# Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve.

Sorry if my terminology is a bit off, most of my maths is self-taught and I am acutely aware I might be asking a question that reveals some fundamental misunderstandings. The reason I want a Bézier curve is that they can be rendered efficiently on modern graphics hardware.

As an example of what I am looking for, I might have a system of differential equations with respect to time:

\begin{align*}\frac{dx}{dt} &= -2x\\ \frac{dy}{dt} &= -2y\\ \frac{dz}{dt} &= -2z\end{align*}

with initial conditions $x=y=z=1$ at $t=0$

Solved with a time-step method (in practice the systems may not be algebraically solvable) for small increments of $t$ and plotted in 3D space the resulting line traces a circular path around the origin. If I wanted to render this on a computer screen, I could collect a series of points and play 'connect the dots' to draw a crude line which would require recalculation with every adjustment to level of detail. If I could approximate the path as a Bézier curve I would only need to calculate it once to get a line which is smooth at any level of detail. The curve will always be plotted in three dimensions, but the system may be higher dimensional.

Is there any way to generate such an approximation algebraically? Should I be looking at generating some points numerically and just fitting a simpler curve to them step by step? Any recommendations on an approach for that?

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You'll be interested in this paper. On the other hand, what I've seen more frequently in practical work is the use of splines instead of Bézier curves for approximating solutions. – J. M. Nov 10 '11 at 14:44
Thanks, the technique this paper describes is exactly what I want and also confirms that I still have a lot of reading to do. :-) – strange loop Nov 11 '11 at 8:35
This is too old to migrate, but you might want to delete and repost on Computational Science – robjohn Jan 23 '13 at 1:12