Combine Plot Graphics

yukterez · August 28, 2013, 5:09pm

I have one Graphics3D and one ParametricPlot3D, which I need to combine. Unfortunately, the Show[] Command does not work here, maybe because I use a Manipulate[] Command. The Prolog-> also does not work.

Plot1, the Graphics 3D, is a set of 3 Points (the initial positions of Plot2):

(* Plot 1 *)
P1 = {0, 0, 0}; P2 = {-1, 0, 2}; P3 = {-3, 4, -2};
Plot1 = Graphics3D[Point[{P1, P2, P3}],
PlotRange -> {{-5, 5}, {-5, 5}, {-3, 7}}, AspectRatio -> 1];

Plot 2, the ParametricPlot3D, is a 3D simulation of the three-body-problem. I need to have the Plot1 plotted right into Plot2:

(Edit: I dont know if the Code gets lost due conversion to plain text. If so, see .nb File at http://yukterez.ist.org/3kp/3k3D.nb

(* Plot 2 *)
G = 667384/
   10^16;(* Gravitationskonstante *)
M1 = 100000000; M2 = 70000000; \
M3 = 50000000;(* Massen 1,2,3 in kg *)
v1x = 1/70; v1y = -1/130; v1z =
  1/20;(* v0 von M1 in m/sek *)
v2x = -1/150; v2y = 1/100; v2z = 
 1/90;(* v0 von M2 in m/sek *)
v3x = 1/200; v3y = 
 1/110; v3z = -1/170;(* v0 von M3 in m/sek *)

P1 = {0, 0, 0}; (* Position 1 XYZ *)

P2 = {-1, 0, 2}; (* Position 2 XYZ *)

P3 = {-3, 4, -2}; (* Position 3 XYZ *)

(* Container *)
Funktion[{{x10_, y10_, z10_}, {x20_, y20_, z20_}, {x30_, y30_, 
    z30_}}, {{vx10_, vy10_, vz10_}, {vx20_, vy20_, vz20_}, {vx30_, 
    vy30_, vz30_}}, {m1_, m2_, m3_}, T_, plotType : ("x" | "v"), 
  plotOptions___] := Module[{nds, Tmax, prolog, funcToPlot},
   
   (* Differentialgleichung *)nds = NDSolve[{
      
      x1'[t] == vx1[t], y1'[t] == vy1[t], z1'[t] == vz1[t],
      
      x2'[t] == vx2[t], y2'[t] == vy2[t], z2'[t] == vz2[t],
      
      x3'[t] == vx3[t], y3'[t] == vy3[t], z3'[t] == vz3[t],
      
      
      vx1'[
        t] == -((G  m2 (x1[t] - x2[t]))/
         Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
             z2[t])^2)^3]) - (G  m3 (x1[t] - x3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3],
      
      
       vy1'[
        t] == -((G  m2 (y1[t] - y2[t]))/
         Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
             z2[t])^2)^3]) - (G m3 (y1[t] - y3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3],
      
      
       vz1'[
        t] == -((G m2 (z1[t] - z2[t]))/
         Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
             z2[t])^2)^3]) - (G m3 (z1[t] - z3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3],
      
      
      vx2'[t] == (G m1  (x1[t] - x2[t]))/
        Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
            z2[t])^2)^3] - (G  m3 (x2[t] - x3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vy2'[t] == (G m1  (y1[t] - y2[t]))/
        Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
            z2[t])^2)^3] - (G  m3 (y2[t] - y3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vz2'[t] == (G m1  (z1[t] - z2[t]))/
        Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
            z2[t])^2)^3] - (G m3 (z2[t] - z3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vx3'[t] == (G m1  (x1[t] - x3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3] + (G m2  (x2[t] - x3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vy3'[t] == (G m1  (y1[t] - y3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3] + (G m2  (y2[t] - y3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vz3'[t] == (G m1  (z1[t] - z3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3] + (G m2  (z2[t] - z3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
      x1[0] == x10, y1[0] == y10, z1[0] == z10,
      
      x2[0] == x20, y2[0] == y20, z2[0] == z20,
      
      x3[0] == x30, y3[0] == y30, z3[0] == z30,
      
      vx1[0] == vx10, vy1[0] == vy10, vz1[0] == vz10,
      
      vx2[0] == vx20, vy2[0] == vy20, vz2[0] == vz20,
      
      vx3[0] == vx30, vy3[0] == vy30, vz3[0] == vz30},
     
     {x1, x2, x3, y1, y2, y3, z1, z2, z3, vx1, vx2, vx3, vy1, vy2, 
      vy3, vz1, vz2, vz3}, {t, 0, T}];
   
