{
  "schema": "AN02-rational-top-forms-exact-geometry/v1",
  "license": "CC0-1.0",
  "graph": {
    "equation": "z*w=1",
    "cycle": "(exp(i*theta),exp(-i*theta)),0<=theta<=2*pi",
    "orientation": "theta increases",
    "figure_scope": "Two complex-coordinate projections of a real curve in C^2, not a rendering of all four real ambient coordinates.",
    "period": "integral w dz = 2*pi*i"
  },
  "morse": {
    "ambient_center": [
      "1/2",
      "0"
    ],
    "parameterization": "z=r*exp(i*theta),w=r^(-1)*exp(-i*theta)",
    "height": "r^2+r^(-2)-r*cos(theta)+1/4",
    "minimum": {
      "r": "1.15277658071830802697117801313064000956936800924748045671093690073074357517761416958052587",
      "theta": "0",
      "quartic": "r^4-(1/2)*r^3-1=0",
      "index": 0
    },
    "saddle": {
      "r": "0.894975488634740861898634657689760984244980477520869493677758036791140968734086330656478662",
      "theta": "pi",
      "quartic": "r^4+(1/2)*r^3-1=0",
      "index": 1
    },
    "radial_hessian": "2+6*r^(-4)",
    "angular_hessian": "r*cos(theta)",
    "figure_scope": "Radial height profiles in the global parameter chart; shaded values are the exact range over theta. This is not the ambient graph embedding.",
    "proof_locator": "Learner Example 4; formal Proposition 6.3 and Corollary 6.4"
  }
}