Assignment # 2: Part 10

Exploring a simple circle:

... and a variation of it:

First of all, let's look at what happens if we vary our radius:



From this, we can infer that modifying our radius won't really effect the overall exploration. For the sake of uniformity, we are going to keep it at a constant - 3.

Now, let us look at what happens with different coefficients on the 'xy' term:


 The coefficients used are:

  • 1 - giving us the purple ellipse
  • 2 - giving us the red seemingly two parallel lines
  • 4 - giving us the green hyperbola
  • 20 - the hyperbola seems to be getting closer (in fact, as this number gets very large, the x-axis and y-axis become asymptotes.

So now lets make our "axy" a variable - namely, "zxy" and let's look at it in 3D.


 Side view

Top View

We can see that the graph is symmetric and seems to be moving from a circle in the middle, to an ellipse, to hyperbolas. Now let's try to find the transitional coefficients between the graphs.

We want to now examine the following equation for different values of 'a':

1. Circle ( a=0 )

The value of 'a' is zero.

2. Circle --> Ellipse ( -2 < a < 2 )

This starts to happen right away with the introduction of the middle "xy" term. Even with the coefficient being .1, the graph is still elliptic. These ellipses have both rotational symmetry (of 180 degrees) and reflexive symmetry about the y = x and y = -x lines.

3. Ellipse --> Two Straight Lines ( a = 2 )

This is where the graph is unique. Why do we have two straight lines? Click here for the proof.

4. Two Straight Lines --> Hyperbola ( a > 2 and a < -2 )

Again, we have a regular hyperbola with a little 45 degree twist - observe in the picture below:


The blue graph is our equation - the red is your standard hyperbola.

Conclusions drawn ...

From my investigation, I have noticed that the middle term (xy) changes our axis by 45 degrees. The hyperbolas are still hyperbolas and ellipses are still ellipses - just bit rotated. The interesing values for it are a = 2 and a = 0 (circle). Overall, it's behavior outside those regions is pretty regular.

Now, if you want to do it yourself, here are some resources:

3D Graph of the region.

2D Graph of the region - mess with variable.

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