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Video transcript

what I hope to introduce you to in this video is the notion of a transformation in mathematics and you're probably used to the word and every day language transformation means something is changing its transforming from one thing to another so what would transformation mean in a mathematical context well it could mean that you're taking something mathematical and you're changing you to do something else mathematical that's exactly what it is it's talking about taking a set of coordinates or a set of points and then changing them into a different set of coordinates or a different set of points for example this right over here this is a quadrilateral we've plotted it on the coordinate plane this is a set of points not just the four points that represent the vertices of the quadrilateral but all the points along the sides too there's a bunch of points along this you could argue there's an infinite or there are an infinite number of points along this quadrilateral this right over here the point x equals zero y equals negative four this is a point on the quadrilateral now we can apply a transformation to this and the first one I'm going to show you is a translation which just means moving all the points in the same direction and the same amount in that same direction and I'm using the Khan Academy translation widget to do it so let's translate let's translate this and I can do it by grabbing on to one of the vertices and notice I've now shifted it to the right by two every point here not just the orange points has shifted to the right by two this one has shifted to the right by two this point right over here has shifted to the right by two every point has shifted in the same direction by the same amount that's what a translation is now I've shifted let's see if I put it here everything every point has shifted to the right one and up one they've all shifted by the same amount in the same direction that is a translation but you can imagine a translation is not the only kind of transformation in fact there's an unlimited variation there's a unlimited number of different transformations so for example I could do a rotation so I have another set of points here that's represented by quadrilateral I guess we call it C D or B dde and i could rotate it and i'll rotate it i would rotate it around the point so for example i could rotate it around the point d so this is what i started with if I let me see if I can do this I could rotate it like actually let me see so if I start like this I could rotate it 90 degrees I could rotate 90 degrees so I could rotate it I could rotate it like that looks pretty close to a 90 degree rotation so every point that was on the the original or in the original set of points I've now shifted it relative to that at that point that I'm rotating around I've now rotated it 90 degrees so this point has now mapped to this point over here this point has now mapped at this point over here now I'm just picking the vertices because those are a little bit easier to think about this point has mapped at this point the point of rotation actually since that was one of since D is actually the point of rotation that one actually has not shifted and just so you get some terminology the the set of points after you apply the transformation this is called the image of the transformation so I had I had I had quadrilateral BCDE I applied a 90-degree counterclockwise rotation around the point D and so this new set of points this is the image of our original quadrilateral after the transformation and I don't have to just let me undo this I don't have to rotate around just one of the points that are on the original set that are on our quadrilateral I could rotate around I could rotate around the origin I could do something like that notice it's a different rotation now it's a different rotation I could rotate around any any point now let's look at another transformation and that would be the notion of a reflection and you know what reflection means in everyday life you kind of imagine the reflection of an image in a in a mirror or on the water and that's exactly what we're going to do over here so if we reflect we reflect across a line let me do that so this was this 1 2 3 4 5 this this this not a regular pentagon let's let's let's reflect it and to reflect it let me actually let me actually let me make a line like this I could reflect it across a whole series of lines whoops let me see if I can so let's reflect it across this now what does it mean to reflect across something one way I imagine is if this was we're going to get it's mirror image and you kind of imagine this is kind of the the line of symmetry that the image and the original set of the original shape they should be mirror images across this line we could see that let's let's do the reflection there you go and you see we haven't a mirror image this is this far away from the line this that's corresponding point in the image is on the other side of the line but the same distance this point over here is this distance from the line and this point over here is the same distance but on the other on the other side now all of the transformations that I've just showed you the translation the the reflection the rotation these are called rigid transformations and once again you can just think about what is rigid mean in in everyday life it means something that's not flexible it means something that you can't you can't kind of stretch short or scale up or scale down it kind of maintains its shape and that's what rigid transformations are fundamentally about if you want to think a little bit more mathematically a rigid transformation is one in which lengths and angles are preserved you can see in this transformation right over here the distance between this point and this point between points T and R and the distance between their corresponding image points that distance is the same the angle here angle R T Y the measure of this angle over here if you look at the corresponding angle in the image it's going to be this same angle so you might end the same thing is true if you're doing a translation you can imagine these these are kind of acting like rigid objects you can't stretch them and they're not flexible they're maintaining their shape now what would be examples of transformations that are not rigid transformations well you can imagine scaling things up and down if I were to zoom if I were to scale this out where it has maybe the angles are preserved but the lengths aren't preserved that would not be a rigid transformation if I were to just stretch one side of it if I were to just if I were to just pull this point while the other point stayed where they are I'd be kind of ordering it or stretching it that would not be a rigid transformation so hopefully this gets you it's actually very very interesting I mean when you use an art program or actually use a lot of computer graphics we play a video game that most of what the video game is doing is actually doing transformations sometimes in two dimensions sometimes in three dimensions and once you get into more advanced math especially things like linear algebra as a whole field it's really focused around transformations in fact some of the computers with really good graphics processors a graphics processor is just a piece of hardware that is really good at performing mathematical transformations so that you can immerse yourself in a 3d reality or whatever else so this is really really interesting stuff