Abstract
A method for constructing a directional cubic spline for a set of points on a plane is formulated and proposed. The spline is compared with the Schoenberg B-spline, Akima and Catmull–Rom splines. It is shown that for unequally spaced points, in comparison with the B-spline, it gives significantly lower overshoots and is practically free of strong kinks, which are characteristic of Akima splines. The spline does not give loops and oscillations, which are a characteristic drawback of parametric splines, in particular, Hermitian ones, which include the Catmull–Rom spline. A fast method for optimizing the spline guiding coefficient is proposed, the purpose of which is to minimize the discontinuities of the second derivative of the function at its intermediate points. An example of optimization of a directional third-order spline is given. A fourth-order directional spline, which is free of kinks, is also proposed. The method of optimization of the directional spline of the fourth order is formulated, the algorithm of its optimization is stated. The optimization criteria are the spline length and the smallest distance between its global maximum and minimum. It is shown that, in comparison with the Schoenberg spline, the fourth-order directional spline has smaller outliers. A method for automatic blunting of sharp peaks of curves is proposed, which can be applied to all types of splines.
Highlights
Для наборов с малым числом точек удовлетворительные результаты дают интерполяционные методы Лагранжа, Ньютона, Стирлинга [2,3,4]
A fast method for optimizing the spline guiding coefficient is proposed, the purpose of which is to minimize the discontinuities of the second derivative of the function at its intermediate points
The method of optimization of the directional spline of the fourth order is formulated, the algorithm of its optimization is stated
Summary
Оптимизация преследует цель сделать менее заметными изломы направленного сплайна, которые определяются абсолютной величиной разности значений вторых производных в точке сопряжения соседних полиномов. Вычислительный эксперимент показал, что в большинстве случаев при крайних значениях направляющего коэффициента α = 0 и α = 1, когда углы касательной к кривой в промежуточных точках и одного из отрезков линейного сплайна совпадают, направленный сплайн обычно имеет не только выраженные изломы, но и большие выбросы. Найдем значение функции A. y = D(A. x) и A. z = [D(A. x + ε) − A. y]/ε на левом конце интервала поиска. Что минимум функции находится в точке αSTU = x = 0,425. Поскольку ожидалось, что якобы имеющий преимущества при интерполяции монотонных функций, а именно таковой в данном примере является функция y(x), сплайн Акимы должен был бы демонстрировать лучшие показатели не только в сравнении с направленным сплайном, но и с Bсплайном. На стыках соседних отрезков [x,+(, x,] для полиномов (3) потребуем выполнения условий непрерывности сплайна и ее первой и второй производных в точках (1). Далее коэффициенты c,, e, можно найти по формулам (17), (18)
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