A bicubic_patch is a 3D curved surface created from a mesh of triangles. POV-Ray supports a type of bicubic patch called a Bezier patch. A bicubic patch is defined as follows:

BICUBIC_PATCH:
    bicubic_patch
    {
        PATCH_ITEMS...
        <Point_1>,<Point_2>,<Point_3>,<Point_4>,
        <Point_5>,<Point_6>,<Point_7>,<Point_8>,
        <Point_9>,<Point_10>,<Point_11>,<Point_12>,
        <Point_13>,<Point_14>,<Point_15>,<Point_16>
        [OBJECT_MODIFIERS...]
    }
PATCH_ITEMS:
    type Patch_Type | u_steps Num_U_Steps | v_steps Num_V_Steps |
    flatness Flatness

flatness : 0.0
u_steps  : 0
v_steps  : 0

Another flat, finite object available with POV-Ray is the disc. The disc is infinitely thin, it has no thickness. If you want a disc with true thickness you should use a very short cylinder. A disc shape may be defined by:

DISC:
    disc
    {
        <Center>, <Normal>, Radius [, Hole_Radius]
        [OBJECT_MODIFIERS...]
    }

HOLE RADIUS : 0.0

The vector <Center> defines the x, y, z coordinates of the center of the disc. The <Normal> vector describes its orientation by describing its surface normal vector. This is followed by a float specifying the Radius. This may be optionally followed by another float specifying the radius of a hole to be cut from the center of the disc.

Note: The inside of a disc is the inside of the plane that contains the disc. Also note that it is not constrained by the radius of the disc.

The mesh object can be used to efficiently store large numbers of triangles. Its syntax is:

MESH:
    mesh
    {
        MESH_TRIANGLE...
        [MESH_MODIFIER...]
    }
MESH_TRIANGLE:
    triangle
    {
        <Corner_1>, <Corner_2>, <Corner_3>
        [uv_vectors <uv_Corner_1>, <uv_Corner_2>, <uv_Corner_3>]
        [MESH_TEXTURE]
    } |
    smooth_triangle
    {
        <Corner_1>, <Normal_1>,
        <Corner_2>, <Normal_2>,
        <Corner_3>, <Normal_3>
        [uv_vectors <uv_Corner_1>, <uv_Corner_2>, <uv_Corner_3>]
        [MESH_TEXTURE]
    }
MESH_TEXTURE:
  texture { TEXTURE_IDENTIFIER }
  texture_list {
    TEXTURE_IDENTIFIER TEXTURE_IDENTIFIER TEXTURE_IDENTIFIER
  }

MESH_MODIFIER:
    inside_vector <direction> | hierarchy [ Boolean ] |
    OBJECT_MODIFIER

hierarchy : on

Copies of a mesh object refer to the same triangle data and thus consume very little memory. You can easily trace a hundred copies of a 10000 triangle mesh without running out of memory (assuming the first mesh fits into memory). The mesh object has two advantages over a union of triangles: it needs less memory and it is transformed faster. The memory requirements are reduced by efficiently storing the triangles vertices and normals. The parsing time for transformed meshes is reduced because only the mesh object has to be transformed and not every single triangle as it is necessary for unions.

The mesh object can currently only include triangle and smooth triangle components. That restriction may change, allowing polygonal components, at some point in the future.

Triangle mesh objects (mesh and mesh2) can now be used in CSG objects such as difference and intersect, because, after adding inside_vector, they do have a defined 'inside'. This will only work for well-behaved meshes, which are completely closed volumes. If meshes have any holes in them, this might work, but the results are not guaranteed.

To determine if a point is inside a triangle mesh, POV-Ray shoots a ray from the point in some arbitrary direction. If this vector intersects an odd number of triangles, the point is inside the mesh. If it intersects an even number of triangles, the point is outside of the mesh. You can specify the direction of this vector. For example, to use +z as the direction, you would add the following line to the triangle mesh description (following all other mesh data, but before the object modifiers).

  inside_vector <0, 0, 1>

This change does not have any effect on unions of triangles... these will still be always hollow.

The new mesh syntax is designed for use in conversion from other file formats.

  MESH2 :
      mesh2{
        VECTORS...
        LISTS...   |
        INDICES... |
        MESH_MODIFIERS
      }
  VECTORS :
      vertex_vectors 
      {
        number_of_vertices,
        <vertex1>, <vertex2>, ...
      }|
      normal_vectors 
      {
        number_of_normals,
        <normal1>, <normal2>, ...
      }|
      uv_vectors 
      {
        number_of_uv_vectors,
        <uv_vect1>, <uv_vect2>, ...
      }
  LISTS :
      texture_list 
      {
        number_of_textures,
        texture { Texture1 },
        texture { Texture2 }, ...
      }|
  INDICES :
      face_indices 
      {
        number_of_faces,
        <index_a, index_b, index_c> [,texture_index [,
                                    texture_index, texture_index]],
        <index_d, index_e, index_f> [,texture_index [,
                                    texture_index, texture_index]],
        ...
      }|
      normal_indices 
      {
        number_of_faces,
        <index_a, index_b, index_c>,
        <index_d, index_e, index_f>,
        ...
      }|
      uv_indices {
        number_of_faces,
        <index_a, index_b, index_c>,
        <index_d, index_e, index_f>,
        ...
      }
  MESH_MODIFIER :
      inside_vector <direction> | OBJECT_MODIFIERS

mesh2 has to be specified in the order VECTORS..., LISTS..., INDICES.... The normal_vectors, uv_vectors, and texture_list sections are optional. If the number of normals equals the number of vertices then the normal_indices section is optional and the indexes from the face_indices section are used instead. Likewise for the uv_indices section.

