On Tue, 26 Oct 1999, James Henstridge wrote:
> I suppose so. The other option would be to use the other formula for
> drawing a bezier curve. It looks something like:
>
> 3
> ___
> x(t) = \ ( 3 ) * x * t^i * (1-t)^(3-i)
> / ( i ) i
> ---
> i=0
>
> 3
> ___
> y(t) = \ ( 3 ) * y * t^i * (1-t)^(3-i)
> / ( i ) i
> ---
> i=0
>
> (t runs from 0 to 1)
>
> Where the first coefficient is the binomial coefficient (ie. 1 for i = 0
> or 3, 3 for i = 1 or 2). This way you could decide on a fixed number of
> points (say 10 or 20), and calculate them. By precomputing the powers of
> the various values of t, to calculate each point you would only need 24
> multiplications and 6 additions. Also, the process is constant time
> rather than a recursive algorithm. This is potentially not as high a
> quality, but may be sufficient for on screen display.
>
> Another place to look is the algorithms used in libart from gnome-libs (in
> the art_vpath_bpath.c file). It looks like it uses a similar approach to
> dia's renderer, except that it looks at `smoothness' to decide when to
> break out of the recursion rather than spacing of the points.
Dia doesn't look at the point spacing. Consider this bezier-curve:
P1 P2
* *
u /|x \
/ | \
*------------*
P0 y v P3
P0-P3 are the bezier points.
u = P1-P0
v = P3-P0
y = u projected on v
x = u-y;
Then the value of |x| decides if the curve is "smooth" enough to draw as a
line. Then the same is checked on the other |x| (P3-P2 projected on
P0-P3).
/ Alex
