FreeType
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rasterizer. The rasterizer is of quite general purpose and could
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easily be integrated into other programs.
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I. Introduction
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II. Rendering Technology
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1. Requirements
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2. Profiles and Spans
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a. Sweeping the Shape
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b. Decomposing Outlines into Profiles
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c. The Render Pool
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d. Computing Profiles Extents
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e. Computing Profiles Coordinates
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f. Sweeping and Sorting the Spans
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I. Introduction
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===============
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A rasterizer is a library in charge of converting a vectorial
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representation of a shape into a bitmap. The FreeType rasterizer
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has been originally developed to render the glyphs found in
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TrueType files, made up of segments and second-order Béziers.
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Meanwhile it has been extended to render third-order Bézier curves
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also. This document is an explanation of its design and
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implementation.
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While these explanations start from the basics, a knowledge of
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common rasterization techniques is assumed.
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II. Rendering Technology
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========================
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1. Requirements
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---------------
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We assume that all scaling, rotating, hinting, etc., has been
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already done. The glyph is thus described by a list of points in
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the device space.
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- All point coordinates are in the 26.6 fixed float format. The
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used orientation is:
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^ y
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| reference orientation
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*----> x
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0
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`26.6' means that 26 bits are used for the integer part of a
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value and 6 bits are used for the fractional part.
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Consequently, the `distance' between two neighbouring pixels is
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64 `units' (1 unit = 1/64th of a pixel).
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Note that, for the rasterizer, pixel centers are located at
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integer coordinates. The TrueType bytecode interpreter,
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however, assumes that the lower left edge of a pixel (which is
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taken to be a square with a length of 1 unit) has integer
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coordinates.
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^ y ^ y
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| (1,1) | (0.5,0.5)
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+-----------+ +-----+-----+
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| | | o-----+-----> x
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| | | (0,0) |
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o-----------+-----> x +-----------+
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(0,0) (-0.5,-0.5)
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TrueType bytecode interpreter FreeType rasterizer
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A pixel line in the target bitmap is called a `scanline'.
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- A glyph is usually made of several contours, also called
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`outlines'. A contour is simply a closed curve that delimits an
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outer or inner region of the glyph. It is described by a series
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of successive points of the points table.
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Each point of the glyph has an associated flag that indicates
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whether it is `on' or `off' the curve. Two successive `on'
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points indicate a line segment joining the two points.
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One `off' point amidst two `on' points indicates a second-degree
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(conic) Bézier parametric arc, defined by these three points
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(the `off' point being the control point, and the `on' ones the
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start and end points). Similarly, a third-degree (cubic) Bézier
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curve is described by four points (two `off' control points
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between two `on' points).
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Finally, for second-order curves only, two successive `off'
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points forces the rasterizer to create, during rendering, an
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`on' point amidst them, at their exact middle. This greatly
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facilitates the definition of successive Bézier arcs.
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The parametric form of a second-order Bézier curve is:
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P(t) = (1-t)^2*P1 + 2*t*(1-t)*P2 + t^2*P3
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(P1 and P3 are the end points, P2 the control point.)
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The parametric form of a third-order Bézier curve is:
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P(t) = (1-t)^3*P1 + 3*t*(1-t)^2*P2 + 3*t^2*(1-t)*P3 + t^3*P4
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(P1 and P4 are the end points, P2 and P3 the control points.)
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For both formulae, t is a real number in the range [0..1].
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Note that the rasterizer does not use these formulae directly.
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They exhibit, however, one very useful property of Bézier arcs:
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Each point of the curve is a weighted average of the control
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points.
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As all weights are positive and always sum up to 1, whatever the
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value of t, each arc point lies within the triangle (polygon)
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defined by the arc's three (four) control points.
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In the following, only second-order curves are discussed since
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rasterization of third-order curves is completely identical.
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Here some samples for second-order curves.
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* # on curve
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* off curve
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__---__
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#-__ _-- -_
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--__ _- -
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--__ # \
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--__ #
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-#
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Two `on' points
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Two `on' points and one `off' point
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between them
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*
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# __ Two `on' points with two `off'
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\ - - points between them. The point
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\ / \ marked `0' is the middle of the
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- 0 \ `off' points, and is a `virtual
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-_ _- # on' point where the curve passes.
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-- It does not appear in the point
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* list.
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2. Profiles and Spans
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---------------------
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The following is a basic explanation of the _kind_ of computations
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made by the rasterizer to build a bitmap from a vector
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representation. Note that the actual implementation is slightly
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different, due to performance tuning and other factors.
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However, the following ideas remain in the same category, and are
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more convenient to understand.
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a. Sweeping the Shape
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The best way to fill a shape is to decompose it into a number of
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simple horizontal segments, then turn them on in the target
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bitmap. These segments are called `spans'.
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__---__
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_-- -_
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_- -
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- \
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/ \
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/ \
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__---__ Example: filling a shape
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_----------_ with spans.
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_--------------
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----------------\
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/-----------------\ This is typically done from the top
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/ \ to the bottom of the shape, in a
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| | \ movement called a `sweep'.
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V
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__---__
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_----------_
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_--------------
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----------------\
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/-----------------\
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/-------------------\
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|---------------------\
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In order to draw a span, the rasterizer must compute its
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coordinates, which are simply the x coordinates of the shape's
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contours, taken on the y scanlines.
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/---/ |---| Note that there are usually
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/---/ |---| several spans per scanline.
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| /---/ |---|
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| /---/_______|---| When rendering this shape to the
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V /----------------| current scanline y, we must
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/-----------------| compute the x values of the
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a /----| |---| points a, b, c, and d.
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- - - * * - - - - * * - - y -
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/ / b c| |d
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/---/ |---|
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/---/ |---| And then turn on the spans a-b
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/---/ |---| and c-d.
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/---/_______|---|
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/----------------|
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/-----------------|
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a /----| |---|
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- - - ####### - - - - ##### - - y -
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/ / b c| |d
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b. Decomposing Outlines into Profiles
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For each scanline during the sweep, we need the following
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information:
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o The number of spans on the current scanline, given by the
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number of shape points intersecting the scanline (these are
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the points a, b, c, and d in the above example).
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o The x coordinates of these points.
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x coordinates are computed before the sweep, in a phase called
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`decomposition' which converts the glyph into *profiles*.
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Put it simply, a `profile' is a contour's portion that can only
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be either ascending or descending, i.e., it is monotonic in the
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vertical direction (we also say y-monotonic). There is no such
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thing as a horizontal profile, as we shall see.
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Here are a few examples:
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this square
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1 2
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---->---- is made of two
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| | profiles | |
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^ v ^ + v
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----<----
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up down
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this triangle
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P2 1 2
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|\ is made of two | \
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^ | \ \ | \
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| | \ \ profiles | \ |
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| | \ v ^ | \ |
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| \ | | + \ v
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P1 ---___ \ ---___ \
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---_\ ---_ \
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<--__ P3 up down
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A more general contour can be made of more than two profiles:
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__ ^
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/ | / ___ / |
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/ | / | / | / |
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| | / / => | v / /
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| | | | | | ^ |
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^ | |___| | | ^ + | + | + v
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| | | v | |
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| | | up |
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|___________| | down |
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<-- up down
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Successive profiles are always joined by horizontal segments
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that are not part of the profiles themselves.
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For the rasterizer, a profile is simply an *array* that
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associates one horizontal *pixel* coordinate to each bitmap
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*scanline* crossed by the contour's section containing the
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profile. Note that profiles are *oriented* up or down along the
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glyph's original flow orientation.
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In other graphics libraries, profiles are also called `edges' or
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`edgelists'.
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c. The Render Pool
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FreeType has been designed to be able to run well on _very_
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light systems, including embedded systems with very few memory.
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A render pool will be allocated once; the rasterizer uses this
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pool for all its needs by managing this memory directly in it.
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The algorithms that are used for profile computation make it
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possible to use the pool as a simple growing heap. This means
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that this memory management is actually quite easy and faster
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than any kind of malloc()/free() combination.
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Moreover, we'll see later that the rasterizer is able, when
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dealing with profiles too large and numerous to lie all at once
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in the render pool, to immediately decompose recursively the
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rendering process into independent sub-tasks, each taking less
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memory to be performed (see `sub-banding' below).
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The render pool doesn't need to be large. A 4KByte pool is
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enough for nearly all renditions, though nearly 100% slower than
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a more comfortable 16KByte or 32KByte pool (that was tested with
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complex glyphs at sizes over 500 pixels).
