[Tex/LaTex] Interpolate grid data to obtain 3d smooth surface

pgfplots

I use this code

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}

\begin{document}

\pgfplotsset{
ytick={0,0.2,...,1},
xtick={0,0.2,...,1},
xlabel={$x$},
ylabel={$y$},
zlabel={$z$},
xmajorgrids=true,
ymajorgrids=true,
major grid style={dashed},
}

\begin{tikzpicture}
    \begin{axis}[view/h=45,xmin=0,xmax=1,ymin=0,ymax=1,colorbar horizontal, xlabel style={sloped}, ylabel style={sloped}, colorbar style={xlabel=$z$, xlabel style={at={(axis description cs:0.5,-1)}}}]
        \addplot3[surf,shader=faceted interp] table [col sep=comma] {scatter.csv};
    \end{axis}
\end{tikzpicture}

\end{document}

with this scatter.csv

0.000000,0.000000,50
0.000000,0.020110,2
0.000000,0.021307,0
0.000000,0.250000,5
0.000000,0.333333,3
0.000000,0.500000,22
0.000000,1.000000,1

0.000300,0.000000,2
0.000300,0.020110,0
0.000300,0.021307,0
0.000300,0.250000,0
0.000300,0.333333,0
0.000300,0.500000,0
0.000300,1.000000,0

0.001908,0.000000,2
0.001908,0.020110,0
0.001908,0.021307,0
0.001908,0.250000,0
0.001908,0.333333,0
0.001908,0.500000,0
0.001908,1.000000,0

0.003472,0.000000,2
0.003472,0.020110,0
0.003472,0.021307,0
0.003472,0.250000,0
0.003472,0.333333,0
0.003472,0.500000,0
0.003472,1.000000,0

0.010204,0.000000,3
0.010204,0.020110,0
0.010204,0.021307,0
0.010204,0.250000,0
0.010204,0.333333,0
0.010204,0.500000,0
0.010204,1.000000,0

0.028409,0.000000,2
0.028409,0.020110,0
0.028409,0.021307,0
0.028409,0.250000,0
0.028409,0.333333,0
0.028409,0.500000,0
0.028409,1.000000,0

0.035714,0.000000,0
0.035714,0.020110,0
0.035714,0.021307,2
0.035714,0.250000,0
0.035714,0.333333,0
0.035714,0.500000,0
0.035714,1.000000,0

0.045455,0.000000,1
0.045455,0.020110,0
0.045455,0.021307,0
0.045455,0.250000,0
0.045455,0.333333,0
0.045455,0.500000,2
0.045455,1.000000,0

0.083154,0.000000,0
0.083154,0.020110,0
0.083154,0.021307,0
0.083154,0.250000,0
0.083154,0.333333,0
0.083154,0.500000,2
0.083154,1.000000,0

0.092784,0.000000,0
0.092784,0.020110,0
0.092784,0.021307,0
0.092784,0.250000,0
0.092784,0.333333,0
0.092784,0.500000,2
0.092784,1.000000,0

0.119578,0.000000,2
0.119578,0.020110,0
0.119578,0.021307,0
0.119578,0.250000,0
0.119578,0.333333,0
0.119578,0.500000,0
0.119578,1.000000,0

1.000000,0.000000,1
1.000000,0.020110,0
1.000000,0.021307,0
1.000000,0.250000,0
1.000000,0.333333,0
1.000000,0.500000,4
1.000000,1.000000,68

I obtain the following 3D plot

enter image description here

which is correct, but ugly. I know this results from how my data points are distributed (not uniformly). How can I obtain a smooth surface?

Best Answer

The builtin capabilities of pgfplots allow to visualize a data matrix as either triangular surface patches or bilinear surface patches (the latter using patch type=bilinear).

Given a suitable data input, it also allows biquadratic and bicubic (i.e. smooth) patches. However, that data input is considerably more complicated. A brief overview of this method is shown in Creating Bezier surfaces using procedural graphics. Note, however, that this method does not work using your input table; the data format would need to be enrichment and reorganization.

In order to improve the quality of your picture, you have two choices: (a) to increase the sampling density of your plot and stick with piecewise linear/piecewise bilinear shading or (b) to keep the sampling density, but increase the order of each patch to biquadratic or bicubic by means of some advanced external tool and import that it into pgfplots.

