graphics - Obtaining a 3D animation as a drop in a liquid surface

Someone could get a similar solution like this animation?

I believe the solution is useful for demonstrations in class to other users who teach Math.

Spreading of a Thin Liquid Drop Under the Influence of Gravity, Rotation and Non-Uniform Surface Tension

Answer

Thanks to J.M.

drop = SetAlphaChannel[#, ColorNegate@#] &@
  Binarize@Rasterize@
    ParametricPlot[{r Cos[t] (1 - Sin[t]), -3 + 
       r (5/2 (Sin[t] - 1) + 3)}, {t, 0, 2 Pi}, {r, 0, 1}, 
     BoundaryStyle -> None, Axes -> False, Frame -> False]

Something to start with:

circle = Table[
  Translate[
    Point[{##, 0} & @@@ CirclePoints[r, 10 + 20 r]], 

    {0, 0, Dynamic[f[#, t]] &@r}
  ], 
  {r, .5, 20, .5}
];

f[r_, t_] := UnitBox[(r - t)/(2 Pi) - .5] Sin[r - t]; 
t = -4 Pi; 

Column@{
  Trigger[Dynamic[t], {-4 Pi, 20}] ,

  Graphics3D[
   {AbsolutePointSize@2, circle,
    Dynamic @ If[t < -1.9 Pi, 
      Inset[drop, {0, 0, -(t + 2 Pi)}, Automatic, Scaled[{.03, .05}]], 
      {}
    ]
    },
   ViewVertical -> {0, 0, 1}, ImageSize -> 700, 
   PlotRange -> {20 {-1, 1}, 20 {-1, 1}, 10 {-1, 1}}, 
   ViewAngle -> Pi/16, Boxed -> False, ViewPoint -> {5, 0, 3}, 

   BoxRatios -> Automatic]}

Comments

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

Blog

Search This Blog