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Drawing real plane algebraic curves in OSCAR

Anne Frühbis-Krüger, Michael Joswig, Lars Kastner

Abstract

We show how the computer algebra system OSCAR can be used to obtain topologically correct or visually pleasing drawings of real plane algebraic curves.

Drawing real plane algebraic curves in OSCAR

Abstract

We show how the computer algebra system OSCAR can be used to obtain topologically correct or visually pleasing drawings of real plane algebraic curves.
Paper Structure (9 sections, 8 figures)

This paper contains 9 sections, 8 figures.

Table of Contents

  1. Introduction
  2. How to draw curves
  3. Schematic rendering
  4. The OSCAR implementation
  5. Elimination
  6. Homotopy continuation
  7. Ordered fields
  8. A More Challenging Example
  9. Outlook

Figures (8)

  • Figure 1: The "apple" of Seidel+Wolpert:2005. Piecewise-linear approximation in red and approximation by degree three Bézier curves in blue. "Diagonal" marked in gray for better comparison with Figure \ref{['fig:apple_graph']}
  • Figure 2: A schematic but topologically correct rendering of the "apple"; cf. Figure \ref{['fig:apple']}. Singular points marked red, other critical points are blue
  • Figure 3: Original illustration copied from gudkov1971. On the left, the product of the ellipses $C_2$ and $\tilde{C_2}$, their perturbation (dotted line) and further lines from the construction. Our construction of this setting is depicted in the rightmost picture.
  • Figure 4: A topologically correct representation of the perturbed curve and the lines from Figure \ref{['figure:gudkov_fig1']} based on explicit computation.
  • Figure 5: Original illustration copied from gudkov1971
  • ...and 3 more figures