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.


  • 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.