If the question is: "how can I get a smooth surface while keeping the input data", the answer is: pgfplots offers at most patch type=bilinear (combined with \usepgfplotslibrary{patchplots}):

enter image description here

\documentclass{standalone}
\usepackage{pgfplots}
\usepgfplotslibrary{patchplots}
\pgfplotsset{compat=1.11}

\begin{document}

\pgfplotsset{
ytick={0,0.2,...,1},
xtick={0,0.2,...,1},
xlabel={$x$},
ylabel={$y$},
zlabel={$z$},
xmajorgrids=true,
ymajorgrids=true,
major grid style={dashed},
}

\begin{tikzpicture}
    \begin{axis}[view/h=45,xmin=0,xmax=1,ymin=0,ymax=1,colorbar horizontal, xlabel style={sloped}, ylabel style={sloped}, colorbar style={xlabel=$z$, xlabel style={at={(axis description cs:0.5,-1)}}}]
        \addplot3[surf,
            patch type=bilinear,
            shader=faceted interp] table [col sep=comma] {
0.000000,0.000000,50
0.000000,0.020110,2
0.000000,0.021307,0
0.000000,0.250000,5
0.000000,0.333333,3
0.000000,0.500000,22
0.000000,1.000000,1

0.000300,0.000000,2
0.000300,0.020110,0
0.000300,0.021307,0
0.000300,0.250000,0
0.000300,0.333333,0
0.000300,0.500000,0
0.000300,1.000000,0

0.001908,0.000000,2
0.001908,0.020110,0
0.001908,0.021307,0
0.001908,0.250000,0
0.001908,0.333333,0
0.001908,0.500000,0
0.001908,1.000000,0

0.003472,0.000000,2
0.003472,0.020110,0
0.003472,0.021307,0
0.003472,0.250000,0
0.003472,0.333333,0
0.003472,0.500000,0
0.003472,1.000000,0

0.010204,0.000000,3
0.010204,0.020110,0
0.010204,0.021307,0
0.010204,0.250000,0
0.010204,0.333333,0
0.010204,0.500000,0
0.010204,1.000000,0

0.028409,0.000000,2
0.028409,0.020110,0
0.028409,0.021307,0
0.028409,0.250000,0
0.028409,0.333333,0
0.028409,0.500000,0
0.028409,1.000000,0

0.035714,0.000000,0
0.035714,0.020110,0
0.035714,0.021307,2
0.035714,0.250000,0
0.035714,0.333333,0
0.035714,0.500000,0
0.035714,1.000000,0

0.045455,0.000000,1
0.045455,0.020110,0
0.045455,0.021307,0
0.045455,0.250000,0
0.045455,0.333333,0
0.045455,0.500000,2
0.045455,1.000000,0

0.083154,0.000000,0
0.083154,0.020110,0
0.083154,0.021307,0
0.083154,0.250000,0
0.083154,0.333333,0
0.083154,0.500000,2
0.083154,1.000000,0

0.092784,0.000000,0
0.092784,0.020110,0
0.092784,0.021307,0
0.092784,0.250000,0
0.092784,0.333333,0
0.092784,0.500000,2
0.092784,1.000000,0

0.119578,0.000000,2
0.119578,0.020110,0
0.119578,0.021307,0
0.119578,0.250000,0
0.119578,0.333333,0
0.119578,0.500000,0
0.119578,1.000000,0

1.000000,0.000000,1
1.000000,0.020110,0
1.000000,0.021307,0
1.000000,0.250000,0
1.000000,0.333333,0
1.000000,0.500000,4
1.000000,1.000000,68
        };
    \end{axis}
\end{tikzpicture}

\end{document}

Note that your screenshot reveals an outdated/incorrect PDF viewer. The correct output of your minimal working example as you typed it into the question is

enter image description here

You can verify this by using Acrobat Reader. I know that some old PDF viewers had lousy support for shadings. I would guess that your viewer is xpdf or some variant of libpoppler. There are more recent versions available which have correct visualization of these shadings. I suggest to upgrade to a more recent viewer.

Related Solutions

TikZ: How to Get Smooth Grid Lines for Surface Plots

The answer of pgfplots for "mesh plots with smooth boundary" is its patchplots library, combined with one of the higher-order patch types. Take, for example, patch type=biquadratic : its boundaries are second order polynomials. This allows to provide a coarse-grained grid combined with smoothness.

