Planar Subdivisions



Planar subdivision.

#define CV_SUBDIV2D_FIELDS()    \
    CV_GRAPH_FIELDS()           \
    int  quad_edges;            \
    int  is_geometry_valid;     \
    CvSubdiv2DEdge recent_edge; \
    CvPoint2D32f  topleft;      \
    CvPoint2D32f  bottomright;

typedef struct CvSubdiv2D

Planar subdivision is the subdivision of a plane into a set of non-overlapped regions (facets) that cover the whole plane. The above structure describes a subdivision built on a 2d point set, where the points are linked together and form a planar graph, which, together with a few edges connecting the exterior subdivision points (namely, convex hull points) with infinity, subdivides a plane into facets by its edges.

For every subdivision there exists a dual subdivision in which facets and points (subdivision vertices) swap their roles, that is, a facet is treated as a vertex (called a virtual point below) of the dual subdivision and the original subdivision vertices become facets. On the picture below original subdivision is marked with solid lines and dual subdivision with dotted lines.


OpenCV subdivides a plane into triangles using Delaunay’s algorithm. Subdivision is built iteratively starting from a dummy triangle that includes all the subdivision points for sure. In this case the dual subdivision is a Voronoi diagram of the input 2d point set. The subdivisions can be used for the 3d piece-wise transformation of a plane, morphing, fast location of points on the plane, building special graphs (such as NNG,RNG) and so forth.



Quad-edge of planar subdivision.

/* one of edges within quad-edge, lower 2 bits is index (0..3)
   and upper bits are quad-edge pointer */
typedef long CvSubdiv2DEdge;

/* quad-edge structure fields */
#define CV_QUADEDGE2D_FIELDS()     \
    int flags;                     \
    struct CvSubdiv2DPoint* pt[4]; \
    CvSubdiv2DEdge  next[4];

typedef struct CvQuadEdge2D

Quad-edge is a basic element of subdivision containing four edges (e, eRot, reversed e and reversed eRot):




Point of original or dual subdivision.

    int            flags;      \
    CvSubdiv2DEdge first;      \
    CvPoint2D32f   pt;         \
    int id;


typedef struct CvSubdiv2DPoint
  • id

    This integer can be used to index auxillary data associated with each vertex of the planar subdivision


void cvCalcSubdivVoronoi2D(CvSubdiv2D* subdiv)

Calculates the coordinates of Voronoi diagram cells.

Parameter:subdiv – Delaunay subdivision, in which all the points are already added

The function calculates the coordinates of virtual points. All virtual points corresponding to some vertex of the original subdivision form (when connected together) a boundary of the Voronoi cell at that point.


void cvClearSubdivVoronoi2D(CvSubdiv2D* subdiv)

Removes all virtual points.

Parameter:subdiv – Delaunay subdivision

The function removes all of the virtual points. It is called internally in CalcSubdivVoronoi2D if the subdivision was modified after previous call to the function.


CvSubdiv2D* cvCreateSubdivDelaunay2D(CvRect rect, CvMemStorage* storage)

Creates an empty Delaunay triangulation.

  • rect – Rectangle that includes all of the 2d points that are to be added to the subdivision
  • storage – Container for subdivision

The function creates an empty Delaunay subdivision, where 2d points can be added using the function SubdivDelaunay2DInsert . All of the points to be added must be within the specified rectangle, otherwise a runtime error will be raised.

Note that the triangulation is a single large triangle that covers the given rectangle. Hence the three vertices of this triangle are outside the rectangle rect .


CvSubdiv2DPoint* cvFindNearestPoint2D(CvSubdiv2D* subdiv, CvPoint2D32f pt)

Finds the closest subdivision vertex to the given point.

  • subdiv – Delaunay or another subdivision
  • pt – Input point

The function is another function that locates the input point within the subdivision. It finds the subdivision vertex that is the closest to the input point. It is not necessarily one of vertices of the facet containing the input point, though the facet (located using Subdiv2DLocate ) is used as a starting point. The function returns a pointer to the found subdivision vertex.


