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


*

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



*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.


*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:


*

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


*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.
 A: 
It has been a long time since I asked this question. I will keep it open some more days before closing, hoping someone else will be able to add more options of visualization. 

Recently I have been able to learn another interesting methodology that can be applied to the visualization of the roots. We will define the tuples $(a,b,c,x_1),(a,b,c,x_2)$ as points of a $4$-dimensional space, projected into the $3$-dimensional space. The methodology is explained in this question.

Basically, we allocate the points inside a $4D$ tesseract whose center will be the origin of coordinates in the $4$-dimensional space and the reference of the position of the points in the internal space of the tesseract, and then the tesseract and its content is projected into $3D$. The example below shows every real root for $a,b,c \in [-10,10] (a,b,c \in \Bbb Z$):


And below there is a zoom showing only the root tuples. As it can be seen, the pattern generated by the set of tuples shows interesting symmetries in the $4D$ space that can be observed in the $3D$ projection of the set of points (kind of mesmerizing!). The movement is required because to be able to visualize a complete $4D$ object through its projection, we need to rotate around one of the $4D$ axis. By doing so, it is possible to see the $3D$ "shadow" of the positions of the points in the original $4$-dimensional space:

