Project $1-$ Parametric Equations

In a web browser open up a desmos graphing window. If you haven't already, then sign up for a desmos account.

Inside a desmos folder $($name it Parameterized Line Segment$),$ do the following tasks.

$($a$)$ Define $a=1,b=2,c=3,$ and $d=1$ in $4$ separate cells.

$($b$)$ Plot the point $(a,b)$ in green and the point $(c,d)$ in red. Observe that you can drag the points wherever you like.

$($c$)$ Define $F(t)=(1-t)a+tc$ to represent the $x$ coordinate of the line segment. Similarly, define $G(t)$ to represent the $y$ coordinate of the line segment. Restrict the domain on both functions for $0t1.$ Then hide both curves. Plot the parameterized line segment $(F(t),G(t)).$

$($d$)$ Compute the slope of the line segment in terms of $a,b,c,$ and $d$ :

$m=\frac{1}{(2{)}_{,}}\frac{1-2}{3-1}=-\frac{1}{2}$

$($e$)$ Plot the "road" on which the line segment rests using rectangular coordinates $x$ and $y($and parameters $m,a,$ and $b)$ as a dashed line with thickness $1\mathrm{.}5.$

$($f$)$ Define slider $A=0,$ and give it domain $0A1.$ Click the "cycle" icon $([vec(larr)])$ on the left and change to the "forward" icon $(\frac{}{}).$

$($g$)$ Plot a "particle" that travels along the line segment with $(F(A),G(A)).$ Click "play" on the slider for $A.$ Slow down the animation to $"0\mathrm{.}5x"$

$($h$)$ Label the point by typing $"t=$$$\{A\}"$ with $1\mathrm{.}5$ fontsize.

$($i$)$ Drag the $A=$ slider outside above the folder.

$($j$)$ Then minimize this folder and hide it$.$

$2.$ Inside a desmos folder $($name it Cubic Bezier Curve$),$ do the following tasks.

$($Read about Bezier curves on page $623$ of your textbook.$)$

$($a$)$ Define $a_i$ and $b_i$ for $i=0,1,2,3.$ Plot each point $(a_i,b_i)$ for $i=0,1,2,3,$ with connecting line segments. You should be able to drag any point as desired. Label the points $P_0,P_1,P_2,$ and $P_3$ accordingly.

$($b$)$ Define functions $f(t)$ and $g(t)$ as the parametric equations for the bezier curve $($Section $11\mathrm{.}1,$ p $623).f(t)$ will represent the $x$ coordinate, and $g(t)$ will represent the $y$ coordinate. Then "hide" them by clicking the colored circles.

$($c$)$ Plot the parametric curve defined by $(f(t),g(t))$ for $0t1.$