Normal-Intersections-Mathematics

A visualiser I created when I was working on a very interesting, self-created mathematics problem.

The problem was roughly something like this: 'For any given point in the cartesian plane, find a rule which dictates the number of normals to the curve $y=x^2$ that intersect that point'.

The problem is very nice and only requires elementary methods in calculus, such as differentiation and the understanding of normals.

After solving this problem I came to the answer of a line which separated the cartesian plane into two sections, the region above the line contained all points with 3 normals that intersect, the region below contained all points which had 1 normal, and on the line itself had points which had 2 normals (or 1 normal, and 2 'repeated' normals. That line is shown below.

$y=\frac{3\sqrt[3]{\frac{x^2}{2}}+1}{2}$

As a mathematician, I was intrigued to generalize my solution to work with other polynomials and maybe some transcendental curves. However, I quickly realised this would be very difficult as it would require solving equations only possible through numerical methods, such as some polynomials with degrees n, where n > 4.

I then decided that it would be interesting to visualise these areas via numerical and computational methods.

Here are a few. (There are more beautiful ones, but I will leave them to you to find).

Algorithms learnt

Brent's Method: A numerical root-finding algorithm used to approximate the roots of a polynomial or any continuous function. It combines the bisection method, secant method, and inverse quadratic interpolation for efficient and robust convergence.

Numpy's Method: NumPy's roots() function uses an eigenvalue-based approach for polynomial root-finding, particularly for real polynomial coefficients. I wish I had a better understanding of eigenvalues and matrix manipulation to explain this better. I am excited to learn more about it.

I chose to use Numpy's roots() method as it was faster than Brent's Method when I timed the speeds of both algorithms. However, I do believe that Brent's method is more intuitive and easier to understand and am sure there are situations where it is advantageous to use it.

Normal intersections

$y=x^2$

$y=x^4$

$y=x^4-4x^2$

$y=x^5-x^3$

$y=\frac{1}{x}$, this one is not possible using the code as it is. However it is not hard to change the code in order to make it work. I'll leave it as a challenge

$y=\sin(x)$, but not really... Sorry to disappoint, but sinx is very difficult to get normal numerical solutions. Can you figure out how I did it. Hint: Polynomials are easy to do, trig/transcendentals are hard.

(Most plots were generated using DETAIL=400, took my Macbook Air about a minute to produce each one on Jupyter Notebook)