The purpose of this section is to present the way FreeType manages
vectorial outlines, as well as the most common operations that can be
applied on them.
1. FreeType outline description and structure
a. Outline curve decomposition
An outline is described as a series of closed contours in the 2D
plane. Each contour is made of a series of line segments and
Bézier arcs. Depending on the file format, these can be
secondorder or thirdorder polynomials. The former are also called
quadratic or conic arcs, and they are used in the TrueType format.
The latter are called cubic arcs and are mostly used in the
Type 1 format.
Each arc is described through a series of start, end, and control
points. Each point of the outline has a specific tag which indicates
whether it is used to describe a line segment or an arc. The tags can
take the following values:
FT_CURVE_TAG_ON

Used when the point is "on" the curve. This corresponds to
start and end points of segments and arcs. The other tags specify
what is called an "off" point, i.e. a point which isn't located on
the contour itself, but serves as a control point for a
Bézier arc.

FT_CURVE_TAG_CONIC

Used for an "off" point used to control a conic Bézier
arc.

FT_CURVE_TAG_CUBIC

Used for an "off" point used to control a cubic Bézier
arc.

Use the FT_CURVE_TAG(tag) macro to filter out other,
internally used flags.
The following rules are applied to decompose the contour's points
into segments and arcs:

Two successive "on" points indicate a line segment joining them.

One conic "off" point amidst two "on" points indicates a conic
Bézier arc, the "off" point being the control point, and
the "on" ones the start and end points.

Two successive cubic "off" points amidst two "on" points indicate
a cubic Bézier arc. There must be exactly two cubic
control points and two "on" points for each cubic arc (using a
single cubic "off" point between two "on" points is forbidden, for
example).

Two successive conic "off" points forces the rasterizer to create
(during the scanline conversion process exclusively) a virtual
"on" point amidst them, at their exact middle. This greatly
facilitates the definition of successive conic Bézier arcs.
Moreover, it is the way outlines are described in the TrueType
specification.

The last point in a contour uses the first as an end point to
create a closed contour. For example, if the last two points of a
contour were an "on" point followed by a conic "off" point, the
first point in the contour would be used as final point to create
an "on" – "off" – "on" sequence as described above.

The first point in a contour can be a conic "off" point itself; in
that case, use the last point of the contour as the contour's
starting point. If the last point is a conic "off" point itself,
start the contour with the virtual "on" point between the last and
first point of the contour.
Note that it is possible to mix conic and cubic arcs in a single
contour, even though no current font driver produces such
outlines.
b. Outline descriptor
A FreeType outline is described through a simple structure:
FT_Outline
n_points

the number of points in the outline

n_contours

the number of contours in the outline

points

array of point coordinates

contours

array of contour end indices

tags

array of point flags

Here, points is a pointer to an array of
FT_Vector records, used to store the vectorial coordinates of
each outline point. These are expressed in 1/64th of a pixel, which
is also known as the 26.6 fixed float format.
contours is an array of point indices used to delimit
contours in the outline. For example, the first contour always starts
at point 0, and ends at point contours[0]. The second
contour starts at point contours[0]+1 and ends at
contours[1], etc. To traverse these points in a callback
based manner, use FT_Outline_Decompose().
Note that each contour is closed, and that n_points should
be equal to contours[n_contours1]+1 for a valid outline.
Finally, tags is an array of bytes, used to store each
outline point's tag.
2. Bounding and control box computations
A bounding box (also called bbox) is simply a
rectangle that completely encloses the shape of a given outline. The
interesting case is the smallest bounding box possible, and in the
following we subsume this under the term "bounding box". Because of the
way arcs are defined, Bézier control points are not necessarily
contained within an outline's (smallest) bounding box.
This situation happens when one Bézier arc is, for example,
the upper edge of an outline and an "off" point happens to be above the
bbox. However, it is very rare in the case of character outlines
because most font designers and creation tools always place "on" points
at the extrema of each curved edges, as it makes hinting much
easier.
We thus define the control box (also called cbox)
as the smallest possible rectangle that encloses all points of a given
outline (including its "off" points). Clearly, it always includes the
bbox, and equates it in most cases.
Unlike the bbox, the cbox is much faster to compute.
Control and bounding boxes can be computed automatically through the
functions FT_Outline_Get_CBox() and
FT_Outline_Get_BBox(). The former function is always very
fast, while the latter may be slow in the case of "outside"
control points (as it needs to find the extreme of conic and cubic arcs
for "perfect" computations). If this isn't the case, it is as fast as
computing the control box.
Note also that even though most glyph outlines have equal cbox and
bbox to ease hinting, this is not necessary the case anymore when a
transformation like rotation is applied to them.
3. Coordinates, scaling and gridfitting
An outline point's vectorial coordinates are expressed in the
26.6 format, i.e. in 1/64th of a pixel, hence the coordinates
(1.0,2.5) is stored as the integer pair (x:64,y:192).
After a master glyph outline is scaled from the EM grid to the
current character dimensions, the hinter or gridfitter is in charge of
aligning important outline points (mainly edge delimiters) to the pixel
grid. Even though this process is much too complex to be described in a
few lines, its purpose is mainly to round point positions, while trying
to preserve important properties like widths, stems, etc.
The following operations can be used to round vectorial distances in
the 26.6 format to the grid:
round( x ) == ( x + 32 ) & 64
floor( x ) == x & 64
ceiling( x ) == ( x + 63 ) & 64
Once a glyph outline is gridfitted or transformed, it often is
interesting to compute the glyph image's pixel dimensions before
rendering it. To do so, one has to consider the following:
The scanline converter draws all the pixels whose centers
fall inside the glyph shape. It can also detect dropouts,
i.e. discontinuities coming from extremely thin shape fragments, in
order to draw the "missing" pixels. These new pixels are always located
at a distance less than half of a pixel but it is not easy to predict
where they will appear before rendering.
This leads to the following computations:

compute the bbox

gridfit the bounding box with the following:
xmin = floor( bbox.xMin )
xmax = ceiling( bbox.xMax )
ymin = floor( bbox.yMin )
ymax = ceiling( bbox.yMax )

return pixel dimensions, i.e.
width = (xmax  xmin)/64
and
height = (ymax  ymin)/64
By gridfitting the bounding box, it is guaranteed that all the pixel
centers that are to be drawn, including those coming from dropout
control, will be within the adjusted box. Then the box's
dimensions in pixels can be computed.
Note also that, when translating a gridfitted outline, one should
always use integer distances to move an outline in the 2D
plane. Otherwise, glyph edges won't be aligned on the pixel grid
anymore, and the hinter's work will be lost, producing very low
quality bitmaps and pixmaps.