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Voronoi Diagrams (DP IB Applications & Interpretation (AI))

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    What is a site in a Voronoi diagram?

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  • What is a site in a Voronoi diagram?

    A site is a specific place of interest located at a point on a Voronoi diagram.

    E.g. the sites on a Voronoi diagram could be police stations in a city.

  • Define a Voronoi cell.

    A Voronoi cell is a region or polygon in a Voronoi diagram containing all points closest to a specific site.

    E.g. if the sites in a Voronoi dram represent the location of police stations in a city, then a Voronoi cell will contain all the points that are closer to the police station within that cell than to any other police station in the city.

  • Define a perpendicular bisector.

    A perpendicular bisector is a line that passes through the midpoint of a line segment at a right angle.

  • What are the edges of each region in a Voronoi diagram?

    Each edge of each region in a Voronoi diagram is the perpendicular bisector of the line segment that joins two sites.

  • True or False?

    The perpendicular bisectors of the line segments between three individual points always intersect at a point equidistant from the three points.

    True.

    The perpendicular bisectors of the line segments between three individual points always intersect at a point equidistant from the three points.

    A Voronoi diagram with three cells and a site in each cell, A, B and C. The edges of the cells meet at a point of intersection, which is equidistant from each site.

  • How can you find the location of a missing site in a Voronoi diagram?

    To find a missing site in a Voronoi diagram, use the given edge of one or two regions and find the second site that would make this edge a perpendicular bisector.

  • True or False?

    A point on an edge of a Voronoi diagram is equidistant from two sites.

    True.

    A point on an edge of a Voronoi diagram is equidistant from two sites.

    This is because each edge is a perpendicular bisector of the line that joins two sites.

  • How do you find the equation of a missing edge in a Voronoi diagram?

    To find the equation of a missing edge in a Voronoi diagram:

    • draw a line connecting the two sites where the edge is missing,

    • find the gradient and midpoint of this line,

    • then substitute the negative reciprocal of the gradient and the coordinates of the midpoint into the equation of a line, i.e. y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses, to find the equation of the edge.

  • What is the first step required when adding a new site to an existing Voronoi diagram?

    E.g. a new site, D, is added to a Voronoi diagram.

    A Voronoi diagram with sites A, B and C. A fourth point D is added and the cell edges need to be adapted.

    When adding a new site to an existing Voronoi diagram, the first step is to draw a perpendicular bisector between the two sites in the same cell and erase the previous edge that extends past the point of intersection.

    The same Voronoi diagram with sites A, B, C and D. A new edge (perpendicular bisector) has been added between sites C and D up to an existing boundary. The previous edge between A and C has been erased after that point of intersection.

  • How do you find the shortest distance from a point to its closest site in a Voronoi diagram?

    Use Pythagoras' Theorem to find the distance between the given coordinates and the site in the same region as it.

  • What is "nearest neighbour interpolation"?

    Nearest neighbour interpolation is a method of estimating the success of a new site by looking at the data for the nearest existing site.

    E.g. if each site on a Voronoi diagram represents a weather station recording rainfall, then each point on the Voronoi diagram would be predicted to have the same rainfall as that of the site it is nearest to, i.e. the site in its cell.

  • What is the toxic waste dump problem?

    The toxic waste dump problem is the idea of finding the point on a Voronoi diagram which is furthest from any of the sites.

  • Where is the furthest point from any site always located in a Voronoi diagram?

    In an exam, the furthest point from any site in a Voronoi diagram is always located at one of the cell vertices.

  • What is the centre of the largest empty circle in a Voronoi diagram?

    The centre of the largest empty circle in a Voronoi diagram is the vertex that is furthest from any site.

    (Note: a circle in a Voronoi diagram is 'empty' if it contains no sites.)

  • How is the radius of the largest empty circle determined?

    The radius of the largest empty circle is the distance from the vertex at the centre of the circle to the closest site.

  • True or False?

    You can use a graphing calculator (GDC) to find the coordinates of the vertices in a Voronoi diagram.

    True.

    You can use a graphing calculator (GDC) to find the coordinates of the vertices in a Voronoi diagram.

    To do this, use a GDC to solve the simultaneous equations of two perpendicular bisectors that intersect at that point.

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