Rational Parameterization as a Method for Eliminating Numerical Errors in Graph Plotting

This article addresses the issue of distortion in mathematical function graphs defined in polar or parametric coordinates. The author notes that calculations near the origin (0,0) often result in significant rounding errors comparable to the function values themselves. Additionally, certain functions, such as the square root, exhibit discontinuities or undefined regions near zero, further complicating accurate visualization. To address this, the author proposes a method of rational parameterization, which reformulates mathematical expressions to minimize the impact of computational errors. The article demonstrates how mathematical transformations can achieve more precise and stable graph rendering, avoiding artifacts caused by the limitations of machine arithmetic. This approach is particularly useful for developers of graphics libraries and data visualization systems who need to accurately plot complex curves.