The desmos graph below is one of the many interpretations of the projective plane. In particular, we aim to demonstrate duality between points and lines. There’s three points: red, green and blue. Each of the points are associated with a line with a line. For example the red point $$(a_1, a_2)$$ is associated with the red line $$a_1 x + a_2 y + 1 = 0$$. Try moving the points around and see how the lines change.

## Questions

• If the three points are colinear, what do you notice about the lines?

• If the three lines intersect at a point, what can you say about the points?

• Can you make all three lines parallel? What can you say about the points?

• Try making two lines intersect, and move the third point to the intersection of the two lines. What do you notice?

• What happens when the points get close to the origin? Why does that happen?

## Projective Geometry

What you’re seeing is essentially the real projective plane and the duality between points and lines. If you’re interested, you could read a set of lecture notes on projective geometry by Hitchin.