More on Newton’s diameter theorem

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).

The post More on Newton’s diameter theorem first appeared on John D. Cook.