I have a water droplet which is falling down through its gravity. I've supposed that my droplet is an ellipsoid and I want to find the equation that represents this ellipsoid in XYZ coordinate system. Therefore, I put two perpendicular cameras, one in XY plane and the other in YZ plane to obtain two projection views of this droplet. Then, I wrote corresponding equations that relate these two projected views to the 3D equation of my droplet. Then, I solved these equations, but they returned infinity number of answers (for 3D droplet body). So, I added another equation for droplet volume and solved altogether again. Although, the volume equation remarkably confined the number of feasible answers, it could not unique the answer and returns two possible answers for the droplet as shown in below figure (i.e. both of the proposed droplets have the same projection views on XY and YZ planes AND have the same volume as well)
Fig. 1 Two feasible answers from top view (XZ plane) that we don't have any camera on it. Although these two possible solutions have the same projection views on XY and YZ planes, they are different from top view and this can help us in recognition of the correct answer.
However, only one of these droplets (red or blue) is really my falling droplet (that the cameras have captured its frames). Therefore, I need one more piece of information to be able to distinguish the right one. My question is that which equation OR condition I can use (like the one I did for droplet volume) in order to be able to separate my desired answer among two existing possibilities?
I am using two high speed cameras @ 4500 fps and since I am investigating a collision phenomena, it really needs this high amount of frame rate (or even more). A 2 mm spherical solid particle has been shot by a launcher and collides to my 2 mm falling droplet. Right before colliding, the droplet undergoes deformations because of some forces applied to it through the particle, particle launcher flow, etc. I have a third camera, but it is 400 fps and also it should be "synchronized" with others that is another issue as well. What I mean by asking this question is to find a "condition" OR another "related equation" to give me the capability to distinguish between these two possible answers (like what I've added for droplet volume). I have also thought about putting a mirror behind the droplet with a specific angle such that I can capture two different views of the droplet simultaneously with one camera OR putting an appropriate array of some laser emitters (or any other kind of pointer) on top plane to mark the droplet in both frames (at least in one point) so that I can use marked point(s) to identify the answer. However, I am still seeking to choose the most convenient and applicable method. All of your helpful ideas are really appreciated. Below is a schematic of my experimental setup.
Fig. 2 Experimental setup from top view.