What methods are known to visualize patterns in the set of real roots of quadratic equations?

Question

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations $ax^2+bx+c=0$.

To visualize the pattern of the relationship between the set of real roots, the algorithm in Python calculates the roots $x_1,x_2$ only for the specific values of the intervals $a \in [-a_i,a_i]$, $b \in [-b_i,b_i]$, $c \in [-c_i,c_i]$, $a,b,c \in \Bbb N$. If the limits $a_i,b_i,c_i$ are very big, the set of roots is very dense and it is very difficult to visualize the emerging pattern.

These are three methods I have tried to visualize the relationship between the roots $x_1,x_2$, my questions are at the end of them.

1. Cartesian coordinates $(x,y)=(x_1,x_2)$. E.g. $a_i,b_i,c_i=75$:

2. Polar coordinates $(\theta, r)=(x_1,x_2)$. E.g. $a_i,b_i,c_i=75$: Due to the symmetries the opposite patterns $(x,y)=(x_2,x_1)$ and $(\theta, r)=(x_2,x_1)$ are similar.

3. Spherical coordinates $(\theta, \phi, r)=(x_1,x_2,1)$. E.g. $a_i,b_i,c_i=25$:

So basically there seems to be a pattern in the relationship between both real roots that can be visualized.

I would like to ask the following questions:

1. What methods are known to visualize patterns in the set of real roots of quadratic equations?

2. Are there references to papers or studies about this subject?

Thank you!

UPDATE 2015/08/31

As requested, this is the pattern shown by the Cartesian coordinates example that was included above (1), when $a_i,b_i,c_i=575$ and using only the square roots of the prime numbers of those intervals, $\sqrt{a} \ / a \in [-a_i,a_i]$, $\sqrt{b} \ / b \in [-b_i,b_i]$, $\sqrt{c} \ / c \in [-c_i,c_i]$. This will show the pattern of the roots only for a) $a,b,c$ irrationals (because the square roots of prime numbers are irrational numbers) and still constrained to b) $x_1,x_2 \in \Bbb R$:

UPDATE 2015/09/02

Due to the size of the images I can not add some other animations I have prepared: In this link you will find the graphs of the quadratic complex roots, and also the graphs of the complex and real roots of cubic equations.