Surfaces
A solid 3D object is called linked if it
consists of a single piece, more precisely if any two points in the object can
be connected with a single continuous curve.
A Geometric Surface is external shell
of a 3D solid linked object (Closed Surface). or part of it. In the later
case it is called Open Surface.
DG Kernel Surface, or simply Surface
is a Geometric Surface considered as a collection of patches joined together
by edges.
Patch is a smooth (without internal corners or folded
lines) linked piece of two
dimensional surface. Simplest example of a patch is 3D rectangle or 3D
triangle. Patch can have any number of edges or none at all (sphere). An
Edge
is a smooth curve, which can be closed line or line with ends. Edges are joined by ends into linked sequences called wires. A 3D rectangle has a single wire consisting of four edges. If we cut a triangular hole in a 3D rectangle the resulting shape will be a patch with two wires with four (external) and three (internal) edges respectively.
Edges inside a wire are normally indexed in CCW direction (looking from outside of the object, or in other words looking into direction oppositite to normal to the surface). So if one moves along a wire the patch always remains on the left hand side. Wires are considered as strips of curves (elements) joined by vertices (connectors).
Frequently patches are defined by a number of parameters which can
be modified programmatically or at design time to change its geometry.
A box is a surface with six patches and twelve edges. Note that a geometric
surface can be subdivided into patches in different
ways. Sphere can either be considered as collection of eight equal patches
with eight edges or as is single patch without edges.
Normal to a patch at a point is a ray
perpendicular to the patch at this point.
Point at an end of an edge is called Vertex
(Vertices plural). A box has eight vertices. Each vertex in a box has three
adjacent edges and three patches. Note
that closed edges normally do not have ends.
Set of normals of all patches adjacent to a vertex at the point of the
vertex are considered a property of the vertex. They are called normals
of the vertex.
Above means that a surface can be considered as a hierarchy of patches,
edges (lines) and vertices.
3DS Patches are special kinds of
patches. They are limited by number of edges, which must be at least two (top and bottom
edges have zero length) and maximum four. 3DS Patches can have between two
and four vertices. Top and bottom edges of a 3DS patch are flat curves in plane
orthogonal to z axis.
A vertex is called singular if adjacent patches have different normals at
the point. For example each vertex of a box has three normals so it is singular.
An edge is called singular if either end vertex is singular. It means
that adjacent patches have non-zero angle along the edge
Vertices and edges, which are not singular are called regular.
Surface versus Mesh
Mesh of an object is a lower level representation
of geometrical surface. It is designed for rendering the object quickly in
3D. Mesh is not aware of patch/edge/vertex structure of the surface.
Normally each patch of a surface is approximated by many simplexes of the
mesh. Each patch edge normally consists of many simplex edges. Vertices of a surface are vertices of the underlying mesh, but normally there are many
more vertices in the mesh, which do not belong to the formal hierarchy of
the surface.
Mesh and surface are synchronized. Edges of patches are represented by
simplex edges only. There is no simplex edge which crosses a patch edge.
Patches are represented by simplexes only. There is no simplex, which
geometrically belong to two different patches.
Surfaces and meshes are organized similarly. Patches act similar to
simplexes and edges of surface are similar to simplex edges. One of the
differences is that simplexes have fixed number of edges.
Normally surface is defined first by constructing it from patches and
modifying its parameters. Meshes are generated automatically on a later stage from the
surface information to approximate it. The reverse procedure is possible
too. For example Mesh Entity objects are
constructed from mesh only. Surface structure in this case is generated by analyzing which vertices and edges are
singular (are sharp corners or
edges). Surface generated in this way is called Singular Subdivision.
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