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# 2. Geometry of rotation

## Video transcript

now we know the coordinates of a few special points when they're rotated to create our software tools for setting up shots we need to have formulas for where every point goes when rotated that is if we start with an arbitrary point XY we'd like to know the coordinates of x-prime y-prime of the point where it ends up after rotation the formulas we'll come up with aren't too complicated in fact here they are x prime equals x cosine theta minus y sine theta y prime equals x sine theta plus y cosine theta so knowing X Y and theta you can compute X Prime and Y Prime but where do these formulas come from well there's a couple of different ways to get these formulas one is to use properties of linear transformations a more elementary way to derive these formulas is using the basic definitions of trigonometry and it'll take us a little work to get there so roll up your sleeves and tie back your hair let's call the point we start with P and the point it gets rotated to P Prime we need to construct some other points to help us so let's go back to what we already know and break down the problem first let's rotate the diagram and imagine Oh P is the x axis this looks like the situation we saw in the previous video when we rotated the point 1 0 on the x axis so we drop a perpendicular from P prime to the x axis to define a new point a now let's reverse the rotation and drop a perpendicular from a to the x axis to define a point B similarly drop a perpendicular from P prime to get Point C observe that the x-coordinate of a line OB is greater than the x coordinate of P prime line OC so we must subtract a certain amount the amount we subtract is the length of a new line ad recall the coordinates of P prime we're looking for are defined by OC for the x-coordinate and CP prime for the y-coordinate finally drop a perpendicular from P to the x-axis to create a point e this diagram now has all the information we need let's get some practice using this diagram by deriving a couple of formulas will need later suppose R is the distance from the origin o to P and let Phi be the angle that Opie makes with the x-axis the distance EP is y in the distance oh a is x EP is opposite Phi so sine Phi equals EP over R but EP is y so sine Phi equals y over R and that means y equals R sine Phi similarly Z is adjacent to Phi so cosine Phi equals 0 e over R but Oh ay is just X so cosine Phi equals x over R and that means x equals R cosine Phi wow that was a lot of trigonometry before we proceed see if you can answer a few general questions about this diagram in the next exercise good luck