The pictures on the page you just saw were generated at home. These were generated using a program written at home to generate contours for functions of 3 variables. All you need to do is feed in the function, F (x,y,z) = 0 in the program and it draws the contours. Of course, other parameters need to be set, so that the plot looks like a plot and not like a dot.

There were many parameters that I had to consider. There was the fineness of the graph- the finer the graph (the more the points) the greater the time it took to make the graph, and the gerater the boredom. And then there was the spacing between the contours. The spacing was to be such that the figure was clear. So, basically I had to pull all these things to an optimum, where the picture would be the clearest.

I hope that the pictures are reasonably clear. I did my best in getting the optimum settings.

However, there are some flaws that can be ironed out in the program, and I being no whiz at programming find it a little tough to fix them.

The plot, for instance is good when the slope of the tangent at the point of plotting is high, and is very spare when the tangents are horizontal. This can be seen- the more horizontal parts of the plots seem a little bare.I reckong that plotting by varying y also and superimposing would make more sense, but, I am unable to do that. All that does to the program is make it more complex.

And yet another problem with this program is stream lines. These stream lines do not seem to meet. A stream line seems to coverge onto a vertical line, and new stream lines seem to emerge from that. This is probably because of the nature of these source functions, they being functions of arctan( ). Now, this is a shady function, when x is very small. I had to rule out the case when x was zero, so probably in doing so, this happened.

Streamlines for y = const does not mean a continuous line here. This line is continued by y = another contant later, on the other side of vertical line that passes through the origin, that has no business being there, but it there due to a defect in the program. Since the constant is never right, these lines do not seem to converge at that point. It can be rationalised by the fact that another stream lines does indeed exist which continues this stream line, only the line is different.

Plot Algorithm:

1. Make a function that can find the roots of a given one variable expression, f(x). as x1,x2, x3…xn.
2. Take g(x,y), keeping y = a, a constant. Treat g(x,a) as f(x). Find roots. Plot all (xr,a), 0

3. Vary y now, as a1, a2, … between the limits. Plot (xr,a1)s, (xr,a2)s and so on.
4. We now have a 2d Graph generator for g(x,y) = 0.
5. Take h(x,y,z). Keep z as a constant, and treat it as g(x,y). Plot contour.
6. Plot contours for different values of z.
7. Here z is y,the stream function.

We have stream lines!

If the Input is a function like (x*x + y*y –z ) then we get a nice paraboloid, and circles as contours.

Source code (html file)

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