# Tutorial :How Do I Generate a 3-D Surface From Isolines? ### Question:

I have a set of isoline points (or contour points) such as this:

alt text http://www.rcad.eu/triang&isolines%20example1.jpg

Each point on an isoline has its own respective X, Y, and Z coordinate. Since they are isolines, that means that each point will have a unique X-Y pair, but points on the same line will have the same Z coordinate.

Now, is there any algorithm or any software packages (either in C# or C++ or MATLAB) that I can use to interpolate the isoline points into a full 3-D surface?

P/S: I am not just interested in the final output, I am interested in getting the interpolated surface data so that I can plot the surface myself.

Edit: C++ solutions are welcomed as well.

### Solution:1

In MATLAB you can use either the function `griddata` or the `TriScatteredInterp` class (Note: as of R2013a `scatteredInterpolant` is the recommended alternative). Both of these allow you to fit a surface of regularly-spaced data to a set of nonuniformly-spaced points (although it appears `griddata` is no longer recommended in newer MATLAB versions). Here's how you can use each:

• `griddata`:

``[XI,YI,ZI] = griddata(x,y,z,XI,YI)  ``

where `x,y,z` each represent vectors of the cartesian coordinates for each point (in this case the points on the contour lines). The row vector `XI` and column vector `YI` are the cartesian coordinates at which `griddata` interpolates the values `ZI` of the fitted surface. The new values returned for the matrices `XI,YI` are the same as the result of passing `XI,YI` to `meshgrid` to create a uniform grid of points.

• `TriScatteredInterp` class:

``[XI,YI] = meshgrid(...);  F = TriScatteredInterp(x(:),y(:),z(:));  ZI = F(XI,YI);  ``

where `x,y,z` again represent vectors of the cartesian coordinates for each point, only this time I've used a colon reshaping operation `(:)` to ensure that each is a column vector (the required format for `TriScatteredInterp`). The interpolant `F` is then evaluated using the matrices `XI,YI` that you must create using `meshgrid`.

## Example & Comparison

Here's some sample code and the resulting figure it generates for reconstructing a surface from contour data using both methods above. The contour data was generated with the `contour` function:

``% First plot:    subplot(2,2,1);  [X,Y,Z] = peaks;  % Create a surface  surf(X,Y,Z);  axis([-3 3 -3 3 -8 9]);  title('Original');    % Second plot:    subplot(2,2,2);  [C,h] = contour(X,Y,Z);  % Create the contours  title('Contour map');    % Format the coordinate data for the contours:    Xc = [];  Yc = [];  Zc = [];  index = 1;  while index < size(C,2)    Xc = [Xc C(1,(index+1):(index+C(2,index)))];    Yc = [Yc C(2,(index+1):(index+C(2,index)))];    Zc = [Zc C(1,index).*ones(1,C(2,index))];    index = index+1+C(2,index);  end    % Third plot:    subplot(2,2,3);  [XI,YI] = meshgrid(linspace(-3,3,21));  % Generate a uniform grid  ZI = griddata(Xc,Yc,Zc,XI,YI);          % Interpolate surface  surf(XI,YI,ZI);  axis([-3 3 -3 3 -8 9]);  title('GRIDDATA reconstruction');    % Fourth plot:    subplot(2,2,4);  F = TriScatteredInterp(Xc(:),Yc(:),Zc(:));  % Generate interpolant  ZIF = F(XI,YI);                             % Evaluate interpolant  surf(XI,YI,ZIF);  axis([-3 3 -3 3 -8 9]);  title('TriScatteredInterp reconstruction');  `` Notice that there is little difference between the two results (at least at this scale). Also notice that the interpolated surfaces have empty regions near the corners due to the sparsity of contour data at those points.

### Solution:2

You can use the gridfit tool, found on the MATLAB Central file exchange. One of the examples I give is exactly what you want to do, starting from a list of points taken from isolines, I reconstruct a smooth surface from the data. In fact, the example I used was taken from a topographic map.

### Solution:3

I think what you want is called "Contour stitching", for example discussed in this paper

### Solution:4

MATLAB has a built-in command called SURF that takes three arrays for X,Y,Z and plots a surface. It might be what you're looking for.

Note:If u also have question or solution just comment us below or mail us on toontricks1994@gmail.com
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