牛顿直径定理再探：五次曲线的新视角More on Newton’s diameter theorem
John D. Cook 进一步探讨了牛顿直径定理——该定理指出，对于次数为 n 的多项式方程 f(x, y) = 0 所定义的曲线，当绘制多条平行线与其相交于 n 个点时，这些交点中点轨迹的包络线即为曲线的直径。本文聚焦于五次多项式曲线的情况，展示了如何通过几何构造揭示高次代数曲线的对称性与极值特性。这一方法为理解复杂代数簇的结构提供了有力工具。
John
A few days ago I wrote a post on Newton’s diameter theorem. The theorem says to plot the curve formed by the solutions to f(x, y) = 0 where f is a polynomial in x and y of degree n. Next plot several parallel lines that cross the curve at n points and find the centroid of the intersections on each line. Then the centroids will fall on a line.
The previous post contained an illustration using a cubic polynomial and three evenly spaced parallel lines. This post uses a fifth degree polynomial, and shows that the parallel lines need not be evenly spaced.
In this post
f(x, y) = y³ + y − x (x + 1) (x + 2) (x − 3) (x − 2).
Here’s an example of three lines that each cross the curve five times.
The lines are y = x + k where k = 0.5, −0.5, and −3. The coordinates of the centroids are (0.4, 0.9), (0.4, -0.1), and (0.4, -2.6).
And to show that the requirement that the lines cross five times is necessary, here’s a plot where one of the parallel lines only crosses three times. The top line is now y = x + 2 and the centroid on the top line moved to (0.0550019, 2.055).
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