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Real Plane Curves - A Numerical Approach via Mathematica

Barry H. Dayton

Wolfram Technology Conference 2014
Conference location

Champaign, Illinois, USA

Our objects of study are the real solution sets of 2-variable polynomial equations with floating point Mathematica machine numbers as coefficients.  We work primarily in the affine plane with the addition of infinite points of the curve. Among the various procedures available are point finding, including singular points, isolated points, and infinite points with their asymptotes. Fractional Linear Transformations are supported allowing degree 2 and 3 curves to be transformed to canonical form. Infinite points can be transformed to finite points for inspection. There are plotting and decomposition routines for further analysis. Real plane curves can be decomposed into affine topological components, projective components, algebraic components or ovals.  Our analsys of curves is based on the oval decomposition. The present software works well for nicely scaled polynomials up to degree 6 and adequately for poorly scaled polynomials up to degree four.

*Wolfram Technology

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RealNumericalCurves-BarryHDayton.nb (643.6 KB) - Mathematica Notebook