Looking at the T-Shape I will now explain a few aspects of it, firstly the ‘a’, you add this because I am moving it by that amount or it could be said I am moving it by ‘a’ in other words a general formula for how many squares across I am moving it by. If that made any sense I will now explain why you take ‘gb’ from the formula. This is because I am moving it by the number that ‘b’ represents times by the grid size that ‘g’ represents in the formula, BUT you are decreasing by ‘g’, so as you move the T-Shape up or down it is taken away.
Now as in the previous section of the coursework I simplified each of the squares of the T-Shape into one, simple equation. So as before I will add all the mini formula up and simplify them into one straight forward ‘general’ formula that will work for any sort of translation the T-Shape might make. Now here is the formula:
(t – 2g – 1 + a – gb) + (t – 2g + a – gb) + (t – 2g + 1 + a – gb) + (t – g + a – gb) + (t+ a – gb)
When simplified it looks like this:
5t – 7g + 5a – 5gb
So as you will see I have made a simple formula to show in ‘general’ everything you will ever need when translating T-Shapes. Now all that’s left to do is to prove that my formula works and what better way to do so than by testing it with completely random data. If I pick a random sized grid, say 12 x 12 and then pick a totally random T-Number, say 31, then pick a totally random vector to translate the T-Shape by, say (2, -3). Through extensive testing I have worked out that the T-Number after the T-Shape has been translated is 69. Now lets see if I can prove that using my trusty formula:
t = 31 so the start of the equation will be (5 x 31) which equals 155.
g = 12 so the second part of the equation will be (7 x 12) which equals 84.
a = 2 so the third part of my equation will be (5 x 2) which equals 10.
gb = 12 x –3 so the fourth part of my equation will be 5(12 x –3) which equals
–180.
So, now by adding all the numbers up and putting them into an equation I will be able to prove that my prediction of the T-Number is correct:
(5 x 31) + (7 x 12) + (5 x 2) + 5(12 x –3) = T-Number
155 + 84 + 10 + -180 = 69
Now just in case you have doubts in the back of your mind that I just made up that result, then below I will back up the evidence by drawing out the 12 x 12 grid and drawing on the translation of the T-Shape step-by-step. I will now also explain another aspect of translating the T-Shape; a vector is very simple and works in the much the same way as a graph. The first way it works like a graph is the direction in which the T-Shape is translated and how we know to translate it in that way. On a graph, if a number on the x-axis (or in translating terms the ‘a’ axis) then the number is moved to the right of the centre line, and if the number is negative it is moved to the left of the centre line. The same happens on the y-axis (in translating terms the ‘b’ axis) if the number is negative it is moved down from the horizontal centre line and if the number is positive it is moved up from the horizontal centre line. Translating works in the same way that if the ‘a’ part of the vector is positive it moves right however many spaces the number is equal to and vice versa if the number is negative. The same happens with the ‘b’ part of the vector, negative equals down and positive equals up. So now I have explained that aspect of translating a T-Shape I will go on to add more evidence to prove my prediction. Remember I said the T-Shape was to be translated by a vector of (2, -3), below is the T-Shape translated by a vector of (2, -3):
As you can see from the grid above, the whole 12 x 12 grid is not drawn out, this is because the whole 12 x 12 grid is not needed and I will not get extra marks simply for drawing out the whole grid. As you will also see from the grid my formula has worked, as usual, and I have also proved that with any type of vector large or small, with any sized grid large or small and with any T-Number, my formula can be used to get the answer to all your questions.
