论文标题
信封正在解决四边形和立方体以及任意程度的某些多项式的求解机
Envelopes are solving machines for quadratics and cubics and certain polynomials of arbitrary degree
论文作者
论文摘要
每个人都在学校知道如何求解$ x^2-px+q = 0 $的二次方程式。但是，如果应该解决多个方程式，则这种方法可能会变得乏味，因为对于每对$（p，q）$，必须绘制一个新的抛物线。令人惊叹的是，有一条曲线可用于通过给定的点$（P，Q）$绘制切线线（P，Q）$来求解每个二次方程。 在本文中，我们以基本的方式得出了此方法，并将其推广到$ x^n-px+q = 0 $的等式，以任意$ n \ ge 2 $。此外，该技术可以立即看到此形式的特定方程式的解决方案的数量。最后，我们指出了与平面中点和线的二元性以及Legendre转换概念的联系。
Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought to be solved, as for each pair $(p,q)$ a new parabola has to be drawn. Stunningly, there is one single curve that can be used to solve every quadratic equation via drawing tangent lines through a given point $(p,q)$ to this curve. In this article we derive this method in an elementary way and generalize it to equations of the form $x^n-px+q=0$ for arbitrary $n \ge 2$. Moreover, the number of solutions of a specific equation of this form can be seen immediately with this technique. Concluding the article we point out connections to the duality of points and lines in the plane and to the the concept of Legendre transformation.