   If[Head[nds] =!= NDSolve, Tmax = nds[[1, 1, 2, 1, 1, 2]];
    
    funcToPlot = 
     If[plotType === 
        "x", {{x1[t], y1[t], z1[t]}, {x2[t], y2[t], z2[t]}, {x3[t], 
         y3[t], z3[t]}}, {{vx1[t], vy1[t], vz1[t]}, {vx2[t], vy2[t], 
         vz2[t]}, {vx3[t], vy3[t], vz3[t]}}] /. nds[[1]];
    
    (* Plot Specifications *)
    
    prolog = {PointSize[0.3], Point[{P1, P2, P3}]};
    
    ParametricPlot3D[Evaluate[funcToPlot], {t, 0, Tmax}, 
     PlotStyle -> {Red, Blue, Green},
     
     (* Plot Range und Seitenverhältnis *)
     
     PlotRange -> {{-5, 5}, {-5, 5}, {-3, 7}}, AspectRatio -> 1, 
     (* Prolog -> prolog, *) MaxRecursion -> ControlActive[3, 9], 
     Axes -> True, plotOptions], Text["Yukterez Mod. of demonstrations.wolfram.com/PlanarThreeBodyProblem"]]] // Quiet

Plot2 = Manipulate[Funktion[
   (* Positionen xy *){P1, P2, P3},
   (* Geschwindigkeiten xy *){{v1x, v1y, v1z}, {v2x, v2y, v2z}, {v3x, 
     v3y, v3z}},
   (* Massen *){M1, M2, M3},
   (* Regler *)T, xv,
   ImageSize -> {320, 320}],
  {{xv, "x", "Position/Geschwindigkeit"},
   {"x" -> "Position", "v" -> "Geschwindigkeit"}},
  {{T, 1, "Zeit"}, 1, 200}, ControlPlacement -> {Bottom}]

(* http://yukterez.ist.org/3kp/3k3D.html *)

arnoudb · August 28, 2013, 9:31pm

I am working directly off your notebook (which is different from the pasted code):

Import["http://yukterez.ist.org/3kp/3k3D.nb","NotebookObject"]

I think you need to do something like:

Manipulate[
 Show[{
  Plot1,
  Funktion[..]
 }],
 ...]

And also remove the PlotRange from Plot1 since it won’t work well with the large values of your points (for example Au = 149597870690, which won’t fit in the given plot range).

yukterez · August 28, 2013, 9:57pm

Thanks a lot! I updated the .nb File to simulate sun, earth and moon to test the physics of the code. The Three body simulation now runs with the code

(* Yukterez 2013, Mathematica Syntax *)

Au = 149597870690; (* sun earth distance in m *)
Mu = 384400000; (* moon earth distance in m *)
mSol = 1988*10^26; (* sun mass in kg *)
mE = 5972*10^21; (* earth mass in kg *)
vE = Sqrt[G*mSol/Au]; (* orbital velocitvy earth to sun in m/sek *)
mM = 7346*10^18; (* moon mass in kg *)
vM1 = Sqrt[G*mSol/(Au + Mu)]; (* v orbit moon to sun in m/sek *)
vM2 = Sqrt[G*mE/Mu]; (* v orbit moon to earth in m/sek *)

G = 667384/10^16;(* gravitational constant *)
M1 = mSol; M2 = mE; M3 = mM;(* mass 1,2,3 in kg *)
v1x = 20; v1y = 210; v1z = 90;(* v0 of M1 in m/sek *)
v2x = 20 + vE; v2y = 210; v2z = 90;(* v0 of M2 in m/sek *)
v3x = 20 + vM1; v3y = 210; v3z = 90 + vM2;(* v0 of M3 in m/sek *)
P1 = {0, 0, 0}; (* Position 1 XYZ *)
P2 = {0, Au, 0}; (* Position 2 XYZ *)
P3 = {0, Au + Mu, 0}; (* Position 3 XYZ *)

Plot1 = Graphics3D[Point[{P1, P2, P3}], 
   PlotRange -> {{-1.5 Au, 1.5 Au}, {-1.5 Au, 1.5 Au}, {-1.5 Au, 
      1.5 Au}}, AspectRatio -> 1];

(* Container *)
Funktion[{{x10_, y10_, z10_}, {x20_, y20_, z20_}, {x30_, y30_, 
    z30_}}, {{vx10_, vy10_, vz10_}, {vx20_, vy20_, vz20_}, {vx30_, 
    vy30_, vz30_}}, {m1_, m2_, m3_}, T_, plotType : ("x" | "v"), 
  plotOptions___] := Module[{nds, Tmax, prolog, funcToPlot},
   
