A fundamental Exploration of a parametric equation
(This was my first write-up done on Aug. 31, and Sep. 8, 2009, in which I included some basic case analysis. )
Here is the parametric equation we will try to explore in the write-up.
with t in some range.
When p, a, b, q, c, d are all equal to 1’s,
1) We let t go from 0 to 1.57 (approximate pi/2), then we will see a quart of circle in the first quadrant.
2) Similarly let t go from 0 to 3.14 (approximate pi), the semi-circle appears in the first and fourth quadrant.
3) We directly let t go from 0 to 6.28 (approximate 2pi), a circle shows up in the grid with the center at point (0,0).
Why is this?
It just shows the relation between values of sin(t) and cos(t), which usually are shown in two separate “wave” graphs of sin and cos.
If p, a, b, q, c, d are all equal to 3’s, obviously , the circle becomes larger with radius of length 3.
But how can we move the circle to wherever we want? After some trying, we find adding a certain real number to the x value takes the circle to the right, subtracting a certain real number to the x value takes the circle to the left, similarly adding a certain real number to the y value takes the circle to the up, subtracting a certain real number to the y value takes the circle to the down, respectively, with the distance of the real number.
Interestingly, isn’t this another way to draw a circle with an arbitrary radius centered at the origin or somewhere else in the coordinate plan?
The example of (x, y)=(3sin(t)+2.4, 3(cos(t)-0.3) with t from (0, 6.28) is shown in the below figure.
What if p=2 , a, b, q, c, d are all equal to 1’s? A horizontal ellipse at the origin will be graphed like the following one.
Why? It is because the x values are double when y values keep as the same as in the all 1’s cases.
So we can image that when q=2, p, a, b, c, d are all equal to 1’s, a vertical ellipse will be graphed. If q goes greater, the ellipse will look like being “squeezed” more.
From now on, we move from arcs to circles then to ellipses. Wasn’t that pleasing?
What if a, b, c, d are not all equal to 1’s?
Keeping the other parameters in the same hypothesis for the unit circle case, let b go greater.
When b=1.01 goes to b=1.03, the unit circle I get a little bit slanted and there is the little hole appearing around point (-0.2, 1); up to b=1.1, the hole gets bigger, and the circle gets more slanted; then up to b=1.4, the unit circle does not look like itself any more; then up to b=1.6, the curve looks like the Greek letter gama; then up to around b= 1.9, the space “inside” the gama-like curve gets smaller, and the curve looks being folded into two halves.
What happened to the changes from the circle to the curve? When b gets greater, a/b gets smaller, and the range of the values of sin((a/b)t) gets changing, so it plots different sets of points (x,y).
Notice, when b=2, the “two halves” of the gama-like curve coincide. It is just like the right half of a parabola.
Why? (½)t is in the interval [0, 3.14], sin((½)t) goes from 0 up to 1 and then down to 0 again, and cos(t) goes from 1 to -1 when t goes from 0 to pi, so the points on the “right half of a parabola” curve does go back and forth once.
When b=3 the curve looks Greek letter gama again. When b=5 the right extremity of the gama-like curve seems intending to “point toward outside”, which makes the curve looks like the letter v. When b goes greater from 5 to much greater number, say 80, the curve looks more like a letter v, which gets thinner and thinner. When b gets greater and greater a/b approaches to 0, the v-like curve gets extremely slim, but it will never becomes a line. –- Well, for women who want to lose weight fast this might be a better and cheaper product than “slim-fast” sold in Wal Mart.
What all we have discussed above is confined with t in the interval [0, 6.28]. So let’s observe some more cases when t is in the interval [-6.28, 6.28]. Thus, the domain of x values will be symmetric about the y axis.
Look at the graph when b=1.9 compared with the one we’ve seen above. We will see this new curve is the curve in the above b=1.9 graph plus its reflection about the y axis.
Therefore, we claim when b=2 the curve will be a “complete” parabola comparing to the above curve of “right half of a parabola”. Again, another way to draw a parabola.
Therefore, using the idea of reflection we will understand better why a curve looks like what it is. Now let’s snap some curves with some specific parameters as follows.
b=4 or b=-4
Ok, now let observe a few more complex cases.
When p, b, q, c, d are all equal to 1’s,
So, it seems when the number of the “loops” formed by the curve equals the value of a. But, why? One thing I can say is that when a gets greater, the value of sin(at) goes back and forth faster, i.e., the frequency of the “vibration” of the “combination wave” of sin and cos gets bigger. As for the corresponding equality relationship between number of the “loops” and the value of a when (x, y)=(sin(at), cos(t)) for t in [0, 6.28], my explanation would be that when a=1 the value of sin(at) goes through a “vibration cycle” (maybe namely, “period”); when a=2, the speed of the change of the value of sin(at) is doubled, so the value of sin(at) goes through two “vibration cycles”; similarly for other integers (including both positive and negative; it doesn’t change the graph) which the value of a could be chosen, i.e., when a=n, the speed of the change of the value of sin(at) is doubled, so the value of sin(at) goes through n “vibration cycles”.
What if we let x be sin(t), and change the values of c and d? The rationale for this exploration would be similar as above. What if we change the values of p and q? The point is that the graph would be either stretched or squashed in two directions (x-axis and y-axis). There are still many other good cases we could try to explore. We may find some other interesting graphs. But I think all further explorations would be based on what we have done fundamentally.