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# Einführung in geometrische Transformationen

Video-Transkript

- [Voiceover] 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 in everyday language. Transformation means
something is changing, it's transforming from
one thing to another. What would transformation mean
in a mathematical context? Well, it could mean that
you're taking something mathematical and you're changing it into 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 0I 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. Let's translate, let's translate this, and I can do it by grabbing
onto 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 every point has shifted to the right one and up one, they've all shifted by the same amount in the same directions. That is a translation,
but you could imagine a translation is not the
only kind of transformation. In fact, there is an unlimited variation, there's an unlimited number
different transformations. So, for example, I could do a rotation. I have another set of points here that's represented by quadrilateral, I guess we could call it CD or
BCDE, and I could rotate it, and I 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 original or in the original set of
points I've now shifted it relative to 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
to this point over here, and I'm just picking the
vertices because those are a little bit easier to think about. This point has mapped to this point. The point of rotation, actually, since D is actually the point of rotation that one actually has not shifted, and just 'til you get some terminology, the set of points after you
apply the transformation this is called the image
of the transformation. So, 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. 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 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 imagine the reflection of an image in a mirror or on the water, and that's exactly what
we're going to do over here. If we reflect, we reflect across
a line, so let me do that. This, what is this one,
two, three, four, five, this not-irregular
pentagon, let's reflect it. To reflect it, let me actually, let me actually make a line like this. I could reflect it across
a whole series of lines. Woops, 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 its mirror image, and you imagine this
as the line of symmetry that the image and the
original shape they should be mirror images across
this line we could see that. Let's do the reflection. There you go, and you see
we have a mirror image. This is this far away from the line. This, its 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 side. Now, all of the transformations
that I've just showed you, the translation, the
reflection, the rotation, these are called rigid transformations. Once again you could just think about what does rigid mean in everyday life? It means something that's not flexible. It means something that you can't stretch 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 difference 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 the same angle. The same thing is true if
you're doing a translation. You could imagine these are
acting like rigid objects. You can't stretch them,
they're not flexible they're maintaining their shape. Now what would be examples
of transformations that are not rigid transformations? Well you could imagine
scaling things up and down. 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, or if I were to just pull
this point while the other points stayed where they
are I'd be distorting it or stretching it that would
not be a rigid transformation. So hopefully this gets you, it's actually very, very interesting. When you use an art program, or actually you use a
lot of computer graphics, or you play a video game, 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, there's a whole field that'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. This is really really interesting stuff.