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MathGroup Archive 2005

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Re: 3D Plot of frequency data

  • To: mathgroup at smc.vnet.net
  • Subject: [mg62723] Re: 3D Plot of frequency data
  • From: dh <dh at metrohm.ch>
  • Date: Fri, 2 Dec 2005 05:53:35 -0500 (EST)
  • References: <dmjrcm$5ud$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Leigh,
a linear transformation that transforms
v1 = {0, 0, 0};
v2 = {1, 0, 0};
v3 = {0, 1, 0};
v4 = {0, 0, 1};
into a regular tetrahedron centered at {0,0,0}:
w1 = {0, 0, Sqrt[2/3]};
w2 = {1/2, -1/Sqrt[12], 0};
w3 = {-1/2, -1/Sqrt[12], 0};
w4 = {0, 1/Sqrt[3], 0};

is:

lintrans[v_]:=Module[{
   mat={{1/2, -1/2, 0}, {-1/(2*Sqrt[3]),
   -1/(2*Sqrt[3]), 1/Sqrt[3]},
  {-Sqrt[2/3], -Sqrt[2/3], -Sqrt[2/3]}},
   tr={0, 0, Sqrt[2/3]}
   },
  mat.v+tr
]

Concerning the surface, the implicite equation can be made explicite by 
solving for x3 (for x1 and x2 !=0):
x3= (x1 - x1^2 - x1*x2)/(x1 + x2)

the parametrized (by x1,x2) surface in old coordinates are(note that 
x1=x2= 0 is excluded):
{x1, x2, (x1 - x1^2 - x1*x2)/(x1 + x2)}
and in new coordinates:
lintrans[ {x1, x2, (x1 - x1^2 - x1*x2)/(x1 + x2)} ]
we can plot this by:

g1=ParametricPlot3D[lintrans[ {x1, x2, (x1 - x1^2 - x1*x2)/(x1 + x2)} 
],{x1, 0.001, 1}, {x2, 0.001, 1}]

To display the tetrahedron by a wire frame, we change the definition of 
"thedrahedron":
tetrahedron[{v1_, v2_, v3_, v4_}] := {Line[{v1, v2, v3, v1}], Line[{v1,
     v2, v4, v1}], Line[{v1, v4, v3, v1}], Line[{v4, v2, v3, v1}]};
the tetraedron canthen be plotted by:
g2 = Graphics3D[{Thickness[0.01],
         tetrahedron[lintrans /@ {{0, 0, 0}, {1, 0,
         0} , {0, 1, 0}, {0, 0, 1}}]
         }];

and we may combine the plots by:
Show[g1, g2, PlotRange -> {0, 1}]

have fun, Daniel


leigh pascoe wrote:
> Dear Mathgroup,
> 
> I would like to do a 3D plot of several functions where the coordinates 
> (x, y,z) are subject to the restrictions
> 
> x>=0
> y>=0
> z>=0
> x+y+z<=1
> 
> These restrictions arise naturally when considering the frequencies of 
> four types in a population. The fourth frequency, subject to similar 
> restrictions, is just 1-x-y-z.
> 
> I believe this range of possible values defines a tetrahedron with 
> vertices (0,0,0),(1,0,0), (0,1,0),(0,0,1) that could be drawn using the 
> code recently posted by David Park. i.e.
> 
> v1 = {0, 0, 0};
> v2 = {1, 0, 0};
> v3 = {0, 1, 0};
> v4 = {0, 0, 1};
> 
> tetrahedron[{v1_, v2_, v3_, v4_}] := {Polygon[{v1, v2, v3}], Polygon[{v1, v2, v4}], Polygon[{v1, v4, v3}], Polygon[{v4, v2, v3}]};
> 
> Show[Graphics3D[tetrahedron[{v1, v2, v3, v4}]]];
> 
> If possible (for symmetry) I would like to apply a shear function (say 
> to the y and z directions) so that the resulting tetrahedron would be 
> regular. A point (x1,x2,x3) could then be plotted by measuring along 
> each of the sides of the tetrahedron corresponding to the original axes 
> and finding the intersection point of the perpendiculars. I imagine a 
> transformation can be defined to convert the (x1,x2,x3) or (x',y',z') 
> coordinates to the orthogonal (x,y,z) system if this is necessary.
> 
> Now to the functions. As  first step I would like to plot within this 
> regular tetrahedron the implicit function
> 
> x2 x3-x1(1-x1-x2-x3)=0
> 
> This is a twisted surface that would have edges corresponding to the axes
> x'=y'=0 (z' axis) and
> x'=z'=0 (y' axis)
> and the tetrahedral sides defined by
> 1-x'-y'-z'=0=y' and
> 1-x'-y'-z'=0=z'
> 
> For easy visualisation the tetrahedron should be a wireframe and the 
> function surface should be shown as a grid mesh. I would then like to 
> add some individual points, both on  and off the surface and ultimately 
> some trajectories (lines) defined by a series of points. The ability to 
> visualise points and functions in this manner would be very useful when 
> considering problems related to frequency data. Can anyone help me to do 
> this in Mathematica?
> 
> Thanks
> 
> Leigh
> 
> 
> 
> 


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