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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 produces a simple approximate equation for the curve. There may be several such
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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
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is the degree of the curve, is added to form a triangle which contains the diagram. This method considers all lines which bound the smallest convex polygon which contains the plotted points (see
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If the curve passes through the origin then determine the tangent lines there. For algebraic curves, this can be done by removing all but the terms of lowest order from the equation and solving.
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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.
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given its equation, without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features.
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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
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Similarly, removing all but the terms of highest order from the equation and solving gives the points where the curve meets the
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of the curve. Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve.
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as approximate equations for the horizontal and vertical branches of the curve where they cross at the origin.
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in the equation of the curve. The resulting diagram is then analyzed to produce information about the curve.
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The following are usually easy to carry out and give important clues as to the shape of a curve:
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respectively. If the equation of the curve cannot be solved explicitly for
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in each term is always even or always odd, then the curve is
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extended Newton's diagram to form a technique called the
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Determine the symmetry of the curve. If the exponent of
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equal to 0 in the equation of the curve and solving for
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equal to 0 in the equation of the curve and solving for
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Hilton, Harold (1920). "Chapter III: Curve-Tracing".
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is always even in the equation of the curve then the
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is always even in the equation of the curve then the
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276:. Suppose the curve is approximated by
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177:Determine any bounds on the values of
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229:, finding these derivatives requires
59:value of the red, or the blue, curve
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365:{\displaystyle x^{3}+y^{3}-3axy=0\,}
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127:intercepts are found by setting
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284:near the origin. Then the term
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465:{\displaystyle y^{2}-3ax=0\,}
418:{\displaystyle x^{2}-3ay=0\,}
111:intercepts of the curve. The
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664:Encyclopedia of Mathematics
307:is defined by the equation
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168:symmetric about the origin
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657:Trenogin, V.A. (2001) ,
638:Frost, Percival (1918).
231:implicit differentiation
16:Not to be confused with
479:The analytical triangle
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27:Graph of the function 3
18:Digital curve sketching
630:Plane Algebraic Curves
552:Numerical continuation
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247:Newton's parallelogram
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669:EMS Press
633:. Oxford.
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679:Category
526:See also
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77:geometry
61:vanishes
205:Equate
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532:Curve
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51:. An
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