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<TITLE>Bezier Splines</TITLE>
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<H1 ALIGN=Center>
Bezier Splines
</H1>
<P>
<IMG SRC="splines.gif" WIDTH="172" HEIGHT="179" ALIGN="Left">Technically
PostScript uses cubic Bezier splines. Each control point determines the slope
of the spline at the corresponding end point. TrueType uses quadratic Bezier
splines, in these there is only one control point between two end points
and that point determines the slope of the spline at both end points.
<P>
It is also possible to have higher degree Bezier splines. For a quartic spline
there would be 3 control points, for a quintic 4 control points, etc.
<P>
In general if there are n+1 points labeled P<SUB>0</SUB>, P<SUB>1</SUB>,
... P<SUB>n</SUB>, with P<SUB>0</SUB> and P<SUB>n</SUB> the end points (and
all the others control points) then the equation of the Bezier spline between
them is: <IMG SRC="bezier.gif" WIDTH="137" HEIGHT="56" ALIGN="Middle">. If
there are two points then this is just the line between the two end points,
if three then the quadratic spline used by TrueType, if four then the cubic
spline used by PostScript.
<P>
A cubic Bezier curve may be viewed as:
<BLOCKQUOTE>
x = a<SUB>x</SUB>*t<SUP>3</SUP> + b<SUB>x</SUB>*t<SUP>2</SUP> +
c<SUB>x</SUB>*t +d<SUB>x</SUB><BR>
y = a<SUB>y</SUB>*t<SUP>3</SUP> + b<SUB>y</SUB>*t<SUP>2</SUP> +
c<SUB>y</SUB>*t +d<SUB>y</SUB>
</BLOCKQUOTE>
<P>
Where
<TABLE CELLPADDING="2">
<TR>
<TD> </TD>
<TD>d<SUB>x </SUB>= P0.x</TD>
<TD>d<SUB>y</SUB> = P0.y</TD>
</TR>
<TR>
<TD></TD>
<TD>c<SUB>x</SUB> = 3*P1.x-3*P0.x</TD>
<TD>c<SUB>y</SUB> = 3*P1.y-3*P0.y</TD>
</TR>
<TR>
<TD></TD>
<TD>b<SUB>x</SUB> = 3*P2.x-6*P1.x+3*P0.x</TD>
<TD>b<SUB>y</SUB> = 3*P2.y-6*P1.y+3*P0.y</TD>
</TR>
<TR>
<TD></TD>
<TD>a<SUB>x</SUB> = P3.x-3*P2.x+3*P1.x-P0.x</TD>
<TD>a<SUB>y</SUB> = P3.y-3*P2.y+3*P1.y-P0.y</TD>
</TR>
</TABLE>
<P>
And a quadratic Bezier curve:
<TABLE CELLPADDING="2">
<TR>
<TD> </TD>
<TD>d<SUB>x </SUB>= P0.x</TD>
<TD>d<SUB>y</SUB> = P0.y</TD>
</TR>
<TR>
<TD></TD>
<TD>c<SUB>x</SUB> = 2*P1.x-2*P0.x</TD>
<TD>c<SUB>y</SUB> = 2*P1.y-2*P0.y</TD>
</TR>
<TR>
<TD></TD>
<TD>b<SUB>x</SUB> = P2.x-2*P1.x+P0.x</TD>
<TD>b<SUB>y</SUB> = P2.y-2*P1.y+P0.y</TD>
</TR>
</TABLE>
<H2>
Converting TrueType to PostScript
</H2>
<P>
Any quadratic spline can be expressed as a cubic (where the cubic term is
zero). The end points of the cubic will be the same as the quadratic's.
<BLOCKQUOTE>
CP<SUB>0</SUB> = QP<SUB>0</SUB><BR>
CP<SUB>3</SUB> = QP<SUB>2</SUB>
</BLOCKQUOTE>
<P>
The two control points for the cubic are:
<BLOCKQUOTE>
CP<SUB>1</SUB> = QP<SUB>0</SUB> + 2/3
*(QP<SUB>1</SUB>-QP<SUB>0</SUB>)<BR>
CP<SUB>2</SUB> = CP<SUB>1</SUB> + 1/3 *(QP<SUB>2</SUB>-QP<SUB>0</SUB>)
</BLOCKQUOTE>
<P>
So converting from TrueType to PostScript is trivial. There is a slight error
introduced due to rounding, but it is unlikely to be noticeable.
<H2>
Converting Postscript to TrueType
</H2>
<P>
Most cubic splines cannot be represented exactly by a quadratic (or even
by a series of quadratics). The best that can be done is to approximate the
cubic to within some margin of error. Here is a way to do that:
<P>
Start from the end of the spline and every so often (ie. within the margin
of error) check to see if the one permissible quadratic approximation actually
matches the cubic closely enough (the one quadratic has its end points as
the end points of the interval, and its control point is determined by the
intersections of the lines tangent to the cubic at the start and end of the
interval).
<P>
If this approximation works then keep it as part of the result, and continue
the process by advancing our start point along to the cubic spline to the
end of the quadratic we just created.
<P>
(There are some slight complexities introduced because there may not be a
quadratic approximation at a given point (if the tangents happen to be parallel)
or because the approximation happens to be linear, but these are easily dealt
with).
<P>
It may, of course, happen that the "cubic" we are given is actually a quadratic
(if it's third degree term is 0), the most likely cause is that the font
came from a truetype source. In that case the control point for the quadratic
is at:
<BLOCKQUOTE>
QP<SUB>1</SUB> = CP<SUB>0</SUB> + 3/2 * (CP<SUB>1</SUB> - CP<SUB>0</SUB>)
</BLOCKQUOTE>
<H2>
Open Type, another solution
</H2>
<P>
Adobe and Microsoft decided to produce one font format which could hold either
a true type font or a postscript font. This is called Open Type. It is
essentially a superset of TrueType. Any TrueType font is a valid Open Type
font, but Open Type fonts can also contain postscript. Anything that supports
Open Type will not require converting between PostScript and True Type.
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