The cost for such a format is to provide patches in a nontrival sequence: coordinates

0 1 2 3 4 5 6 7 8

are interpreted as the following points on a (single) rectangle:

3  6  2
7  8  5
0  4  1

the next 9 points make up the next rectangle and so forth. Here, points 0,1,2,3 make up the corners whereas points 4,5,6,7 are used to define the boundary parables. Point 8 is only relevant if you have shader=interp; I think it is unused for mesh plots. If you combine the approach with shader=faceted interp, you get smooth fill colors + mesh lines.

Examples and images can be found in Section 5.6 Patchplots Library.

Of course, the cost is to manually arrange your input coordinate sequence in the format shown above - one set of 9 arranged points for every patch. To simplify this, you can also provide a single list of coordinates, and a connectivity table which tells pgfplots that patch 0 consists of vertices with indices #42 #5# #7 #3 #30 #12 ... (see the patch table option of pgfplots).

The alternatives are (as percusse already mentioned) to combine two plots: one which only fills the surface and one which provides the mesh. However, this has a severe limitation: pgfplots only considers image depths inside of one single plot (compare the discussion and the alternative solution described in Section "5.6.4 Drawing Grids" in the patch plots library.

pgfplots has no option to "draw only each Nth mesh line".

EDIT

I have just realized that your example function is a biquadratic function! As such, you only need one patch to capture its boundary without any loss in precision - and the effort to put input coordinates into the requested sequence is limited to only 9 points. That can be done by means of \addplot table[z expr=<expression>].

Only the color interpolation is of first order (because pdf only supports bilinear interpolation). Consequently, you can get the effect for your example function using shader=faceted interp combined with patch refines - and just one patch as input:

enter image description here

\documentclass{article}

\usepackage{pgfplots}
\usepgfplotslibrary{patchplots}

\pgfplotsset{compat=1.3}

\begin{document}
\thispagestyle{empty}

\begin{tikzpicture}
    \begin{axis}
    \addplot3[patch,patch refines=3,shader=faceted interp,patch type=biquadratic] 
        table[z expr=x^2-y^2]
    {
        x y
        -2 -2
        2 -2
        2 2
        -2 2
        0 -2
        2 0
        0 2
        -2 0
        0 0
    };
    \end{axis}
\end{tikzpicture}
\end{document}

Here is it with patch refines=2:

enter image description here

TikZ – Draw a Smooth Surface

Interesting question. Putting two torus is not a big deal as soon as you have some 3D routines and a lot of 3D solutions can do it. The remark of Charles Staats is relevant.

I tried to make a sufficiently smooth transition between two torus. To do this I take an (x,y,z) parametrization of the left part which is possible for x belonging to [-R-a,-R], compute the interval in which y lies and then obtain the z value. The modification consists to add a x dependant function null in -R and having a null derivative in -R-a. From a mathematical point of view the result is C^1 (I think)

Second Solution. I remember the interpolation in Tchebychev points versus equidistant points. Because for x fixed the curve is closed in some parts of an ellipse, I change the y parametrization so that the points are more well-balanced (it was the reason I put sqrt(y) in the first attempt). So the solution is to take sin(pi/2*y) and then it is possible to use the smooth-spline functionality of Asymptote. Please find the (always dirty and to improve) code

size(200);
import graph3;

currentprojection=perspective(5,4,4);

real R=3;
real a=1;

triple f(pair t) {
  triple z;
  z= ((R+a*cos(t.y))*cos(t.x),(R+a*cos(t.y))*sin(t.x),a*sin(t.y));
  return z;
}
surface s=surface(f,(0,0),(2pi,2pi),Spline);
surface ss=shift((2*R+2*a,0,0))*surface(f,(0,0),(2pi,2pi),Spline);

draw(s,green,render(compression=Low,merge=true));
draw(ss,green,render(compression=Low,merge=true));

