I cannot believe that someone has so much time, either that or your a genius. But seriously this is actually amazing post to lwaiy. I Was watching the logo and then as I was looking this up it hit the corner and I missed it!
Could you find a function for the position vector given time and the starting angle if you were to model the screen as unpixellated and continuous? Or would this be chaotic? In the TV show The Office , there is a humorous scene where all the employees, who are supposed to be paying attention to their boss who is leading a meeting, are watching the bouncing DVD logo on the television screen behind him instead.
They are all hoping that the bouncing DVD logo will move in and out of one of the exact corners of the television screen, and show disappointment every time the logo comes close but just misses it. Meanwhile, their boss, oblivious to all of this, thinks that all of their reactions to the bouncing DVD logo are actually reactions to his agenda. This begs the question: can we mathematically predict when the logo will hit one of the corners?
Let us first consider some simplified examples, and then draw some generalizations from there. Here is a 1 x 1 pixel logo moving inside of a 16 x 10 pixel screen, where the total distance traveled by the logo is recorded in the bottom right:.
But why the numbers 9 and 15? Therefore, a bouncing logo will reach a corner at the least common multiple of the difference of the screen and logo heights and the difference of the screen and logo widths, or.
We can test our equation by looking at a bigger 4 x 2 pixel logo moving inside the same 16 x 10 pixel screen:. Another way to explain the bouncing logo equation is to picture its path as a straight line through a series of reflections. When the logo hits the top or bottom wall, a horizontal mirror along the half-height-logo margin would reflect its rebounded path in a straight line to the original.
Likewise, when the logo hits the left or right wall, a vertical mirror along the half-width-logo margin would reflect its rebounded path in a straight line to the original. We can see that the logo will hit the corner when the reflected screens can be combined to make a square. In other words, the distance and time of one cycle is proportionate to the distance and time of one screen length:. This means that the time it takes for a bouncing logo to cycle through one corner to the next can be calculated with five variables: the time it takes for it to move across one width of the screen, the screen height, the screen width, the logo height, and the logo width.
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