I have never been very good at visualizing and when in 11th standard they told me that z=ax+by+c is the equation of a plane, I couldn't "feel" it. That y=ax+b is a line in 2D world I had no doubt becuz all one had to do was to grab a graph paper and draw some points and voila! there is a dull yet wonderful line.
For 3D graphs, things were not so easy. The best one could do was to depend on a plotting software. Unfortunately, I then knew as much about computing as about driving a train. So I had to just take the text books on their word.
(OK, the last para was not wholly accurate. "The best one could do"? "So I had to"?. Really it should be "I thought The best one could do" and "So I thought I had to". But I am coming to that)
Some weeks ago when I was attending a boring class I thought I should as well try and visualize some 3-D graphs. I tried and to my chagrin found out that it was not so hard after all! So if you are still with me, here is how to:
The first thing one should do to enjoy any good show is to be on the best place to watch. What better place for watching the graph show than the Holy Origin?
Imagine a large field (infintely large, to be precise) with you standing in it. Call the place you stand the origin (don't stray, please). Your right and left hands indicate the positive and negative x axes respectively. Along your height is the positive z axis and stretching directly in front of you is the positive y-axis.
Now here is the trick: imagine an infinite series of vertical semi-transparent graph papers at each point on the y axis. ie, at y=1 there is an x-z 2-D graph paper. It is not actually important that there are an infinite number of them. But you should be able to produce one at any y point as required. Of course, it is terribly important that the graph papers should have a variable transparency. Graph papers are just props and not a part of the show.
That's it! Let's try and visualize z=y.
Here is one very obvious plain fact: z=y is the same as z=c at y=c. Too obvious, rt? But this is all that is needed to visualize z=y. at y=0, the graph is z=0 ie., a line parallel to the x axis. Plot it in the graph paper at y=0 (if you remember, this paper should be splitting you in half). Visualize this plot (if it helps, you can imagine the line as a lengthy tube light or in Star Wars) but try to make the graph paper completely transparent. In the graph paper at y=1, plot the line z=1 ( a line parallel to the x-axis but a little higher than the previous line). Continue this process till you get the feel of it. Behind you, you can draw some plots for negative y. All these lines are parallel to x-axis but as y goes on increasing, the lines will be having greater heights.
Now, just forget about the graph papers but retain the lines plotted. Can you see a staircase like structure? It is a staircase because we plotted only a finite no. of lines. From y=0 we jumped to y=1. If we draw lines at y=0.5, y=1.5 etc. The number of lines will be increased. We can go on adding lines and in the end -- it is the plane!
Make sure that you can successfully visualize z=y before going any further!
Next let's take z=by.
Here the "very obvious plain fact" is: z=by is the same as z=bc at y=c. This means simply that for the plot lines to reach the same height it may take a longer (b<1),>1) or the same (b=1) time. In other words, the plot is the same plane and the intermediate plot the same staircase but with a greater or smaller steep if b is not 1.
What about z=ax+by? We already know the special case when a=0. The only difference in the general case is that the lines(the rungs of the staircase) are no longer lines parallel to the x-axis but are at a definite angle (depending on the value of a) to it. So, our staircase is as though placed on uneven ground. Our plane has a horizontal tilt.
Try it yourself question:
* Visualize z=ax+by+c
If you have reached here unscathed, let's try to visualize something a little bit more complex: z=xy.
Our simplifying statement: z=xy is the same as z=cx at y=c. So at y=0 our line is z=0; at y=1 z=x; at y=1 z=2x so on. Now, in z=cx c is a measure of how (comparitively )fast we reach infinity. ie., z=2x reaches the point z=2 earlier (at x=1) than z=x which reaches the same point at x=2. (In this article I have used the terms like earlier, fast etc in a sloppy manner. If you are mathematically oriented, forgive me as my intention is just to be intuitively expressive). So z=2x is "closer" to the z axis (which reaches infinity the fastest -- at x=0!)
So, the "steps" that make up our "staircase" (we don't actually know if it is a staircase for this graph yet) for z=xy become closer to the z-axis before us as y increases and closer to the negative y-axis behind as y decreases (why?). Our plot lines start (if we are looking forward in our hypothetical plane. if you are not, look ahead!) at z=0 the line is horizontal. As the distance from us increases, the lines become steeper and steeper until, at infinity, the line becomes parallel to the z axis. The graph is hence a twisted plane. One thing to note is that the plane is not uniformly twisted because only at y=infinity does the plane becomes perpendicular. The twist is rather sharp around y=0 but becomes almost unnoticable at higher y's. As it turns out, our "staircase" is a long spiral one. Try to visualize this graph -- it is beautiful. Really!
Try it yourself questions:
* Visualize z=yx^2 and z=xy^2
* Can we visualize z=xy as a very large sheet that is shifted perpendicularly at the ends?
P.S. I have doubts whether I have been very clear. In fact I am positive that some of what I have written is rather confusing at places. Plz help me write a better version of this article.
