So we ask the question again,

A cubic curve with a nodal point at the origin through six points

And Euclid said, How can I define parallel lines?

A half line by its start point, and another point somewhere on it.

So my slope of my tangent of my secant

A cubic curve with a horizontal cusp at the origin through four points

Conic by Directrix, Focus Point

Dot

The Dotted Line

Dots

Cubic Curve with Node by Six Points

Cubic Curve with Cusp by Four Points

A conic with given directrix and focus, through a point

Let's call this line right over here line AB.

Points and lines

Hyperbola by Focuses Point

And what you see, it should just be one curve.

So before when we were doing 3 and 4 points that was just to kind of show you, but you really just need two points for a line.

Perhaps, it is all equally good. It turns out green is the right answer. Let's just look at this.

Hyperbola by Asymptotes Point

Parabola by Directrix Focus

One scenario is where the lines don't intersect at all.

Do you see?

We'll talk more about how do you graph it or how do you know what the interesting points on a parabola actually are.

And now we should be able to use this information to figure out the actual coordinates of the focus and the directrix of what I would call the classic parabola, y is equal to x squared.

Line by Two Points

Polar Line of a Point

Dash Dot

Dash Dot Dot

Intersection of lines

So there's one point that just by looking at it we could say is going to be in our locus.

The polar line of a point with respect to a conic.

What's the connection?

Fair enough.

A parabola defined by its directrix and focus

The polar point of a line with respect to a conic.

A hyperbola with given asymptotes through a point

Polar Point of a Line

On the left hand panel, again, we have the S shaped curve of adoption.

A hyperbola constructed by its focuses and a point that pertains to it

The lines constructed through the intersections of two conics. This is also defined for non intersecting conics.

So the rest of the function is going to look like this.

In mathematics, a straight line is defined by two points.

So let's add a straight edge here, so this is to draw a line.

That's what R2 is.