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We consider the question of characterizing the behavior of parametric curves whose components are cubic polynomials. When there is no chance of confusion, we will refer to such curves as cubic curves with the understanding that each of x(t) and y(t) are themselves cubic polynomials. We classify various types of parametric cubics using their defining(More)
Sometimes an ordinary differential equation (ODE) solver gives the results it should, even if they are unexpected or undesirable. To obtain correct, expected numerical results, both relative and absolute error control tolerances must be chosen judiciously. Here we attempt to clear up some possible misunderstandings regarding the performance of a popular ODE(More)
This paper explores some of the basic and most interesting facts about quadric surfaces. It describes the canonical coordinate transformations required to eliminate cross terms from the equation of a general quadric equation. It explains how to use these coordinates to obtain each of the seventeen canonical quadrics. It further describes the determination(More)
Moving averages of the solution of an initial value problem for a system of ordinary differential equations are used to extract the general behavior of the solution without following it in detail. They can be computed directly by solving delay differential equations. Because they vary much less rapidly and are smoother, they are easier to compute.