{
  "name": "GenerateSphere",
  "main": {
    "name": "GenerateSphere",
    "namespace": "sofa::component::engine::generate",
    "module": "Sofa.Component.Engine.Generate",
    "include": "sofa/component/engine/generate/GenerateSphere.h",
    "doc": "Engine generating a spherical (Bezier) tetrahedral and triangular mesh.\n\nThis class creates a mesh on the sphere as the tessellation of a regular tetrahedron,\n regular octahedron or regular dodecahedron.\n The mesh can be either a triangulation, a tetrahedal mesh (with the sphere center) or a\n rational Bezier triangulation or tetrahedral mesh.",
    "inherits": [
      "DataEngine"
    ],
    "templates": [
      "sofa::defaulttype::Vec3Types"
    ],
    "data_fields": [
      {
        "name": "f_outputTetrahedraPositions",
        "type": "VecCoord",
        "xmlname": "output_TetrahedraPosition",
        "help": "output array of 3d points of tetrahedra mesh"
      },
      {
        "name": "f_tetrahedra",
        "type": "SeqTetrahedra",
        "xmlname": "tetrahedra",
        "help": "output mesh tetrahedra"
      },
      {
        "name": "f_outputTrianglesPositions",
        "type": "VecCoord",
        "xmlname": "output_TrianglesPosition",
        "help": "output array of 3d points of triangle mesh"
      },
      {
        "name": "f_triangles",
        "type": "SeqTriangles",
        "xmlname": "triangles",
        "help": "output triangular mesh"
      },
      {
        "name": "f_radius",
        "type": "Real",
        "xmlname": "radius",
        "help": "input sphere radius"
      },
      {
        "name": "f_origin",
        "type": "Coord",
        "xmlname": "origin",
        "help": "sphere center point"
      }
    ],
    "links": [],
    "methods": [
      {
        "name": "init",
        "return_type": "void",
        "params": [],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "reinit",
        "return_type": "void",
        "params": [],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      },
      {
        "name": "doUpdate",
        "return_type": "void",
        "params": [],
        "is_virtual": true,
        "is_pure_virtual": false,
        "is_static": false,
        "access": "public"
      }
    ]
  },
  "desc": {
    "title": "SofaImplicitSurfaceMeshGenerator",
    "description": "This class generates an implicit surface mesh for use in simulations within the Sofa framework.",
    "parameters": [
      {
        "name": "center",
        "type": "Vec3",
        "description": "The center of the sphere used as a reference point for generating the mesh."
      },
      {
        "name": "radius",
        "type": "float",
        "description": "Radius of the sphere that defines the size of the generated mesh."
      },
      {
        "name": "scale",
        "type": "Vec3",
        "description": "Scaling factor applied to the generated mesh along each axis."
      },
      {
        "name": "resolutionEdge",
        "type": "int",
        "description": "Resolution for edge generation, determining the number of edges in the mesh."
      },
      {
        "name": "resolutionTriangle",
        "type": "int",
        "description": "Resolution for triangle generation, controlling the detail level of triangular faces."
      },
      {
        "name": "resolutionTetrahedron",
        "type": "int",
        "description": "Resolution for tetrahedron generation, influencing the density and detail of internal tetrahedral elements."
      }
    ],
    "outputs": [
      {
        "name": "positionTriangle",
        "type": "VecCoord",
        "description": "Output positions for triangles generated in the mesh."
      },
      {
        "name": "indexTriangle",
        "type": "VecTriangle",
        "description": "Indices defining connectivity of triangle elements in the mesh."
      },
      {
        "name": "positionTetrahedron",
        "type": "VecCoord",
        "description": "Positions for tetrahedral elements generated within the mesh."
      },
      {
        "name": "indexTetrahedron",
        "type": "VecTetra",
        "description": "Indices defining connectivity of tetrahedral elements in the mesh."
      }
    ],
    "functionality_details": [
      {
        "step": "Mesh Initialization",
        "details": "Initializes triangle and tetrahedral elements with default positions based on specified resolutions and sphere parameters."
