Answer to Question #316183 in Differential Geometry | Topology for sket

Question

sketch the astroid in curves also calculate its tangent vector at each point .at which point is the tangent vector zero:

(i) γ(t) = (cos^2 t, sin^2 t)

(ii) γ(t) = (e^t, t^2)

Answer 1

$x\left( t \right) =\cos ^2t,y\left( t \right) =\sin ^2\left( t \right) \Rightarrow x+y=1,x,y\in \left[ 0,1 \right]$

The tangent vector  is 0 if $\left\{ \begin{array}{c} x'\left( t \right) =0\\ y'\left( t \right) =0\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} 2\sin t\cos t=0\\ 2\sin t\cos t=0\\\end{array} \right. \Rightarrow t=\frac{\pi n}{2}$

$x\left( t \right) =e^t,y\left( t \right) =t^2\Rightarrow y\left( x \right) =\ln ^2x$

The tangent vector is 0 if $\left\{ \begin{array}{c} x'\left( t \right) =0\\ y'\left( t \right) =0\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} e^t=0\\ 2t=0\\\end{array} \right. \Rightarrow t\in \emptyset$