# 2D trajectory in minimum amount of time given min/max acceleration per axis

I am having a little problem with determining a trajectory.

I have a 2D curve, say $\{x(\lambda), y(\lambda)\}$ where $\lambda$ is not time; it is only a parameter that describes the evolution of the curve.

The point can move with a max acceleration defined for every axis; so we will have a $maxAcc_x$ and a $maxAcc_y$. Acceleration can be chosen independently for every axis. I'd like to know the minimum time required to draw the curve.

• This seems like a difficult problem to write down an expression for. Are you looking for a way to go about calculating a numerical solution? – muaddib Jul 31 '15 at 21:14
• Yes. I need to know the logic behind. I need to find the estimated time needed for a laser-cutting machine in order to cut a defined shape. I know accelerations because they are defined by motors that drive the laser. – Jepessen Jul 31 '15 at 21:17
• I'm interested in the solution when both maximum accelerations are equal – arthur.sw Jan 7 '16 at 20:05