$\%$ For a complete ellipse:

a $=3$;

b $=2$;

BP$\_$full $=$ arc$\_$length$\_$ellipse$($a$,$ b$,2*$pi$)$; $\%$ For complete ellipse $(360$ degrees$)$

disp$([\text{'}$Complete arc length: $\text{'},$ num$2$str$($BP$\_$full$)])$;

$\%$ For a segment of $60$ degreew

BP$\_$segment $=$ arc$\_$length$\_$ellipse$($a$,$ b$,$ pi$/3)$; $\%$ For $60$ degrees

disp$([\text{'}$Arc length for $60$ degrees: $\text{'},$ num$2$str$($BP$\_$segment$)])$;

$\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%$

$\%$ MATLAB Function $($Simpson$\text{'}$s Rule$)$:

$\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%$

function fx$\_$integral $=$ simpson$34($x$\_$min, x$\_$max, fx$\_$array$)$

$\%$ This function implements Simpson's Rule $(1/3$ and $3/8)$

$\%$ Inputs:

$\%$ x$\_$min: lower limit of integration

$\%$ x$\_$max: upper limit of integration

$\%$ fx$\_$array: array of function values at equally spaced points

$\%$ Output:

$\%$ fx$\_$integral: the integral value $($area under the curve$)$

$\%$ Number of intervals $($n must be even$)$

n $=$ length$($fx$\_$array$)-1$;

h $=($x$\_$max $-$ x$\_$min$)/$ n; $\%$ Step size

$\%$ Check if the number of intervals is even or odd

if mod$($n$,2)==0$

$\%$ Use Simpson's $3/8$ rule for the first three points, then $1/3$ rule

simpson$38\_$part $=(3*$ h $/8)*($fx$\_$array$(1)+3*$fx$\_$array$(2)+3*$fx$\_$array$(3)+$ fx$\_$array$(4))$;

$\%$ Apply Simpson's $1/3$ rule for the rest

simpson$13\_$part $=($h $/3)*($fx$\_$array$(4)+4*$ sum$($fx$\_$array$(5$:$2$:end$-1))+2*$ sum$($fx$\_$array$(6$:$2$:end$-1))+$ fx$\_$array$($end$))$;

fx$\_$integral $=$ simpson$38\_$part $+$ simpson$13\_$part;

else

$\%$ Apply Simpson's $1/3$ rule to all points if odd number of points

fx$\_$integral $=($h $/3)*($fx$\_$array$(1)+4*$ sum$($fx$\_$array$(2$:$2$:end$-1))+2*$ sum$($fx$\_$array$(3$:$2$:end$-2))+$ fx$\_$array$($end$))$;

end

end

$\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%$

$\%$ MATLAB Code for Arc Length:

$\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%\%$

function BP $=$ arc$\_$length$\_$ellipse$($a$,$ b$,$ phi$)$

$\%$ Function to compute arc length of ellipse using Simpson's rule

$\%$ Inputs:

$\%$ a: semimajor axis

$\%$ b: semiminor axis

$\%$ phi: maximum angle in radians $($for a segment of the ellipse$)$

$\%$ Number of data points $($increase for higher accuracy$)$

n$\_$points $=1000$;

$\%$ Create equally spaced points between $0$ and phi

phi$\_$values $=$ linspace$(0,$ phi, n$\_$points$)$;

$\%$ Define the integrand function

integrand $=$ sqrt$(1-(($a$^2-$ b$^2)/$ a$^2)*$ sin$($phi$\_$values$).^2)$;

$\%$ Use simpson$34$ to compute the integral

BP $=$ a $*$ simpson$34(0,$ phi, integrand$)$;

end