Synchronized Translations $(30$ points$)$ In this problem, we will write a function

called shift.m which allows us to translate a three$-$dimensional object along the x$,$ y$,$ or

z axis by some specified amount. We will use this function to create a cool movie showing

moving objects.

On Canvas, there are files called setup.m and show$2.$m$.$ These generate and plot an

object that we$$ll translate in various ways, using show$2.$m to visualize the results. It isn$$t

important to understand the details of the commands in these files, just how to use them.

For example, if you download these files then type

setup

show$2($T$,$S$)$

you$$ll see the object. Note that we will always call show$2.$m with the same variable T as

the first argument. The second argument will be a translation of S obtained through a

call to the function that you$$ll write.

a$)(10$ pts$)$ Write a function shift.m which takes as input a $3\backslash$times n matrix S $($for example,

as obtained from running setup.m$),$ a scalar dist which gives the distance that the object

should translate, and a string $$x$,$y$,$ or $$z$$ which tells which axis you want to translate

along.

This function should return a matrix of the same size as S$,$ but with all elements in

the first row equal to their original values plus dist if the input string is $$x$,$ all elements

in the second row equal to their original values plus dist if the input string is $$y$,$ and

all elements in the third row equal to their original values plus dist if the input string is

$$z$.$

These actions correspond to translations of the object along the specified axis. For

example, the commands

setup

show$2($T$,$shift$($S$,80,$x$))$

should show the object translated by a distance of $80$ along the x$-$axis.

b$)(20$ pts$)$ Here you will write Matlab script called shift$\_$movie.m which shows a movie

with three objects moving in the following ways. Except for the first one, each of these

steps should show multiple frames, which when shown in succession give a movie.

$$ The three objects start on top of each other at their default position.

$$ One object translates by increasing amounts in the $+$x direction, and one translates

by the same magnitudes in the $$x direction, until they are a distance of $80$ away

from their original position.

$$ The object which was translated in the $$x direction should now translate by in$-$

creasing amounts in the $+$z direction up to a height of $80.$ Simultaneously, the

object which was translated in the $+$x direction should translate by the same mag$-$

nitudes in the $$z direction down to z $=80.$ These translations in the z direction

should occur with the objects maintaining their new x coordinate.

$$ This previous step should be reversed, so that the objects return to their original

height while maintaining their new x coordinate.

$$ The objects should return back to their original positions through appropriate trans$-$

lations along the x direction.

To make the movie, you can use successive application of code such as the following:

for counter $=$ start:increment:stop

show$2($T$,$S$)$

hold on

show$2($T$,$shift$($S$,...)$;

show$2($T$,$shift$($S$,...))$;

pause$(0\mathrm{.}05)$

clf

end

the given code for show$2.$m is

function show$2($T$,$S$)$

trisurf$($T$,$S$(1,$:$),$S$(2,$:$),$S$(3,$:$),$'FaceColor','cyan','FaceAlpha', $0\mathrm{.}8)$;

axis$([-120120-120120-120120])$

xlabel$(\text{'}$x$\text{'})$

ylabel$(\text{'}$y$\text{'})$

zlabel$(\text{'}$z$\text{'})$

end

The given code for setup.m is

$\%$tet$1.$m

clear;clc;

load tetmesh

TR $=$ triangulation$($tet$,$X$)$;

$[$FBtri$,$FBpoints$]=$ freeBoundary$($TR$)$;

T $=$ FBtri;

S $=$ FBpoints' $-[$zeros$(1,1079)$;zeros$(1,1079)$;$40*$ones$(1,1079)]$;