This is where we are.

And they're symmetric around the center of the ellipse.

K Means will do is group all of these points into one cluster and group all of these points into a second cluster and group all of those points into a third cluster.

IV, So this point is in the third quadrant.

Let's take this point right here.

There's M data points.

And it's often used as the definition of an ellipse is, if you take any point on this ellipse, and measure its distance to each of these two points.

So let's figure out what are the coordinates of these points?

And then I have this distance over here, so I'm taking any point on that ellipse, or this particular point, and I'm measuring the distance to each of these two foci.

I guess we've always been talking on a massive scale, but now it's an even more massive scale. We're still not done!

Well, it totally is that's super convenient.

All of these coordinate axes I draw are going be R2.

These are the corner points in our data points.

from this point

So it will UNlNTELLlGIBLE that point.

Well, the only point that sits on both lines is the intersection of those lines.

So I can get to any of these vectors.

We know that the train passes this point at t is equal to a.

This one over here would be at the center of the 3 data points over here, but this one pulls back this data point drastically towards it.

The square root of that.

I am going to talk about linear dimensionality reduction.

And now, where these two circles intersect, those are points that are equidistant from both of these centers that I just constructed.

is when two rays share a common vertex. and that common vertex is actually called the vertex of the angle. so A is vertex.

Right there.

So this is the vertex of this parabola.

For all of the points in the xy plane that satisfy the conditions that its distance to the focus is equal to the distance to this line.

So this is x h.

Remember, this isn't going to be the slope just at this point.

And I'm going to explain what all of this means in a second.

I'll call that well, we'll call that a focus because it will be a focus.

Mark to model.

Right?

So they give us four coordinates right over here.

In the last video, we said what is the equation of the line that is equidistant between this focus and this directrix?

But that's not the focus of this video.

So I connect the eye, and then I play the audience's faces.

Let's take it right there.

What makes these clusters so different is not the absolute location of each data point, but the connectedness of these data points.

An alternative plan might be to go straight over here, take the detour around the block, and then go straight to the target location.

And 2 minus negative 1 is 3.

And, actually, this is often used as the definition for an ellipse, where they say that the ellipse is the set of all points, or sometimes they'll use the word locus, which is kind of the graphical representation of the set of all points, that where the sum of the distances to each of these focuses is equal to a constant.

We go along the y axis 1.

A segment constructed from its start and end point

And we'll talk a little bit about that as we plot these points.

If you fix this data point over here, then this guy is perfectly happy with his 2 neighbors this is a smooth path and so is this guy over here.