- [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 0, 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.
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.