Curve Sketching - Newton's Diagram

Newton's diagram (also known as Newton's parallelogram, after Isaac Newton) is a technique for determining the shape of an algebraic curve close to and far away from the origin. It consists of plotting (α, β) for each term Axαyβ in the equation of the curve. The resulting diagram is then analyzed to produce information about the curve.

Specifically, draw a diagonal line connecting two points on the diagram so that every other point is either on or to the right and above it. There is at least one such line if the curve passes through the origin. Let the equation of the line be qα+pβ=r. Suppose the curve is approximated by y=Cxp/q near the origin. Then the term Axαyβ is approximately Dxα+βp/q. The exponent is r/q when (α, β) is on the line and higher when it is above and to the right. Therefore, the significant terms near the origin under this assumption are only those lying on the line and the others may be ignored it produce a simple approximate equation for the curve. There may be several such diagonal lines, each corresponding to one or more branches of the curve, and the approximate equations of the branches may be found by applying this method to each line in turn.

For example, the folium of Descartes is defined by the equation

.

Then Newton's diagram has points at (3, 0), (1, 1), and (0, 3). Two diagonal lines may be drawn as described above, 2α+β=3 and α+2β=3. These produce

as approximate equations for the horizontal and vertical branches of the curve where they cross at the origin.