1.2.1 The Cycloid Problem

The classic cycloid curve is generated by tracing the path of fixed point on the rim of a circle as it rolls on a flat surface. When the circle has radius one, the curve is parameterized by the path [math]\vec{c}\left(t\right)=\left(t-\sin t,1-\cos t\right),t\in\mathbb{R}[/math]

Which of the following describes [math]\vec{c}\left(t\right)[/math]?

[math]\vec{c}\left(t\right)[/math] is differentiable. [math]\vec{c}\left(t\right)[/math] is injective. [math]\vec{c}\left(t\right)[/math] is regular. The image curve of [math]\vec{c}\left(t\right)[/math] is closed. The image curve of [math]\vec{c}\left(t\right)[/math] has self-intersection points The image curve of [math]\vec{c}\left(t\right)[/math] is simple.

Define a vector as follows:[br][math]\vec{v}_h\left(t\right)=\left(x'\left(t\right),0\right)[/math].[br]Describe this vector: [br][list][*]What does this vector look like? [/*][*]Is it defined at all points along the path? [/*][*]When is it long? [/*][*]How long does it get? [/*][*]When does it vanish? [/*][*]In which direction does it point at each value of [math]t[/math]?[/*][/list]

Repeat the previous exercise for the vector [math]\vec{v}_v\left(t\right)=\left(0,y'\left(t\right)\right)[/math].

1.2.1 The Cycloid Problem

Information: 1.2.1 The Cycloid Problem