Let f(x, y) be an nth diploma polynomial in x and y. Basically, a straight line will cross the zero set of f in n places [1].
Newton outlined a diameter to be any line that crosses the zero set of f precisely n occasions. If
f(x, y) = x² + y² − 1
then the zero set of f is a circle and diameters of the circle within the ordinary sense are diameters in Newton’s sense. However Newton’s notion of diameter is extra normal, together with strains the cross the circle with out going by the middle.
Newton’s theorem of diameters says that when you take a number of parallel diameters (in his sense of the phrase), the centroids of the intersections of every diameter with the curve f(x, y) = 0 all line on a line.
As an example this theorem, let’s take a look at the elliptic curve
y² = x³ − 2x + 1,
i.e. the zeros of f(x, y) = y² − (x³ − 2x + 1). It is a third diploma curve, and so typically a straight line will cross the curve thrice [2].
The orange, inexperienced, and pink strains are parallel, every intersecting the blue elliptic curve thrice. The dot on every line is the centroid of the intersection factors, the middle of mass when you think about every intersection to be a unit level mass. The centroids all lie on a line, a vertical line on this instance although typically the road might have any slope.
I hadn’t seen this theorem till I ran throughout it lately when skimming [3]. Search outcomes recommend the theory isn’t extensively recognized, which is shocking for a consequence that goes again to Newton.
Associated posts
[1] Bézout’s theorem says a curve of diploma m and a curve of degee n will at all times intersect in mn factors. However that features advanced roots, provides a line at infinity, and counts intersections with multiplicity. So a line, a curve of diploma 1, will intersect a curve of diploma n at n factors on this prolonged sense.
[2] See the outline of Bézout’s theorem within the earlier footnote. Within the elliptic curve instance, the parallel strains meet at some extent at infinity. A line that misses the closed element of the elliptic curve and solely passes by the second element has 1 actual level of intersection however there could be 2 extra if we had been working in ℂ² slightly than ℝ².
In algebraic phrases, the system of equations
y² = x³ − 2x + 1
3y = 2x + ok
has three actual options for small values of ok, however for sufficiently giant values of |ok| two of the options will likely be advanced.
[3] Arithmetic: Its Content material, Strategies, and That means. Edited by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent’ev. Quantity 1.
