Applet: the farthest point Voronoi diagram

Given a finite set of points in the plane, P={p1,…,pn}, called sites, the fathest point Voronoi diagram of P is the descomposition of the plane into regions, each of them being the geometric locus of the points of the plane whose distance to a given sitepiis greather than the distance to any of the remaining sites.

The following applet allows visualizing the farthest point Voronoi diagram of a finite set of sites in the plane, its structure and its construction.

Instructions for using the applet. The applet opens a window which allows visualizing the plane with the following options:

      • What can be visualitzed. A button for each of option allows to show or hide the following elements:
        • Sites: show or hide the set P of sites. Color: black.
        • Voronoi diagram: show or hide the farthest point Voronoi diagram of P. Color: green.
        • Smallest Enclosing Circle: show or hide the minimim spanning circle of P. Color: blue.
        • Convex Hull: show or hide the convex hull of P. Color: pink.
        • Coordinates: show or hide the cartesian coordinates of each point. Color: the same as the point.
      • Adding, deleting or moving sites. In order to add, delete or move a site, select the desired option in the Pointssection of the bottom left menu. Once selected, do:
        • Add: click the mouse wherever you want to create the new site.
        • Esborrar: click on the site to be deleted.
        • Moure: click and drag the site.
      • Highlighting a Vornoi vertex, edge or face. Voronoi vertices, edges and faces may be highlighted by selecting the corresponding option in the Selectsection of the bottom menu. The behaviour is as follows:
        • Vertex: pointing the mouse on a Voronoi vertex highlights the selected vertex in red color. Simultaneously, the edge to which it points in the DCEL structure of the diagram gets highlighted in blue.
        • Edge: pointing the mouse on a Voronoi edge highlights in orange color the selected edge and the two sites determining it. In addition, its initial edge and vertex get highlighted in red, and its final edge and vertex get highlighted in blue, according to the DCEL structure of the diagram.
        • Face: pointing the mouse on a Voronoi face highlights in red color all its edges, as well as the corresponding site.
      • Visualizing the construction of the farthest point Voronoi diagram. It is possible to visualize the construction od the diagram step by step. To activathe this option, click on the Stepsbutton, located in the top right part of the window. Starting from step 0, you can advance step by step by clicking the button each time. At each step you will see:
        • The new Voronoi vertex, center of the shown circle, and the portion of the fathest point Voronoi diagram computed so far.
        • The sites p, previous(p) and next(p) in the boundary of the convex hull of P maximizing the radius of the circle, in blue and big.
        • The sites of the boundary of the convex hull of P which still need to be processes, in red and big.
      • Navigation. It is possible to zoom and pan:
        • Zoom: use the scroll bar of the lower menu.
        • Pan: push the scroll arrows located in the low right part of the window.
      • Clear. There is a Clear botton to eliminate all the sites.
      • Close. In order to exit the application, use the back button of your browser or simply close the window of your browser.

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