If I have a parametric cubic curve in 3 dimensions defined by:
$$\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$$ where $t\in [0; 1]$, how can I determine the co-ordinates of the smallest 3D cuboid $\vec P_1$ and $\vec P_2$ that entirely contain the curve over this domain?
To my understanding, each dimension is entirely independent, and so my idea is to do the following, but I am not sure if it is entirely correct/if there is a better way. For each dimension:
- Find the derivative and set to 0 to determine any local maxima and minima in the domain
- Determine the co-ordinate of the endpoints of the domain
- Determine the minimum and maximum of all the values obtained above
- Use these two values (min and max) as the scalar component for that dimension of $\vec P_1$ and $\vec P_2$ respectively
Thereby obtaining a set of minimums and maximums defining the two opposite corners of a cuboid.
Would this work?