Спирала

Тази статия е за математическата крива. За други значения вижте Спирала (пояснение).

Спиралата (от гр. σπειρα – „извивка“, от лат. spira – „извивка“, „пръстен“) е равнинна трансцендентна крива, която описва намотки около една (или повече) точка, приближавайки или отдалечавайки се от нея. За целта функцията, с която е зададена спиралата, трябва да е монотонна функция. Спиралата е мощен символ още от времената на древното мегалитно изкуство; тройната спирала играе значителна роля в келтската символика.

Има два вида спирали: алгебрични спирали и псевдоспирали. По повод терминологията е нужно следното уточнение: всички видове спирали са трансцендентни криви, за разлика от алгебричните криви, които се задават с полиноми.

Спиралите обикновено се задават в полярни координати чрез функции, които показват връзката между ъгъла на въртене $\theta$ и радиус-вектора на спиралата r. Ако от своя страна обаче тази функция има вид на полином, спиралата се нарича алгебрична спирала. Псевдоспиралите могат да се изразят с уравнения от вида $k=as^{m}$ , където k е радиусът на кривината, s е дължината на дъгата, отчитана от началото на спиралата, а a е коефициент.

Алгебрични спирали	Псевдоспирали
Архимедова спирала: $r=a+b\theta$ Спирала на Галилей: $r=a\theta ^{2}-d$ Спирала на Ферма: $r=a{\sqrt {\theta }}$ Параболична спирала: $r=a{\sqrt {\theta }}+d$ Хиперболична спирала: $r={\frac {a}{\theta }}$ Жезъл на Коутс: $r={\frac {a}{\sqrt {\theta }}}$	Логаритмична спирала: $r=ae^{b\theta }\;$ или $k=as$ Клотоида: $k={\frac {a}{s}}$

Коефициентите a, b, d в горните формули се отразяват на графиката на спиралите, като определят единствено посоката на въртене и разстоянието между две съседни намотки.

Вижте също

Винтова линия – аналог на спиралата в тримерното пространство

Източници

„Математический энциклопедический словарь“, Ю. В. Прохоров, „Советская энциклопедия“, Москва, 1988
„Математическая энциклопедия“ (5 тома), Изд. „Советская энциклопедия“, 1985
"The Penguin Dictionary of Mathematics", John Daintith, R.D. Nelson, Penguin Books, 1989

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