      },
      {
        "step": "Resolution Handling",
        "details": "Adapts to different resolution levels by generating vertices, edges, triangles, and tetrahedrons accordingly. For higher resolutions, additional points are calculated within the volume of the mesh for finer detail."
      },
      {
        "step": "Edge Point Calculation",
        "details": "Calculates positions for edge points using interpolation between adjacent triangle or tetrahedral points based on the given resolution settings."
      },
      {
        "step": "Triangle and Tetrahedron Point Calculation",
        "details": "Generates internal points within triangles and tetrahedrons to increase mesh detail. For specific resolutions, specialized calculations are performed to ensure accurate geometry."
      }
    ]
  },
  "maths": {
    "maths": "The provided code generates an implicit surface mesh that can be used in simulations within the Sofa framework. The primary goal is to create a detailed and accurate representation of a geometric structure, which typically could be a sphere or some other complex shape defined by its vertices (points) and faces (triangles/tetrahedra). This process involves several mathematical and computational steps:\n\n1. **Input Parameters**: \n    - `radius`: The radius of the sphere that is being generated.\n    - `origin`: The center point of the sphere in 3D space.\n    - `degree`: The degree of polynomial used to define the implicit surface, which impacts the complexity and smoothness of the mesh.\n\n2. **Mesh Generation**:\n   - The code first generates an initial set of vertices (`posTrian` for triangular meshes and `posTetra` for tetrahedral meshes) that lie on the surface of a sphere centered at `origin` with radius `radius`. \n    - It then adds additional points to create higher-order elements (Bezier curves or surfaces), especially in cases where the degree is greater than 2.\n\n3. **Bezier Curves and Surfaces**:\n   - For higher degrees, it creates Bezier curves along edges (for triangles) or faces (for tetrahedra). These are generated by interpolating positions between existing vertices to create smoother geometry. The weights (`bezierWeights` for triangular and `bezierTetraWeights` for tetrahedral elements) are computed based on specific mathematical formulas, such as binomial coefficients and interpolation rules.\n\n4. **Implicit Surface Representation**:\n   - An implicit surface is defined by a function F(x,y,z)=0 where the zero level set represents the surface of interest. In this context, the generated mesh approximates this implicit surface through a combination of vertices (points) and higher-order elements that ensure continuity and smoothness.\n\n5. **Normalization and Translation**:\n   - After generating these points, they are scaled by `radius` and translated to be centered at `origin`. This ensures that the final mesh is appropriately positioned and sized according to the input parameters.\n\n6. **Mesh Integrity**:\n   - The code also handles special cases such as triangles and tetrahedra within a sphere, ensuring that each face/triangle and volume/tetrahedron maintains its geometric integrity while being part of the larger implicit surface mesh.\n\nOverall, this component leverages mathematical concepts like interpolation, Bezier curves/surfaces, and implicit surfaces to generate detailed geometries suitable for simulation purposes in the Sofa framework."
  },
  "summary": {
    "abstract": "Generates spherical tetrahedral and triangular meshes based on input parameters such as radius and origin.",
    "sheet": "# GenerateSphere\n\n## Overview\nGenerateSphere is an engine that generates spherical (Bezier) tetrahedral and triangular meshes. It creates a mesh on the sphere as the tessellation of a regular tetrahedron, octahedron, or dodecahedron.\n\n## Parameters and Data\n- **f_radius**: Input sphere radius (`Real` type).\n- **f_origin**: Sphere center point (`Coord` type).\n- **f_outputTetrahedraPositions**: Output array of 3D points for tetrahedral mesh (`VecCoord` type).\n- **f_tetrahedra**: Output mesh tetrahedra (`SeqTetrahedra` type).\n- **f_outputTrianglesPositions**: Output array of 3D points for triangle mesh (`VecCoord` type).\n- **f_triangles**: Output triangular mesh (`SeqTriangles` type)."
  }
}