Visualization of Points and Segments of Real Algebraic Plane Curves

This thesis presents an exact and complete approach for visualization of segments and points of real plane algebraic curves given in implicit form f(x, y) = 0. A curve segment is a distinct curve branch consisting of regular points only. Visualization of algebraic curves having self-intersection and isolated points constitutes the main challenge. Visualization of curve segments involves even more difficulties since here we are faced with a problem of discriminating different curve branches, which can pass arbitrary close to each other. Our approach is robust and efficient (as shown by our benchmarks), it combines the advantages both of curve tracking and space subdivision methods and is able to correctly rasterize segments of arbitrary-degree algebraic curves using double, multi-precision or exact rational arithmetic.

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