{"id":100,"date":"2014-11-18T08:56:09","date_gmt":"2014-11-18T13:56:09","guid":{"rendered":"https:\/\/blogs.mathworks.com\/graphics\/?p=100"},"modified":"2014-11-18T08:57:18","modified_gmt":"2014-11-18T13:57:18","slug":"what-is-a-surface","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/graphics\/2014\/11\/18\/what-is-a-surface\/","title":{"rendered":"What is a Surface?"},"content":{"rendered":"

What is a Surface?<\/h3>

What exactly is MATLAB doing when we say the following?<\/p>

surf(peaks)<\/pre>
The answer seems obvious. We're telling MATLAB to draw a continuous surface through the points which are defined in the array which the function peaks returns. But where does that surface go in between those points? It turns out that there are some subtle issues hiding here, so let's look a bit closer.<\/p>
We'll start with a smaller array so that we can see what's really going on.<\/p>
rng(0)\r\nz = randn(5,7)\r\n<\/pre>
\r\nz =\r\n\r\n    0.5377   -1.3077   -1.3499   -0.2050    0.6715    1.0347    0.8884\r\n    1.8339   -0.4336    3.0349   -0.1241   -1.2075    0.7269   -1.1471\r\n   -2.2588    0.3426    0.7254    1.4897    0.7172   -0.3034   -1.0689\r\n    0.8622    3.5784   -0.0631    1.4090    1.6302    0.2939   -0.8095\r\n    0.3188    2.7694    0.7147    1.4172    0.4889   -0.7873   -2.9443\r\n\r\n<\/pre>
Let's give that to surf and see what it draws.<\/p>
f1 = figure;\r\nh = surf(z);\r\nh.FaceColor = 'interp'<\/span>;\r\nh.Marker = 's'<\/span>;\r\nh.MarkerFaceColor = [.7 .2 .3];\r\nh.MarkerEdgeColor = 'none'<\/span>;\r\ntitle('surf'<\/span>)\r\n<\/pre>\"\" 
Those little grey squares are the values in the array z. It looks like the surface is interpolating linearly between those values. Is that actually what's happening? And is it the best choice?<\/p>
MATLAB has a useful function named interp2<\/a> which will interpolate between the values in a 2D array. We can use interp2 to do the same sort of interpolation that surf is doing, but with more control over the interpolation.<\/p>
First we'll create a high-res version of the X & Y coordinates of that surface. I've just increased the resolution by a factor of 10, but you could use any number.<\/p>
[n, m] = size(z);\r\n[x, y] = meshgrid(1:m,1:n);         % low-res grid<\/span>\r\n[x2,y2] = meshgrid(1:.1:m,1:.1:n);  % high-res grid<\/span>\r\n<\/pre>
Then we'll use interp2 to interpolate the Z values up to that high-res mesh. And then we'll give that high-res array of Z values to surf.<\/p>
clf(f1);\r\nz2 = interp2(x,y,z, x2,y2); % interpolate up<\/span>\r\nf = surf(x2,y2,z2);\r\nf.EdgeColor = 'none'<\/span>;\r\nf.FaceColor = 'interp'<\/span>;\r\nf.FaceLighting = 'gouraud'<\/span>;\r\ntitle('interp2'<\/span>)\r\nhold on<\/span>\r\n<\/pre>\"\" 
I've turned the edges off because they would be the edges of the high-res grid. We want to add the edges of the low-res grid. We'll have to do that in two steps.<\/p>
First the column edges.<\/p>
[x3,y3] = meshgrid(1:m,1:.1:n);\r\nz3 = interp2(x,y,z, x3,y3);\r\nc = surf(x3,y3,z3);\r\nc.FaceColor = 'none'<\/span>;\r\nc.MeshStyle = 'column'<\/span>;\r\nhold on<\/span>\r\n<\/pre>\"\" 
