The real projective plane ℝℙ² is the set of lines through the origin in ℝ³. A projective point [ x : y : z ] represents the line through the origin and (x, y, z), so [ 1 : 2 : 3 ] and [ 2 : 4 : 6 ] denote the same point. A homogeneous polynomial p(x, y, z) of degree d has a well-defined zero set in ℝℙ² because scaling (x, y, z) by λ scales p by λd.
Each disk shows ℝℙ² via the disk model: take the closed upper hemisphere of the sphere S² (relative to a chosen pole) and orthogonally project to its equatorial disk. The interior of the disk represents projective points away from the chart's line at infinity; the boundary circle represents that line, with antipodal points identified.
The three views correspond to placing the pole at [ 0:0:1 ], [ 0:1:0 ], or [ 1:0:0 ]. A curve that looks like a circle in one view can look like a hyperbola in another — the affine type depends on how the curve meets the line at infinity, but the projective curve is the same. The animation interpolates between the three views by rotating the sphere.
Singular points — where p and ∇p both vanish (nodes, cusps, isolated double points) — are marked with small open circles. Hover over any disk to read the projective coordinates of the point under your cursor.
x, y,
z with operators + − * / ^; implicit multiplication is
accepted. Non-homogeneous inputs are automatically homogenized with z.