triple f11 (pair z)
{
  real y,ty,x,yy;
  triple zz;
  if ((z.y>=0) & (z.y<=1))
    {
      x=z.x+1/2*(z.x+R)^2/a;
      y=sin(pi*(z.y)/2)*sqrt((R+a)^2-x*x);
      zz=(-z.x,y,sqrt(a^2-(R-sqrt((x)^2+y^2))^2));
    }
  if ((z.y<0) & (z.y>=-1))
    {
      x=z.x+1/2*(z.x+R)^2/a;
      y=-sin(pi*abs(z.y)/2)*sqrt((R+a)^2-x*x);
      zz=(-z.x,y,sqrt(a^2-(R-sqrt((x)^2+y^2))^2));
    }
  if ((z.y>1) && (z.y<=2))
    {
      x=z.x+1/2*(z.x+R)^2/a;
      yy=2-z.y;
      y=sin(pi*(yy)/2)*sqrt((R+a)^2-x*x);
      zz=(-z.x,y,-sqrt(a^2-(R-sqrt((x)^2+y^2))^2));
    }
  if ((z.y>1) && (z.y<=2))
    {
      zz=(-z.x,y,-sqrt(a^2-(R-sqrt((x)^2+y^2))^2));
    }
  if ((z.y>=-2) && (z.y<-1))
    {
      x=z.x+1/2*(z.x+R)^2/a;
      yy=-2-z.y;
      y=-sin(pi*(abs(yy))/2)*sqrt((R+a)^2-x*x);
      zz=(-z.x,y,-sqrt(a^2-(R-sqrt((x)^2+y^2))^2));
    }
  return zz;
}

triple f21(pair z)
{
  triple z=f11(z);
  return (-z.x,z.y,z.z);
}
int N=20;
surface st=surface(f11,(-R-a,-2),(-R,2),N,2*N,Spline);
surface sstb=shift((2*R+2*a,0,0))*surface(f21,(-R-a,-2),(-R,2),N,2*N,Spline);
surface pont1=surface(st,sstb);
draw(pont1,green, render(compression=Low,merge=true));

You can observe that the two torus are drawn and a 'bridge' is added. The code is faster and the result looks like

enter image description here

First solution. However I cannot use the surface(...,Spline) routine to have a true smooth surface. I think it is due in fact to some artefacts of the Bézier patch parametrization (De Boor reference). Perhaps that using some monotonic option to Spline is the solution but I have to admit that I do not have the time. Im y opinion, with the standard surface(....,100,100), the result is satisfactory. I am sure that it is possible to improve it.

Please find the (dirty and to clean) code

size(200);
import graph3;

currentprojection=perspective(5,4,4);

real R=3;
real a=1;

triple f(pair t) {
  triple z;
  z= ((R+a*cos(t.y))*cos(t.x),(R+a*cos(t.y))*sin(t.x),a*sin(t.y));
  return z;
}


bool active(pair pos) {return (R+a*cos(pos.y))*cos(pos.x)<=3.2;}
bool active2(pair pos) {return (R+a*cos(pos.y))*cos(pos.x)>=-3.2;}

surface s=surface(f,(0,0),(2pi,2pi),200,200, active);
surface ss=shift((2*R+2*a,0,0))*surface(f,(0,0),(2pi,2pi),200,200,active2);

draw(s,green,render(compression=Low,merge=true));
draw(ss,green,render(compression=Low,merge=true));

triple f11 (pair z)
{
  real y,ty,x;
  if (z.y>=0)
    {
      x=z.x+1/2*(z.x+R)^2/a;
      y=sqrt(z.y)*sqrt((R+a)^2-x*x);
    }
  else
    {
      x=z.x+1/2*(z.x+R)^2/a;
      y=-sqrt(abs(z.y))*sqrt((R+a)^2-x*x);
    }
  return (-z.x,y,sqrt(a^2-(R-sqrt((x)^2+y^2))^2));
}

triple f12(pair z)
{
  triple z=f11(z);
  return (z.x,z.y,-z.z);
}

triple f21(pair z)
{
  triple z=f11(z);
  return (-z.x,z.y,z.z);
}
triple f22(pair z)
{
  triple z=f11(z);
  return (-z.x,z.y,-z.z);
}
int N=100;
surface st=surface(f11,(-R-a,-1),(-R,1),N,N);
surface sst=surface(f12,(-R-a,-1),(-R,1),N,N);
surface sstb=shift((2*R+2*a,0,0))*surface(f21,(-R-a,-1),(-R,1),N,N);
surface sstbm=shift((2*R+2*a,0,0))*surface(f22,(-R-a,-1),(-R,1),N,N);
surface pont1=surface(st,sst,sstb,sstbm);
draw(pont1,green);//blue);

and the result enter image description here

Best Answer

Related Solutions

TikZ: How to Get Smooth Grid Lines for Surface Plots

TikZ – Draw a Smooth Surface

Related Question