I have 2 ellipsoids:

Ax^2/a^2+Ay^2/b^2+Az^2/c^2=1; (*a,b,c>0 constants*)
Ellipsoid A rotates around axis [wax;way;waz] (unit vector) with an infinite speed;
Ellipsoid B rotates around axis [wbx;wby;wbz] (unit vector) with an infinite speed;
[Dx;Dy;Dz] is the vector from ellipsoid A center to ellipsoid B center;
The velocity of ellipsoid A center is [VAx;VAy;VAz];
The velocity of ellipsoid B center is [VBx;VBy;VBz];

How can I find the collision time?


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Interesting question, however it is better suited for math (or physics) stackexchanges. It also needs a rewrite with LaTeX. Here is an incomplete answer.

From what you stated it seems the two ellipsoids are similar. Let's say $A>B>0$ so ellipsoid $B$ is inside $A$ at the start. Also assume $a\ge b\ge c >0$. Note that the vector from center of $A$ to center of $B$ is $D=(V_B-V_A)t$.

Let's first simplify the problem and say the ellipsoids spin randomly as opposed to spin around fixed axes. Then as soon as $|D|$ becomes equal to sum of longest axes of the ellipsoids they must intersect (if they have not intersected already), that is $|D|=a/\sqrt A + a/\sqrt B$. This gives an overly simplified indication for the collision time, $t_c \le (a/\sqrt A + a/\sqrt B)/|V_B-V_A|$.

Perhaps you meant something like this

Two ellipsoids at time $t=0$ are at $(x-A_1)^2/a^2+(y-B_1)^2/b^2+(z-C_1)^2/c^2=1$ and $(x-A_2)^2/a^2+(y-B_2)^2/b^2+(z-C_2)^2/c^2=1$. They start to spin at a very high rate, $\omega=\infty$, around certain axes through their centers $(L_1,M_1,N_1)$ and $(L_2,M_2,N_2)$, while at the same time their centers move uniformly towards the origin at speeds $S_1$ and $S_2$ . I want an estimate for the time to collision (if there is one).

  • $\begingroup$ what is sqrt(A)? $\endgroup$ – gy ab May 22 '17 at 15:38
  • $\begingroup$ Reading your equation of ellipsoid I though it is $A x^2/a^2 +.. =1$. Now I realize you meant $x_A^2 /a^2 +..=1$. In that case the two ellipsoids are identical and collide at $t=0$ already. Perhaps you meant two ellipsoids, as in $x^2/a^2+y^2/b^2+z^2/c^2=1$ for $A$ and $x^2/d^2+y^2/e^2+z^2/f^2=1$ for $B$. If so, then more the reason to rewrite the question carefully. $\endgroup$ – Maesumi May 22 '17 at 16:18
  • $\begingroup$ the 2 ellipsoids are identical, but they are located far away from each other and translate with VA and VB toward each other, their rotation angle can be any angle that gives the minimal time of collision, so you can assume an infinite rotation speed. exactly. I am not a professional mathematician, rather a fan of math. I am working on this question for 3 weeks, unsuccessfully. so any help will be blessed. $\endgroup$ – gy ab May 22 '17 at 16:31
  • $\begingroup$ I suggest simplifying the problem by considering spheres, or 2d, circles, no-spin, spining around $x$, etc and solving it. Then gradually building up the additional complications. It can become quite complicated. Then post the developed version of it to the mathematics exchange. Once the equations are posed clearly then perhaps Mathematica can be used to solve them. A rephrasing is suggested above. I assumed the centers move toward the origin. $\endgroup$ – Maesumi May 22 '17 at 17:26

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