And then the row edges.<\/p>
[x3,y3] = meshgrid(1:.1:m,1:n);\r\nz3 = interp2(x,y,z, x3,y3);\r\nr = surf(x3,y3,z3);\r\nr.FaceColor = 'none'<\/span>;\r\nr.MeshStyle = 'row'<\/span>;\r\n<\/pre>\"\" 
And finally we'll add the markers at the original data points.<\/p>
m = surf(x,y,z);\r\nm.FaceColor = 'none'<\/span>;\r\nm.MeshStyle = 'none'<\/span>;\r\nm.Marker = 's'<\/span>;\r\nm.MarkerFaceColor = [.7 .2 .3];\r\nm.MarkerEdgeColor = 'none'<\/span>;\r\nhold off<\/span>\r\n<\/pre>\"\" 
That looks pretty close to what surf did, doesn't it?<\/p>
The interp2 function takes an argument named method that we didn't use before. We'd like to see what it looks like when we choose different values for method. We'll do that by converting those previous steps into the following function so that we can call it multiple times.<\/p>
function<\/span> interpsurf(z, method)\r\n  if<\/span> nargin < 2\r\n    method = 'linear'<\/span>;\r\n  end<\/span>\r\n  [n, m] = size(z);\r\n  [x, y] = meshgrid(1:m,1:n);         % low-res grid<\/span>\r\n  [x2,y2] = meshgrid(1:.1:m,1:.1:n);  % high-res grid<\/span>\r\n  z2 = interp2(x,y,z, x2,y2, method); % interpolate up<\/span>\r\n  % Draw the faces with no edges using the high-res grid<\/span>\r\n  f = surf(x2,y2,z2);\r\n  f.EdgeColor = 'none'<\/span>;\r\n  f.FaceColor = 'interp'<\/span>;\r\n  f.FaceLighting = 'gouraud'<\/span>;\r\n  hold on<\/span>\r\n  % Add the column edges using a mix of low-res and high-res<\/span>\r\n  [x3,y3] = meshgrid(1:m,1:.1:n);\r\n  z3 = interp2(x,y,z, x3,y3, method);\r\n  c = surf(x3,y3,z3);\r\n  c.FaceColor = 'none'<\/span>;\r\n  c.MeshStyle = 'column'<\/span>;\r\n  % Add the row edges<\/span>\r\n  [x3,y3] = meshgrid(1:.1:m,1:n);\r\n  z3 = interp2(x,y,z, x3,y3, method);\r\n  r = surf(x3,y3,z3);\r\n  r.FaceColor = 'none'<\/span>;\r\n  r.MeshStyle = 'row'<\/span>;\r\n  % Add markers at the original points<\/span>\r\n  m = surf(x,y,z);\r\n  m.FaceColor = 'none'<\/span>;\r\n  m.EdgeColor = 'none'<\/span>;\r\n  m.Marker = 's'<\/span>;\r\n  m.MarkerFaceColor = [.7 .2 .3];\r\n  m.MarkerEdgeColor = 'none'<\/span>;\r\n  hold off<\/span>\r\n<\/pre>
Now we can try the four different options for the method argument and compare the resulting surfaces.<\/p>
f2 = figure('Position'<\/span>,[100 100 760 600]);\r\nsubplot(2,2,1)\r\ninterpsurf(z,'linear'<\/span>)\r\ntitle linear<\/span>\r\naxis tight<\/span>\r\nsubplot(2,2,2)\r\ninterpsurf(z,'spline'<\/span>)\r\ntitle spline<\/span>\r\naxis tight<\/span>\r\nsubplot(2,2,3)\r\ninterpsurf(z,'nearest'<\/span>)\r\ntitle nearest<\/span>\r\naxis tight<\/span>\r\nsubplot(2,2,4)\r\ninterpsurf(z,'cubic'<\/span>)\r\ntitle cubic<\/span>\r\naxis tight<\/span>\r\n<\/pre>\"\" 
As you can see, all of these functions pass through the original points. But they behave quite differently in between those points. That's showing the difference between the various interpolation methods.<\/p>
The different interpolation methods have some notable characteristics.<\/p>