CvSubdiv2DPoint* cvSubdiv2DEdgeDst(CvSubdiv2DEdge edge)

Returns the edge destination.

Parameter:edge – Subdivision edge (not a quad-edge)

The function returns the edge destination. The returned pointer may be NULL if the edge is from dual subdivision and the virtual point coordinates are not calculated yet. The virtual points can be calculated using the function CalcSubdivVoronoi2D .


CvSubdiv2DEdge cvSubdiv2DGetEdge(CvSubdiv2DEdge edge, CvNextEdgeType type)

Returns one of the edges related to the given edge.

  • edge – Subdivision edge (not a quad-edge)
  • type

    Specifies which of the related edges to return, one of the following:

    • CV_NEXT_AROUND_ORG next around the edge origin ( eOnext on the picture below if e is the input edge)
    • CV_NEXT_AROUND_DST next around the edge vertex ( eDnext )
    • CV_PREV_AROUND_ORG previous around the edge origin (reversed eRnext )
    • CV_PREV_AROUND_DST previous around the edge destination (reversed eLnext )
    • CV_NEXT_AROUND_LEFT next around the left facet ( eLnext )
    • CV_NEXT_AROUND_RIGHT next around the right facet ( eRnext )
    • CV_PREV_AROUND_LEFT previous around the left facet (reversed eOnext )
    • CV_PREV_AROUND_RIGHT previous around the right facet (reversed eDnext )

The function returns one of the edges related to the input edge.


CvSubdiv2DEdge cvSubdiv2DNextEdge(CvSubdiv2DEdge edge)

Returns next edge around the edge origin

Parameter:edge – Subdivision edge (not a quad-edge)

The function returns the next edge around the edge origin: eOnext on the picture above if e is the input edge)


CvSubdiv2DPointLocation cvSubdiv2DLocate(CvSubdiv2D* subdiv, CvPoint2D32f pt, CvSubdiv2DEdge* edge, CvSubdiv2DPoint** vertex=NULL)

Returns the location of a point within a Delaunay triangulation.

  • subdiv – Delaunay or another subdivision
  • pt – The point to locate
  • edge – The output edge the point falls onto or right to
  • vertex – Optional output vertex double pointer the input point coinsides with

The function locates the input point within the subdivision. There are 5 cases:

  • The point falls into some facet. The function returns CV_PTLOC_INSIDE and *edge will contain one of edges of the facet.
  • The point falls onto the edge. The function returns CV_PTLOC_ON_EDGE and *edge will contain this edge.
  • The point coincides with one of the subdivision vertices. The function returns CV_PTLOC_VERTEX and *vertex will contain a pointer to the vertex.
  • The point is outside the subdivsion reference rectangle. The function returns CV_PTLOC_OUTSIDE_RECT and no pointers are filled.
  • One of input arguments is invalid. A runtime error is raised or, if silent or “parent” error processing mode is selected, CV_PTLOC_ERROR is returnd.


CvSubdiv2DEdge cvSubdiv2DRotateEdge(CvSubdiv2DEdge edge, int rotate)

Returns another edge of the same quad-edge.

  • edge – Subdivision edge (not a quad-edge)
  • rotate

    Specifies which of the edges of the same quad-edge as the input one to return, one of the following:

    • 0 the input edge ( e on the picture below if e is the input edge)
    • 1 the rotated edge ( eRot )
    • 2 the reversed edge (reversed e (in green))
    • 3 the reversed rotated edge (reversed eRot (in green))

The function returns one of the edges of the same quad-edge as the input edge.


CvSubdiv2DPoint* cvSubdivDelaunay2DInsert(CvSubdiv2D* subdiv, CvPoint2D32f pt)

Inserts a single point into a Delaunay triangulation.


The function inserts a single point into a subdivision and modifies the subdivision topology appropriately. If a point with the same coordinates exists already, no new point is added. The function returns a pointer to the allocated point. No virtual point coordinates are calculated at this stage.