A superellipse is defined by the inequality

$|($x$)/($a$)|^($n$)+|($y$)/($b$)|^($n

where x and y are the Cartesian coordinates, and a and b are the length of long and short axes

with n$>2$ the deformation parameter. An example of a superellipse centered at r$\_(0)$ with a$=$

$2,$b$=1$ and n$=2\mathrm{.}5$ is shown in the figure below, where e$\_($x$)$ and e$\_($y$)$ are the unit vectors

pointing to the direction of long and short axes, respectively, and e$\_($x$)\_(|)\_($e$)\_($y$).$ All vectors here

are column vectors.

Questions:

Write a program "CalVolume $($a$,$b$,$n$)"$ to numerically calculate the area S of a

superellipse for given a and b$.$ For a$=2,$b$=1,$ plot S as a function of nin$[2,10],$

and compare your result with the analytical formula

S$=((4^(1-(1)/($n$)))/($a$)$bs qrt$\backslash$pi $\backslash$Gamma $(1+(1)/($n$)))/(\backslash$Gamma $((1)/(2)+(1)/($n$)))$

where $\backslash$Gamma $(*)$ is the Gamma function.

$(20$ marks$)$

Write a Matlab program "DrawSuperellipse $($r$\_(0),$a$,$b$,$n$,$e$\_($x$),$e$\_($y$))"$to draw a superellipse

entered at r$\_(0)$ for a given a$,$b$,$n$,$e$\_($x$)$ and e$\_($y$).$ The input of the program is

$\{$r$\_(0),$a$,$b$,$n$,$e$\_($x$),$e$\_($y$)\}.$

$(20$ marks$)$

Write a program "Distance $($r$\_(1),$a$\_(1),$b$\_(1),$n$\_(-),$e$\_($x$1),$e$\_($y$1),$r$\_(2),$a$\_(2),$b$\_(2),$n$\_(),$e$\_($x$2),$e$\_($y$2))"$ using the

Newton's method to calculate the shortest distance between two points on two

superellipses $\{$r$\_(1),$a$\_(1),$b$\_(1),$n$\_(1),$e$\_($x$1),$e$\_($y$1)\}$ and $\{$r$\_(2),$a$\_(2),$b$\_(2),$n$\_($n$),$e$\_($x$2),$e$\_($y$2)\}$ where a$\_(1)=$b$\_(1)=$

a$\_(2)=$b$\_(2)=1$ for $2.$ The program should return a value of $0$ if two

superellipses are overlapped. Use your program to calculate the shortest distance

between two superellipses with the location and orientation listed in Table $1,$ and draw

the two closest points on the two superellipses.

Table $1$

Please assist in coming up with the matlab script for Question $3$ only using Newton Method only $(1$D OR $2$D$),$ thank you so much in advance. The correct shortest distances are approximately $1\mathrm{.}9178$ units when n $=2\mathrm{.}5,0\mathrm{.}64638$ units when n$=3\mathrm{.}0,0\mathrm{.}33277$ units when n$=3\mathrm{.}4.$ Maybe come up with matlab scripts of first derivative $,$ second derivative $,$ distance $,$ newton $,$ starting point, transform $,$ closestpoint,etc?