One point.
Many ways in.
Contract the position to the origin. The blow-up keeps the line you approached along.
Square window: x,y from −1 to1. Circular register encodes real lines at2θ:0° and180° share one position. Small dials: signed r (−1…1), θ (0…180°).
Reduced motion is enabled. Inspect still positions and directions with the controls.
What is being resolved?
We show real points of B={((x,y),[u:v]): xv=yu}, the blow-up of the affine plane at the origin. With u=cosθ,v=sinθ and (x,y)=r(u,v), the incidence equation always holds. Away from the origin, [u:v]=[x:y] is uniquely determined. At r=0, every projective line remains possible above the same base point.
The left guide is a scientific overlay showing the chosen line, not extra data stored by the base point. The right circular register is a schematic for RP1 via angle2θ; it is not a drawing of complex P1. θ and θ+180° name the same line. In these signed coordinates, (r,180°) and (−r,0°) are the same incidence point. |r| is distance; signed r depends on the chosen representative.
In the u≠0 chart, t=v/u and y=xt. In the v≠0 chart, s=u/v and x=ys. Both agree on their overlap, with st=1. Exact horizontal/vertical directions make one chart unavailable; N/A means switch charts, not an infinite physical reading. The chart remains valid at the origin when its denominator is nonzero.
This bounded real teaching slice does not depict the whole complex surface, calculate negative self-intersection, execute a flip, or prove MMP termination, abundance or any manuscript/Lean claim. The instrument is independently authored; its units and circular layout are normalized and schematic. Manual contraction ends at r=0 without wrapping.
UMass: incidence construction and exceptional directions ↗
Stacks Project: exceptional fibre of the plane blow-up ↗