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d. Computing Profiles Extents
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Remember that a profile is an array, associating a _scanline_ to
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the x pixel coordinate of its intersection with a contour.
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Though it's not exactly how the FreeType rasterizer works, it is
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convenient to think that we need a profile's height before
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allocating it in the pool and computing its coordinates.
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The profile's height is the number of scanlines crossed by the
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y-monotonic section of a contour. We thus need to compute these
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sections from the vectorial description. In order to do that,
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we are obliged to compute all (local and global) y extrema of
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the glyph (minima and maxima).
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P2 For instance, this triangle has only
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two y-extrema, which are simply
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| \ P2.y as a vertical maximum
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| \ P3.y as a vertical minimum
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| \
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| \ P1.y is not a vertical extremum (though
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| \ it is a horizontal minimum, which we
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P1 ---___ \ don't need).
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---_\
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P3
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Note that the extrema are expressed in pixel units, not in
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scanlines. The triangle's height is certainly (P3.y-P2.y+1)
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pixel units, but its profiles' heights are computed in
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scanlines. The exact conversion is simple:
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- min scanline = FLOOR ( min y )
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- max scanline = CEILING( max y )
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A problem arises with Bézier Arcs. While a segment is always
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necessarily y-monotonic (i.e., flat, ascending, or descending),
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which makes extrema computations easy, the ascent of an arc can
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vary between its control points.
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P2
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*
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# on curve
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* off curve
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__-x--_
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_-- -_
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P1 _- - A non y-monotonic Bézier arc.
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# \
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- The arc goes from P1 to P3.
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\
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\ P3
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#
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We first need to be able to easily detect non-monotonic arcs,
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according to their control points. I will state here, without
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proof, that the monotony condition can be expressed as:
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P1.y <= P2.y <= P3.y for an ever-ascending arc
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P1.y >= P2.y >= P3.y for an ever-descending arc
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with the special case of
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P1.y = P2.y = P3.y where the arc is said to be `flat'.
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As you can see, these conditions can be very easily tested.
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They are, however, extremely important, as any arc that does not
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satisfy them necessarily contains an extremum.
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Note also that a monotonic arc can contain an extremum too,
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which is then one of its `on' points:
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P1 P2
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#---__ * P1P2P3 is ever-descending, but P1
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-_ is an y-extremum.
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-
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---_ \
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-> \
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\ P3
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#
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Let's go back to our previous example:
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P2
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*
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# on curve
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* off curve
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__-x--_
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_-- -_
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P1 _- - A non-y-monotonic Bézier arc.
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# \
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- Here we have
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\ P2.y >= P1.y &&
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\ P3 P2.y >= P3.y (!)
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#
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We need to compute the vertical maximum of this arc to be able
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to compute a profile's height (the point marked by an `x'). The
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arc's equation indicates that a direct computation is possible,
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but we rely on a different technique, which use will become
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apparent soon.
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Bézier arcs have the special property of being very easily
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decomposed into two sub-arcs, which are themselves Bézier arcs.
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Moreover, it is easy to prove that there is at most one vertical
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extremum on each Bézier arc (for second-degree curves; similar
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conditions can be found for third-order arcs).
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For instance, the following arc P1P2P3 can be decomposed into
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two sub-arcs Q1Q2Q3 and R1R2R3:
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P2
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*
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# on curve
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* off curve
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original Bézier arc P1P2P3.
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__---__
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_-- --_
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_- -_
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- -
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/ \
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/ \
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# #
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P1 P3
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P2
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*
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Q3 Decomposed into two subarcs
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Q2 R2 Q1Q2Q3 and R1R2R3
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* __-#-__ *
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_-- --_
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_- R1 -_ Q1 = P1 R3 = P3
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- - Q2 = (P1+P2)/2 R2 = (P2+P3)/2
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/ \
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/ \ Q3 = R1 = (Q2+R2)/2
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# #
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Q1 R3 Note that Q2, R2, and Q3=R1
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are on a single line which is
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tangent to the curve.
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We have then decomposed a non-y-monotonic Bézier curve into two
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smaller sub-arcs. Note that in the above drawing, both sub-arcs
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are monotonic, and that the extremum is then Q3=R1. However, in
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a more general case, only one sub-arc is guaranteed to be
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monotonic. Getting back to our former example:
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Q2
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*
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__-x--_ R1
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_-- #_
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Q1 _- Q3 - R2
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# \ *
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-
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\
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\ R3
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#
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Here, we see that, though Q1Q2Q3 is still non-monotonic, R1R2R3
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is ever descending: We thus know that it doesn't contain the
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extremum. We can then re-subdivide Q1Q2Q3 into two sub-arcs and
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go on recursively, stopping when we encounter two monotonic
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subarcs, or when the subarcs become simply too small.
