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Консультация № 185180
14.01.2012, 22:55
65.45 руб.
0
3
2
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Линия задана параметрически:
x=2*(t-cost)
y=2*(1-sint)
Требуется найти уравнение этой линии в виде F(x, y) и построить ее в прямоугольной декартовой системе координат.
Линия задана параметрически:
x=2*(t-cost)
y=2*(1-sint)
Требуется найти уравнение этой линии в виде F(x, y) и построить ее в прямоугольной декартовой системе координат.
Обсуждение
14.01.2012, 23:38
общий
это ответ
Здравствуйте, Александра!
Так задается циклоида с радиусом 2.
Выразим t через y
1-sint=y/2
sint=1-y/2
t=arcsin (1-y/2)
t-cos t=x/2
cost=t-x/2
(1-y/2)2+(t-x/2)2=1
(1-y/2)2+(arcsin (1-y/2)-x/2)2=1
На рисунке изображена циклоида
x = rt - rsint,
y = r - rcost;
Наша циклоида такая же, но со сдвигом и перевернутая.
поворотные точки будут при y=0, то есть в точках, где sin t=1
t=pi/2+2pi n
x=2(pi/2+2pi n)=pi+4pin
Так задается циклоида с радиусом 2.
Выразим t через y
1-sint=y/2
sint=1-y/2
t=arcsin (1-y/2)
t-cos t=x/2
cost=t-x/2
(1-y/2)2+(t-x/2)2=1
(1-y/2)2+(arcsin (1-y/2)-x/2)2=1
На рисунке изображена циклоида
x = rt - rsint,
y = r - rcost;
Наша циклоида такая же, но со сдвигом и перевернутая.
поворотные точки будут при y=0, то есть в точках, где sin t=1
t=pi/2+2pi n
x=2(pi/2+2pi n)=pi+4pin
15.01.2012, 02:23
общий
это ответ
Выразим из второго уравнения:
sin(t) = (2-y)/2
cos(t) = [$8730$](4y-y2)/2
Тогда из первого уравнения получаем:
x = 2 arcsin((2-y)/2) - [$8730$](4y-y2) = - 2 arccos((2-y)/2) + [tnr][$960$][/tnr] - [$8730$](4y-y2)
Получилась перевернутая циклоида, сдвинутая на [tnr][$960$][/tnr]
(Обычная циклоида имеет вид x = 2 arccos((2-y)/2) - [$8730$](4y-y2))
sin(t) = (2-y)/2
cos(t) = [$8730$](4y-y2)/2
Тогда из первого уравнения получаем:
x = 2 arcsin((2-y)/2) - [$8730$](4y-y2) = - 2 arccos((2-y)/2) + [tnr][$960$][/tnr] - [$8730$](4y-y2)
Получилась перевернутая циклоида, сдвинутая на [tnr][$960$][/tnr]
(Обычная циклоида имеет вид x = 2 arccos((2-y)/2) - [$8730$](4y-y2))
Об авторе:
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
"Если вы заметили, что вы на стороне большинства, —
это верный признак того, что пора меняться." Марк Твен
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