   (* Differential Equation *)
   
   nds = NDSolve[{
      
      x1'[t] == vx1[t], y1'[t] == vy1[t], z1'[t] == vz1[t],
      
      x2'[t] == vx2[t], y2'[t] == vy2[t], z2'[t] == vz2[t],
      
      x3'[t] == vx3[t], y3'[t] == vy3[t], z3'[t] == vz3[t],
      
      
      vx1'[
        t] == -((G  m2 (x1[t] - x2[t]))/
         Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
             z2[t])^2)^3]) - (G  m3 (x1[t] - x3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3],
      
      
       vy1'[
        t] == -((G  m2 (y1[t] - y2[t]))/
         Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
             z2[t])^2)^3]) - (G m3 (y1[t] - y3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3],
      
      
       vz1'[
        t] == -((G m2 (z1[t] - z2[t]))/
         Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
             z2[t])^2)^3]) - (G m3 (z1[t] - z3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3],
      
      
      vx2'[t] == (G m1  (x1[t] - x2[t]))/
        Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
            z2[t])^2)^3] - (G  m3 (x2[t] - x3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vy2'[t] == (G m1  (y1[t] - y2[t]))/
        Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
            z2[t])^2)^3] - (G  m3 (y2[t] - y3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vz2'[t] == (G m1  (z1[t] - z2[t]))/
        Sqrt[((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2 + (z1[t] - 
            z2[t])^2)^3] - (G m3 (z2[t] - z3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vx3'[t] == (G m1  (x1[t] - x3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3] + (G m2  (x2[t] - x3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vy3'[t] == (G m1  (y1[t] - y3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3] + (G m2  (y2[t] - y3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
       vz3'[t] == (G m1  (z1[t] - z3[t]))/
        Sqrt[((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2 + (z1[t] - 
            z3[t])^2)^3] + (G m2  (z2[t] - z3[t]))/
        Sqrt[((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2 + (z2[t] - 
            z3[t])^2)^3],
      
      
      x1[0] == x10, y1[0] == y10, z1[0] == z10,
      
      x2[0] == x20, y2[0] == y20, z2[0] == z20,
      
      x3[0] == x30, y3[0] == y30, z3[0] == z30,
      
      vx1[0] == vx10, vy1[0] == vy10, vz1[0] == vz10,
      
      vx2[0] == vx20, vy2[0] == vy20, vz2[0] == vz20,
      
      vx3[0] == vx30, vy3[0] == vy30, vz3[0] == vz30},
     
     {x1, x2, x3, y1, y2, y3, z1, z2, z3, vx1, vx2, vx3, vy1, vy2, 
      vy3, vz1, vz2, vz3}, {t, 0, T}];
   
   If[Head[nds] =!= NDSolve, Tmax = nds[[1, 1, 2, 1, 1, 2]];
    
    funcToPlot = 
     If[plotType === 
        "x", {{x1[t], y1[t], z1[t]}, {x2[t], y2[t], z2[t]}, {x3[t], 
         y3[t], z3[t]}}, {{vx1[t], vy1[t], vz1[t]}, {vx2[t], vy2[t], 
         vz2[t]}, {vx3[t], vy3[t], vz3[t]}}] /. nds[[1]];
    
    (* Plot Specifications *)
    prolog = {PointSize[0.3], Point[{P1, P2, P3}]};
    
    ParametricPlot3D[Evaluate[funcToPlot], {t, 0, Tmax}, 
     PlotStyle -> {Red, Blue, Green},
     
     (* Plot Range and Aspect Ratio *)
     PlotRange -> {{-1.5 Au, 1.5 Au}, {-1.5 Au, 1.5 Au}, {-1.5 Au, 
        1.5 Au}}, AspectRatio -> 1,(* Prolog->prolog *)
     MaxRecursion -> ControlActive[3, 9], Axes -> True, plotOptions], 
    Text["Yukterez Mod."]]] // Quiet

Plot2 = Manipulate[Show[Funktion[
    (* positions xyz *){P1, P2, P3},
    (* velocities xyz *){{v1x, v1y, v1z}, {v2x, v2y, v2z}, {v3x,
       v3y, v3z}},
    (* mass 1,2,3 *){M1, M2, M3},
    T, xv,
    ImageSize -> {320, 320}], Plot1],
  {{xv, "x", "Position/Velocity"},
   {"x" -> "Position", "v" -> "Velocity"}},

(* Interval *)
  {{T, 1000, "Time"}, 1000, 10^9}, ControlPlacement -> {Bottom}]

(* by Simon Tyran, Yukterez 2013. Inspired by Michael Trott *)

mathma18 · September 19, 2013, 8:10pm

May be you ran the Newton’s eqilibrium equation formulation including inverse square law already to verify angular momentum conservation, the ellipse orbit etc.. Suggest taking mass of earth = 1, radius =1 , or better in astronomical units and adjust constants like G accordingly to obtain reasonable values of distances and times.

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Combine Plot Graphics

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