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I am trying to model in Mathematica the concentration (density) of an intersection of two graphs, eggs[x,y,z,t] and sperm[x,y,z,t]. The issue arises when one tries to implement an equation describing the graph's boundaries (sperm[x,y,z,t] == eggs[x,y,z,t]) into either DensityPlot3D or say the RegionFunction -> Function[{x,y,z,t},func] specifier. The methodology works for ContourPlot3D, however, DensityPlot3D is much more suitable for the task, as I am trying to animate the solutions and have the color of the graph be the concentration variable (4D output of sperm and egg equations).

Here is my code for the ContourPlot3D process:

In[1]:= sperm[x_,y_,z_,t_]= 1/Sqrt[4*Pi*t]*E^((-x^2-y^2-z^2)/(4*t));
eggs[x_,y_,z_,t_]=1/Sqrt[4*Pi*t]*E^((-(x+3)^2-(y+3)^2-(z+3)^2)/(4*t));
Animate[ContourPlot3D[sperm[x, y, z, t] == eggs[x, y, z, t], {x, -10, 10}, {y, -10, 10}, {z, -10, 10}], {t, 0, 10}]

The output normally crashes the system and thus cannot be uploaded here.

Here is my attempt to apply this method to the DensityPlot3D:

In[2]:= Animate[DensityPlot3D[sperm[x, y, z, t] == eggs[x, y, z, t], {x, -10, 10}, {y, -10, 10}, {z, -10, 10}], {t, 0, 10}]

The output is only a blank 3D plot.

After researching, I had found that RegionFunction (and RegionPlot3D) only seems to use inequalities, and the first argument in DensityPlot3D uses a function that is not in the form of an equation. I had tried to bypass this by doing something like

f = Solve[sperm[x, y, z, t] == eggs[x, y, z, t], {x, y, z, t}, Reals]

and inputting that variable as the first argument, but this did not help, as the result was still considered to be an equation (a complicated one at that).

So, my question is: How can I implement the equation as a boundary for DensityPlot3D?

EDIT: So it turns out the first argument can actually take in equations, as I was able to achieve an animation for s[x, y, t] == e[x, y, t].

Here is the code:

In[1]:= s[x_, y_, t_] = 1/Sqrt[4*Pi*t]*E^((-(x + 0)^2 - (y + 0)^2)/(4*t));
e[x_, y_, t_] = 1/Sqrt[4*Pi*t]*E^((-(x + 3)^2 - (y + 3)^2)/(4*t)); 
Animate[DensityPlot[s[x, y, t] == e[x, y, t], {x, -10, 10}, {y, -10, 10}, ColorFunction -> "TemperatureMap"], {t, 0, 10}]

I cannot figure out how to save the animation on here, and the picture is not uploading here.

Nevertheless, the problem then is simply the following: Why is DensityPlot3D a blank graph for the "4D" case?

Bobby Sanchez
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  • for starters work the problem at a specific `t` before messing with `Animate` – agentp Nov 11 '17 at 13:38
  • @agentp Of course. Theoretically, one can animate any process given a manipulated variable, and in my case the problem exists with the actual interpretation of the first argument, _f_. So, my question applies to the "simplest" case with say `t = 2`: `DensityPlot3D[sperm[x, y, z, 2] == eggs[x y, z, 2], {x, -10, 10}, {y, -10 ,10}, {z, -10 ,10}]`. I just used `Animate` to elucidate what exactly I am trying to do, as many a time responses don't answer the question, but rather produce the desired result with a different workaround *because* the questioner provided context of their goal. Apologies. – Bobby Sanchez Nov 11 '17 at 18:12

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