Activity #$2$: More linear interpolation $-$ individual

In the team lab, your team put together a program that interpolated between two position values based on the time values when each positon was observed. This was a one$-$dimensional $(1$D$)$ interpolation, since you were interpolating only a single value, the distance traveled by the ISS. You are now going to extend that program to one that will linearly interpolate between two points in $3$D$.$

Let's say we are tracking the change of a satellite's position with time. So$,$ at time to the position is $(x0,y0,z0)$ and at time t $2$ the position is $(x2,y2,z2).$ What is the position $(x1,y1,z1)$ at some time t$1$ between to and $t2?$

Refer again to the Linear Interpolation material associated with the team lab. That material describes the development of the equation representing linear interpolation of a dependent variable $y$ versus an independent variable $x.$ For the current problem, what varies linearly with what? What are the dependent variable$($s$)?$ What are the independent variable$($s$)?$

Let's assume that each of the position variables $(x,y,z)$ varies linearly with time $(t).$ Therefore, time $(t)$ is the independent variable in each case. This means we can perform linear interpolation three separate times to get what we need. This can be done in three steps: $1)$ linearly interpolate between $(0,x0$ and $(t2,x2)$ for $t1$ with $x1$ as the result; $2)$ repeat for $(0,y0$ and $(t2,y2$ for $t1$ with $y1$ as the result; $3)$ repeat for $(t0,z0)$ and $(t2,z2)$ for $t1$ with $z1$ as the result. The result will be $(x1,y1,z1)$ associated with time $t1.$

a$)$ Write a program named more$\_$linear$\_$interpolation.py that will take two observed $3$D positions at two points in time, and then will calculate the $3$D position at a third point in time. Let's consider only times between the two observed times. You should output the $x,y,$ and $z$ values for that position on separate lines. Begin by identifying the variables you will use, the names for those variables, and the computations that should occur for those variables. Then, write a program that will output the $3$D position of the interpolated point on $3$ separate lines.

For this initial program, use the following data values:

At time $12$ seconds, observed position was $(8,6,7)$ meters

At time $85$ seconds, observed position was $(-5,30,9)$ meters

You want to find the position at time $30$ seconds

Example output:

At time $30\mathrm{.}0$ seconds:

$x1=4\mathrm{.}794520547945206m$

$y1=11\mathrm{.}917808219178081m$

$z1=7\mathrm{.}493150684931507m$

b$)$ Now, add to your file more$\_$linear$\_$interpolation.py from above. Modify your program in the following ways:

i$.$ When printing the position, follow the output by a line of dashes $("$

ii$.$ Instead of just computing the interpolation at one point and printing the result, you will now compute it at $5$ points. You may copy and paste the portion of your code that is needed to recompute interpolation $5$ times.

iii. Create variables for the starting time of interpolation, and the ending time of interpolation. Display the results from interpolating at $5$ points, evenly spaced from the beginning time to the ending time, inclusive.

iv$.$ Interpolate, starting at time $30$ seconds and ending at time $60$ seconds, printing the result each time. The line of dashes will separate each computation. $($Note: later we will see how we can do this more efficiently, without copying$-$and$-$pasting code, but for now, copy$-$and$-$paste your code.$)$

Example output $($first two times only$)$:

At time $30\mathrm{.}0$ seconds:

$x1=4\mathrm{.}794520547945206m$

$y1=11\mathrm{.}917808219178081m$

$z1=7\mathrm{.}493150684931507m$

At time $37\mathrm{.}5$ seconds:

$x2=3\mathrm{.}4589041095890414m$

$y^2=14\mathrm{.}383561643835616m$

$z2=7\mathrm{.}698630136986301$m

$(($PLEASE SOLVE IT SO IT CAN PRODUCED THESE OUTPUTS $))$