Bäcklund image of the Amsler surface: the path (s=0, a=1) → (s=½, a=10⁴)
The seed f₁(x,y)=Y(xy), (uY′)′=sin Y, Y(0)=πs, is computed accurately (Taylor series + RK4), including xy<0. The Bäcklund image ω=2p−f₁ is obtained without linearization (nonlinear Goursat marching, four quadrants from the origin), and the K=−1 surface is reconstructed by marching the Gauss–Weingarten frame — the x-connection is marched by RK4 from exact axis data (no blow-up for a in [0.4, 10⁴]), and the y-grid is refined geometrically toward y = 0 so that the boundary layer of width 1/a is resolved. The window is symmetric, [−L,L]², so the s=0 and s=1 kinks are shown in full, with the cuspidal edge. The slider coordinates τ and α are remapped to s and a by sinh laws (finest control near s = ½ and near a = 1), which gives finer slider resolution in the central deformation band while s spans 10⁻⁴ … 0.9999 with smooth approach to both limit kinks.
Drag to rotate; zoom with the slider, the −/+ buttons, the mouse wheel or a two-finger pinch. Colour shows ω (light gradient) or ω mod 2π; the pink band is the cuspidal edge sin ω = 0. In the ω mod 2π view the mirror Amsler limit (ω → −f₁ mod 2π, up to a boundary layer of width 1/a around the x-axis) has no colour jump. The scenarios run ping-pong along the path; s→0 gives the exact kink (pseudosphere with two horns), and the endpoint (s=½, a=10⁴) is the mirror Amsler surface up to O(1/a).