Finding Rational Parametric Curves of Relative Degree One or Two

Abstract

SummaryA plane algebraic curve, the complete set of solutions to a polynomial equation f(x,y)=0, can in many cases be drawn using parametric equations x= x(t), y= y(t). Using algebra, attempting to parametrize by means of rational functions of t, one discovers quickly that it is not the degree of f but the “relative degree,” that describes how difficult the computations become. When the relative degree is one, the parametrization technique is well-known (and quite simple). When it is two, solutions can still be directly computed using the quadratic formula. Here, we demonstrate a general method for relative degree two, focusing on specific examples.