Space Lieutenant James Johnson hurtled through fleets of enemy ships, blasting them to smithereens as he raced through the edge of the universe.
[Cheesy, macho, deep space lieutenant voice] “These fools,” barked Johnson.
“They always come straight for me!
Either that, or they…sorta…just zig-zag at me…
I…am…an untouchable…machine”
Kris paused the game he’d been working on.
At any moment, the position of each of his enemy ships was given by a set of x and y coordinates.
[Show axes over the computer screen]
And so far, he’d only programmed two paths for the enemy ships.
First, he’d have ships fall vertically down the screen.
And to do that, every half second, Kris made the y coordinates of these ships…
Every half second, Kris made the y coordinates of these ships decrease.
Second, he also made some ships move down in a zig-zag pattern.
To do that, every half second, Kris also changed the…
Kris also changed the x coordinate of some of his ships.
But, his game had become too easy, and Kris needed some enemy ships to start following a more challenging path…
So, he decided to have an all-new fleet of ships trace out a deadly circle – leaving explosive mines for
Space Lieutenant James Johnson to dodge…or be annihilated by…
Now, Kris already knew that the equation of a circle is…
Kris already knew that this is the equation of a circle.
And he knew that he wanted the circle’s radius to be 100 units on screen.
So first, he tried to figure out how he needed to change each ship’s x coordinate every half second.[Press Here]
To do that, he rearranged his equation and got…this [x=±1002-y2]
However… he didn’t know what value to replace y with…
Not deterred, Kris rearranged the equation again to get the value of y
But this time, he got…
This time, Kris found that y was equal to this [y=±1002-x2]
So, Kris needed to know the x value to update the y value…
And the y value to update the x value…
But, he didn’t know either of them!
With his straight and zig-zag paths, this wasn’t a problem: he could add something to his x and y coordinates separately.
But now, the circle equation defined x and y in terms of each other…
If Kris was ever going to fix his game, and give Space Lieutenant James Johnson a real challenge, he’d need to separate out x and y in the circle equation.
And, if he was going to do that, cartesian equations would no longer be enough.
Instead, he would have to learn about a whole new system for building equations…
He would have to understand…parametric equations…
