Interactive projective plane

As is well known, a natural model for the projective plane is a hemisphere with antipodal points on the border identified. The image is obtained by projection of the hemisphere onto the disc:

Type polynomial from \(\mathbb{Z}\left[X, Y\right]\) into the text field to graph zero set of that polynomial. The equation should be expressed as the sum of monomials, where each monomial is in form rXnYm with r in \(\mathbb{Z}\) and n, m in \(\mathbb{N}\) (which stands for \(rX^nY^m\)). Parentheses are not supported for now. You can also choose from some examples from the drop-down menu.

You can change coordinate system (or equivalently, transform curve by projective transformation), by dragging vertices of coordinate triangle. Using scroll wheel while holding pointer above one of the vertex multiplies the corresponding column of transformation matrix with scalar. Double clicking on one of vertex changes sign of third coordinate of the corresponding column. With these three operations, it is possible to define all matrix in \(\mathsf{PGl}_3\mathbb{R}\).

Example

Equation x²+2y²-1=0 defines an ellipse in the plane. If we move base vectors (blue dots), we will first get a parabola and then a hyperbola. This shows us that non-degenerate conic sections are equivalent in the projective plane.