Note: that the numbers of uv_indices must equal number of faces.

The indexes are ZERO-BASED! So the first item in each list has an index of zero.

The polygon object is useful for creating rectangles, squares and other planar shapes with more than three edges. Their syntax is:

POLYGON:
    polygon
    {
        Number_Of_Points, <Point_1> <Point_2>... <Point_n>
        [OBJECT_MODIFIER...]
    }

The float Number_Of_Points tells how many points are used to define the polygon. The points <Point_1> through <Point_n> describe the polygon or polygons. A polygon can contain any number of sub-polygons, either overlapping or not. In places where an even number of polygons overlaps a hole appears. When you repeat the first point of a sub-polygon, it closes it and starts a new sub-polygon's point sequence. This means that all points of a sub-polygon are different.

If the last sub-polygon is not closed a warning is issued and the program automatically closes the polygon. This is useful because polygons imported from other programs may not be closed, i.e. their first and last point are not the same.

All points of a polygon are three-dimensional vectors that have to lay on the same plane. If this is not the case an error occurs. It is common to use two-dimensional vectors to describe the polygon. POV-Ray assumes that the z value is zero in this case.

A square polygon that matches the default planar image map is simply:

  polygon {
    4,
    <0, 0>, <0, 1>, <1, 1>, <1, 0>
    texture {
      finish { ambient 1 diffuse 0 }
      pigment { image_map { gif "test.gif"  } }
    }
    //scale and rotate as needed here
  }

The sub-polygon feature can be used to generate complex shapes like the letter "P", where a hole is cut into another polygon:

  #declare P = polygon {
    12,
    <0, 0>, <0, 6>, <4, 6>, <4, 3>, <1, 3>, <1,0>, <0, 0>, 
    <1, 4>, <1, 5>, <3, 5>, <3, 4>, <1, 4>
  }

The first sub-polygon (on the first line) describes the outer shape of the letter "P". The second sub-polygon (on the second line) describes the rectangular hole that is cut in the top of the letter "P". Both rectangles are closed, i.e. their first and last points are the same.

The feature of cutting holes into a polygon is based on the polygon inside/outside test used. A point is considered to be inside a polygon if a straight line drawn from this point in an arbitrary direction crosses an odd number of edges (this is known as Jordan's curve theorem).

Another very complex example showing one large triangle with three small holes and three separate, small triangles is given below:

  polygon {
    28,
    <0, 0> <1, 0> <0, 1> <0, 0>          // large outer triangle
    <.3, .7> <.4, .7> <.3, .8> <.3, .7>  // small outer triangle #1
    <.5, .5> <.6, .5> <.5, .6> <.5, .5>  // small outer triangle #2
    <.7, .3> <.8, .3> <.7, .4> <.7, .3>  // small outer triangle #3
    <.5, .2> <.6, .2> <.5, .3> <.5, .2>  // inner triangle #1
    <.2, .5> <.3, .5> <.2, .6> <.2, .5>  // inner triangle #2
    <.1, .1> <.2, .1> <.1, .2> <.1, .1>  // inner triangle #3
  }

The triangle primitive is available in order to make more complex objects than the built-in shapes will permit. Triangles are usually not created by hand but are converted from other files or generated by utilities. A triangle is defined by

TRIANGLE:
    triangle
    {
        <Corner_1>, <Corner_2>, <Corner_3>
        [OBJECT_MODIFIER...]
    }

where <Corner_n> is a vector defining the x, y, z coordinates of each corner of the triangle.

Because triangles are perfectly flat surfaces it would require extremely large numbers of very small triangles to approximate a smooth, curved surface. However much of our perception of smooth surfaces is dependent upon the way light and shading is done. By artificially modifying the surface normals we can simulate a smooth surface and hide the sharp-edged seams between individual triangles.

SMOOTH_TRIANGLE:
    smooth_triangle
    {
        <Corner_1>, <Normal_1>, <Corner_2>,
        <Normal_2>, <Corner_3>, <Normal_3>
        [OBJECT_MODIFIER...]
    }

where the corners are defined as in regular triangles and <Normal_n> is a vector describing the direction of the surface normal at each corner.

These normal vectors are prohibitively difficult to compute by hand. Therefore smooth triangles are almost always generated by utility programs. To achieve smooth results, any triangles which share a common vertex should have the same normal vector at that vertex. Generally the smoothed normal should be the average of all the actual normals of the triangles which share that point.