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We will finally find the vertical extremum. Note that the
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iterative process of finding an extremum is called `flattening'.
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e. Computing Profiles Coordinates
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Once we have the height of each profile, we are able to allocate
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it in the render pool. The next task is to compute coordinates
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for each scanline.
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In the case of segments, the computation is straightforward,
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using the Euclidean algorithm (also known as Bresenham).
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However, for Bézier arcs, the job is a little more complicated.
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We assume that all Béziers that are part of a profile are the
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result of flattening the curve, which means that they are all
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y-monotonic (ascending or descending, and never flat). We now
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have to compute the intersections of arcs with the profile's
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scanlines. One way is to use a similar scheme to flattening
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called `stepping'.
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Consider this arc, going from P1 to
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--------------------- P3. Suppose that we need to
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compute its intersections with the
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drawn scanlines. As already
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--------------------- mentioned this can be done
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directly, but the involved
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* P2 _---# P3 algorithm is far too slow.
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------------- _-- --
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_-
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_/ Instead, it is still possible to
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---------/----------- use the decomposition property in
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/ the same recursive way, i.e.,
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| subdivide the arc into subarcs
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------|-------------- until these get too small to cross
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| more than one scanline!
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-----|--------------- This is very easily done using a
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| rasterizer-managed stack of
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| subarcs.
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# P1
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f. Sweeping and Sorting the Spans
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Once all our profiles have been computed, we begin the sweep to
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build (and fill) the spans.
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As both the TrueType and Type 1 specifications use the winding
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fill rule (but with opposite directions), we place, on each
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scanline, the present profiles in two separate lists.
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One list, called the `left' one, only contains ascending
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profiles, while the other `right' list contains the descending
|
|
profiles.
|
|
|
|
As each glyph is made of closed curves, a simple geometric
|
|
property ensures that the two lists contain the same number of
|
|
elements.
|
|
|
|
Creating spans is thus straightforward:
|
|
|
|
1. We sort each list in increasing horizontal order.
|
|
|
|
2. We pair each value of the left list with its corresponding
|
|
value in the right list.
|
|
|
|
|
|
/ / | | For example, we have here
|
|
/ / | | four profiles. Two of
|
|
>/ / | | | them are ascending (1 &
|
|
1// / ^ | | | 2 3), while the two others
|
|
// // 3| | | v are descending (2 & 4).
|
|
/ //4 | | | On the given scanline,
|
|
a / /< | | the left list is (1,3),
|
|
- - - *-----* - - - - *---* - - y - and the right one is
|
|
/ / b c| |d (4,2) (sorted).
|
|
|
|
There are then two spans, joining
|
|
1 to 4 (i.e. a-b) and 3 to 2
|
|
(i.e. c-d)!
|
|
|
|
|
|
Sorting doesn't necessarily take much time, as in 99 cases out
|
|
of 100, the lists' order is kept from one scanline to the next.
|
|
We can thus implement it with two simple singly-linked lists,
|
|
sorted by a classic bubble-sort, which takes a minimum amount of
|
|
time when the lists are already sorted.
|
|
|
|
A previous version of the rasterizer used more elaborate
|
|
structures, like arrays to perform `faster' sorting. It turned
|
|
out that this old scheme is not faster than the one described
|
|
above.
|
|
|
|
Once the spans have been `created', we can simply draw them in
|
|
the target bitmap.
|
|
|
|
------------------------------------------------------------------------
|
|
|
|
Copyright 2003, 2007 by
|
|
David Turner, Robert Wilhelm, and Werner Lemberg.
|
|
|
|
This file is part of the FreeType project, and may only be used,
|
|
modified, and distributed under the terms of the FreeType project
|
|
license, LICENSE.TXT. By continuing to use, modify, or distribute this
|
|
file you indicate that you have read the license and understand and
|
|
accept it fully.
|
|
|
|
|
|
--- end of raster.txt ---
|
|
|
|
Local Variables:
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|
coding: utf-8
|
|
End:
|