# An Attempt to Simplify Quadratic Curves Using Complex Numbers Source: Machine translated. Refer to the original post to verify accuracy. --- author: Su Jianlin date: January 2, 2013 lang: en markdown_url: 01851-An-Attempt-to-Simplify-Quadratic-Curves-Using-Complex-Numbers.md next_url: 01850-The-Development-of-Feynman-s-Path-Integral-Thought-IV.html post_id: 1851 prev_url: 01866-Messing-with-Math-Formula-Plugins-Again.html site_title: English (unofficial) translations of posts at kexue.fm source_url: "https://kexue.fm/archives/1851" title: An Attempt to Simplify Quadratic Curves Using Complex Numbers --- When a quadratic form is in the case of a two-dimensional plane, it is equivalent to the simplification of a quadratic curve. The simplification of quadratic curves mainly involves translation and rotation, which are exactly what complex numbers are “good at.” Therefore, using complex numbers as a tool to simplify quadratic curves seems like a very obvious idea. However, I have not seen much content on this, and I had previously overlooked this approach myself. Below, I will explore this idea a bit. Since I only intend to provide some heuristic guidance, I will only consider the incomplete form $`Ax^2+2Bxy+Cy^2=1`$ (which does not include parabolas). For a complex number $`z=x+yi`$, we have: ``` math x=\frac{1}{2}(z+\bar{z}), \quad y=\frac{1}{2i}(z-\bar{z}) ``` By substituting these two expressions, we can obtain the complex expression for any plane curve. For the aforementioned quadratic curve, we get: ``` math (A-C+Bi)z^2+(A-C-Bi)\bar{z}^2+2(A+C)z\bar{z}=4 ``` For the final expression of the quadratic form, we hope to have only squared terms. The complex expressions for the squared terms are: ``` math z\bar{z}=x^2+y^2, \quad z^2+\bar{z}^2=2(x^2-y^2) ``` Therefore, what we need to do is to transform the complex expression of the quadratic curve into a linear combination of $`z\bar{z}`$ and $`z^2+\bar{z}^2`$. From the expression: ``` math (A-C+Bi)z^2+(A-C-Bi)\bar{z}^2+2(A+C)z\bar{z}=4 ``` it is easy to see that if we let: ``` math Z=z\sqrt{A-C+Bi} ``` the above equation can be rewritten as: ``` math Z^2+\bar{Z}^2+\frac{2(A+C)}{\sqrt{(A-C)^2+B^2}}Z\bar{Z}=4 ``` This is the form we expected. This is truly a pleasant coincidence! Thus, starting from complex numbers and performing a bit of calculation, we easily arrive at the simplest form of the quadratic curve: ``` math \begin{aligned} 2(X^2-Y^2)+\frac{2(A+C)}{\sqrt{(A-C)^2+B^2}}(X^2+Y^2)=4 \\ \left(\frac{A+C}{\sqrt{(A-C)^2+B^2}}+1\right)X^2+\left(\frac{A+C}{\sqrt{(A-C)^2+B^2}}-1\right)Y^2=2 \end{aligned} ``` Furthermore, when using matrices to simplify quadratic curves, the specific form of the rotation transformation is relatively difficult to provide, but complex numbers are different; they tell us directly: ``` math Z=z\sqrt{A-C+Bi} ``` It can be seen that this kind of geometry of numbers has considerable advantages in certain cases. Although matrix algebra can completely encompass the results of higher-dimensional numbers (quaternions, octonions), studying the laws of the numbers themselves is still quite enlightening. Just as complex numbers occupy a fundamental position in modern quantum mechanics, which was completely unimaginable to their inventors, perhaps one day, a certain type of number will play an even greater role in describing the universe. This is the charm of mathematics! ***Reprinting:** Please include the address of this article: [https://kexue.fm/archives/1851](./01851-An-Attempt-to-Simplify-Quadratic-Curves-Using-Complex-Numbers.html)* ***For more details on reprinting, please refer to:** [Scientific Space FAQ](./06508-Scientific-Space-Browsing-Guide-FAQ.html)*