在复数平面中通过线-圆周（LC）几何构造定位一元二次方程的根
Locating the roots of a quadratic equation in one variable through a Line-Circumference (LC) geometric construction in the plane of complex numbers
AI总结：
本文提出一种基于复平面中直线与圆周（LC）几何构造求解一元二次方程根的方法，该方法可扩展至三次及以上多项式的求根初始近似。
摘要：
本文描述了一种几何方法，通过在复平面中由已知系数 $c_1$、$c_2$ 构造的直线 $L$ 和圆周 $C$，来求解形如 $x^2+c_1 x+c_2=0$ 的一元复变量二次方程的根 $r_1$、$r_2$。该线-圆周（LC）几何结构在其组成元素 $L$ 和 $C$ 的交点处包含了所求的根 $r_1$、$r_2$。直线 $L$ 通过 Mobius 变换映射到圆周 $C$ 上。$L$ 的位置和倾角可直接由系数 $c_1$、$c_2$ 计算得出，而 $C$ 则是通过将常数项 $c_2$ 除以 $L$ 上的每个点来构造的。本文描述了二次 LC 方法的技术细节，随后通过一个数值示例展示了二次 LC 方法的工作原理。本文所述的二次 LC 方法虽然比传统的二次公式更复杂，但可扩展用于求 $n \geq 3$ 次的一元多项式根的初始近似值。作为附加内容，本文还研究了连接二次 LC 结构中关键点的直线段的一个有趣性质。
英文摘要：
This paper describes a geometrical method for finding the roots $r_1$, $r_2$ of a quadratic equation in one complex variable of the form $x^2+c_1 x+c_2=0$, by means of a Line $L$ and a Circumference $C$ in the complex plane, constructed from known coefficients $c_1$, $c_2$. This Line-Circumference (LC) geometric structure contains the sought roots $r_1$, $r_2$ at the intersections of its component elements $L$ and $C$. Line $L$ is mapped onto Circumference $C$ by a Mobius transformation. The location and inclination angle of $L$ can be computed directly from coefficients $c_1$, $c_2$, while $C$ is constructed by dividing the constant term $c_2$ by each point from $L$. This paper describes the technical details for the quadratic LC method, and then shows how the quadratic LC method works through a numerical example. The quadratic LC method described here, although more elaborate than the traditional quadratic formula, can be extended to find initial approximations to the roots of polynomials in one variable of degree $n \geq 3$. As an additional feature, this paper also studies an interesting property of the rectilinear segments connecting key points in a quadratic LC structure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
