{
  "ambient_real_dimension": 4,
  "surface_real_dimension": 2,
  "sphere_volume_coefficient": "b_2=pi",
  "surface_graph": "Gamma(y)=(y,y^2)",
  "graph_area_density": "1+4|y|^2",
  "euclidean_ball_cutoff": "|y|^2=b(r)=(sqrt(1+4r^2)-1)/2",
  "area": "pi*(b+2*b^2)=pi*(2*r^2-b)",
  "projection_area": "pi*b",
  "parallel_planes": {
    "polynomial": "t(t-h)",
    "surface_area": "pi*r^2+pi*(r^2-h^2)_+",
    "repeated_polynomial": "t^2(t-h)^3",
    "log_laplacian_mass": "2*pi^2*(2*r^2+3*(r^2-h^2)_+)"
  },
  "normal_regularization": {
    "logarithm": "0.5*log(|t|^2+delta^2)",
    "laplacian": "2*delta^2/(|t|^2+delta^2)^2",
    "ball_mass": "2*pi^2*(r^2-delta^2*log(1+r^2/delta^2))"
  },
  "ellipsoid_sections": {
    "m": 2,
    "radius": 0.6,
    "centers": [
      {
        "a": 0,
        "b": 0,
        "sigma_exact": "1/1"
      },
      {
        "a": 2,
        "b": 0,
        "sigma_exact": "1/3"
      },
      {
        "a": 5,
        "b": 0,
        "sigma_exact": "1/6"
      }
    ],
    "equation": "|t-a|^2+(1+|theta|)^2*|y-b|^2<r^2",
    "real_section_only": true,
    "three_drawn_sections_do_not_claim_a_cover": true
  },
  "exact_proof_locators": [
    "SR7-SR13",
    "SR16-SR23",
    "L152.1-L152.6",
    "L152.10-L152.14"
  ],
  "human_source": "Lars Hormander, Analysis of Linear PDE I, Theorem4.1.12; II, Theorem15.3.3",
  "original_plot_sources": true,
  "blender_useful": false,
  "blender_reason": "Exact scalar plots and labeled real sections preserve the actual four-dimensional metric; a three-dimensional embedding would not."
}