# Any royal road to Bezier curves...?

Tom Anderson twic at urchin.earth.li
Mon Nov 21 19:50:51 CET 2005

On Sun, 20 Nov 2005, Warren Francis wrote:

> Basically, I'd like to specify a curved path of an object through space.
> 3D space would be wonderful, but I could jimmy-rig something if I could
> just get 2D...  Are bezier curves really what I want after all?

No. You want a natural cubic spline:

http://mathworld.wolfram.com/CubicSpline.html

This is a fairly simple curve, which can be fitted through a series of
points (called knots) in space of any dimensionality, without the need to
specify extra control points (unlike a Bezier curve), and which has the
nice property of minimising the curvature of the curve - it's the shape
you'd get if you ran a springy wire through your knots. It usually looks
pretty good too.

There are other kinds of splines - Catmull-Rom, B-spline (a generalisation
of a Bezier curve), Hermite - but they mostly don't guarantee to pass
through the knots, which might make them less useful to you.

In the opposite direction on the mathematical rigour scale, there's what i
call the blended quadratic spline, which i invented as a simpler and more
malleable alternative to the cubic spline. It's a piecewise parametric
spline, like the cubic, but rather than calculating a series of pieces
which blend together naturally, using cubics and linear algebra, it uses
then interpolates ('blends') between them to draw the curve. It looks very
like a cubic spline, but the code is simpler, and the pieces are local -
each piece depends only on nearby knots, rather than on all the knots, as
in a cubic spline - which is a useful property for some jobs. Also, it's
straightforward to add the ability to constrain the angle at which the
curve passes through a subset of the knots (you can do it for some knots,
while leaving others 'natural') by promoting the pieces to cubics at the
constrained knots and constraining the appropriate derivatives. Let me
know if you want more details on this. To be honest, i'd suggest using a
proper cubic spline, unless you have specific problems with it.

tom

--
... a tale for which the world is not yet prepared