So on parallel coordinates, we're going to take the Cartesian coordinates.

We have a horizontal x-axis and a vertical y-axis, and

we're going to take these axes and we're going to

make them parallel instead of orthogonal, at right angles, as they usually are.

So I'm going to take the x-axis, I'm going to put it here, and

I'm going to take the y-axis and I'm going to put it here.

And so I'll label this axis x and this axis y.

So now the two axes are not orthogonal.

They're parallel to each other and they don't extend from the same origin.

So the origin here is horizontally at the bottom and

increasing x goes up this axis, increasing y goes up this axis.

So now I've got the data points, and I need to figure out where these data points

occur, so if I take the y-coordinates of each data point,

I can map each point onto its corresponding position on the y-axis.

I'm just basically dragging them across horizontally, because their position in y

in this chart corresponds to their height along this y-axis.

If I want to do the same for

the x-axis, I'm going to basically take the x position of each point, and then I'm

going to drag it to the corresponding x position on this vertical x-axis, so

the horizontal length here corresponds to the vertical length here, and

so blue and red are at the same horizontal x coordinate, and so

they overlap each other on the x-axis.

And in green is a little bit farther to the right, so

it's going to be a little bit higher on the parallel x-axis.

Yellow is a little bit farther to the right so

it's going to be a little bit higher.

And then this blue green color dot is the farthest right so

it'll be the highest on the x-axis.

So now I've got this one set of points that's now appearing as two

sets of points, and I've got a correspondence between color,

and color's a good perceptual indicator of category, but

it's very difficult to perceive what's going on here.

So what we're going to do is, instead of displaying these as points on the axis,

I'm going to connect these points with lines, and

I'm going to delete the original points, and

now we get this nice duality between points in the coordinate system,

here, and lines in the coordinate system, here.

So, this orange point, here, has an x-coordinate and a y-coordinate.

It corresponds to this line connecting its x-coordinate to its y-coordinate.

And so each one of these five points in the Cartesian x y coordinate

system corresponds to a line in the parallel x y coordinate system here.