Other Voronoi/Delaunay Structures

Other Voronoi/Delaunay Structures

Overview Alpha hulls (a subset of Delaunay graph) Extension of Voronoi Diagrams

Convex Hull What is it good for? The bounding region of a point set Not so good for describing shapes

Convex Hull Subtractive definition Taking away all empty half planes Edge pp ii, pp jj lies on the hull if it lies on the boundary of an empty half plane

Alpha Hull Subtractive definition Taking away all empty half planes circles of radius α Edge pp ii, pp jj lies on the hull if it lies on the boundary of an empty half plane circle.

Alpha Hull (α controls the level of details) α=0 α=

Alpha Hull Alpha hull is a subset of the Delaunay graph Each hull edge has an empty circle Let αα mmmmmm (pp ii, pp jj ), αα mmaaaa (pp ii, pp jj ) be the minimum and maximum radius of all empty circles of edge pp ii, pp jj. The edge is on the hull if αα mmmmmm pp ii, pp jj < α < αα mmmmmm (pp ii, pp jj )

Alpha Hull αα mmmmmm pp jj αα mmmmmm pp jj pp ii αα mmaaaa pp ii αα mmaaaa

Computing Alpha Hull Compute the Voronoi Diagram of point set For each Voronoi edge Compute αα mmmmmm, αα mmaaaa If α is in range, output the dual Delaunay edge. O(n log n) Subsequent computation of alpha hulls with different α takes only O(n) (or faster )

Alpha Hull Interior of alpha hull is a subset of the Delaunay triangulation An element (point, edge, face) of Delaunay triangulation is on or inside α-hull if the radius of its smallest empty circle is smaller than α

Alpha Hull in 3D α=0 α=

Voronoi Diagram A finite set of point sites pp ii Euclidean distance dd: dd xx, pp ii = xx pp ii Voronoi diagram is the set of xx with multiple nearest sites

Voronoi Diagram A finite set of point sites pp ii Euclidean distance dd: dd xx, pp ii = xx pp ii Voronoi diagram is the set of xx with multiple nearest sites

Weighted Voronoi Diagram A finite set of point sites pp ii with weights rr ii Additively weighted distance dd: dd xx, pp ii = xx pp ii rr ii Voronoi diagram is the set of xx with multiple nearest sites

Weighted Voronoi Diagram dd xx, pp ii measures signed distance from x to a circle centered at pp ii with radius rr ii dd xx rr ii pp ii

Weighted Voronoi Diagram The bisector of two sites in the weighted distance metric Cell of p2 Cell of p2 Cell of p2 Cell of p1 Cell of p1 A hyperbola (if one circle is not completely within another) Does not exist

Weighted Voronoi Diagram A weighted Voronoi cell May be empty May be non-convex Always contains the site

Power Diagram A finite set of point sites pp ii with weights rr ii Power distance dd: dd xx, pp ii = xx pp ii 2 rr ii 2 Voronoi diagram is the set of xx with multiple nearest sites

Power Diagram dd xx, pp ii measures: xx outside circle pp ii, rr ii : squared length of tangent segment from xx to the circle xx inside circle pp ii, rr ii : negative squared length of halfchord perpendicular to diameter at xx dd xx dd xx pp ii pp ii

Power Diagram The bisector of two sites in the power metric is always a straight line Not always between the sites

Power Diagram A power cell May be empty Always convex May not contain the site Applet!

Voronoi Diagram A finite set of point sites pp ii Euclidean distance dd: dd xx, pp ii = xx pp ii Voronoi diagram is the set of xx with multiple nearest sites

Voronoi Diagram of Segments A finite set of line segments ll ii Euclidean distance dd: dd xx, ll ii = min pp ll ii xx pp Voronoi diagram is the set of xx with multiple nearest segments

Voronoi Diagram of Segments The bisector of two (disjoint) segments is made up of straight and parabolic pieces

Voronoi Diagram of Segments When the segments from a closed polygon, the diagram is known as medial axis

Medial Axis Captures shape and topology of objects 2D objects 3D objects

Voronoi Diagram A finite set of point sites pp ii Euclidean distance dd: dd xx, pp ii = xx pp ii Voronoi diagram is the set of xx with multiple nearest sites

Furthest-point Voronoi Diagram A finite set of point sites pp ii Euclidean distance dd: dd xx, pp ii = xx pp ii Voronoi diagram is the set of xx with multiple furthest sites VVVVVV pp ii = xx dd xx, pp ii > dd xx, pp jj, ii jj}

Furthest-point Voronoi Diagram A cell is also an intersection of half-planes defined by bisector lines It uses the half-planes that do not contain the site Cell of p2 Cell of p1 pp 1 pp 2

Furthest-point Voronoi Diagram A cell May be empty (if the site is not on the convex hull) Always convex Never contains the site

Furthest-point Voronoi Diagram Can be used to find the smallest circle containing the set The center of this circle is on the furthest-point Voronoi diagram Applet!