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Chapter 3: Coordinate Geometry, Voronoi Diagrams, Vectors, Lines, The…

- - - - cells
        
        for each site, there is a corresponding region consisting of all coordinates of the plane closer to that site. These regions are called cells.
      - cell boundaries (voronoi edge)
        
        cell boundaries are the edges that surround and form the cell.
        
        the boundaries are formed by perpendicular bisectors of the line segments joining the sites.
      - Sites
        
        sites are just points on the plane
      - voronoi vertex
        
        the center of the largest empty circle passing through all 3 surrounding sites
      - How to construct a voronoi diagram
        
        The best approach is to draw all the perpendicular bisectors at the same time then create the diagram
        
        but how to draw perpendicular bisectors?
        
        find the midpoint,m between all of the sites and draw a perpendicular line on the midpoint which joins the 2 sites.
        
        formula for m =
        
        do the same with every site (even with newly added sites)
        
        but it becomes difficult in deciding where the cell regions and boundaries of the voronoi diagram are. This can be solved with the inceremntal algorithm.
        
        The Incremental Algortithm
        
        the incremental algorithm allows the edges of a Voronoi diagram to be easily found
        
        Method with example
        1. Plot sites A (1,4) and B (1,0) and draw the perpendicular bisector
        
        2. Add the new site C (5,1) and find the perpendicular bisectors of [AC] and [BC]
        incremental algotrithm
        i) Begin with perpendicular bisector which lies between the new site (C) and the site which lies in the current cell (B).
        
        ii) Move along this line until you reach an intersection with another perpendicular bisector between the new site and an existing one.
        iii) Leave the intersectoin along the other new perpendicular bisector in the direction that lies entirley in the cell surrounding another site
        
        iv) The algorithm stops either when you return to your starting point or if there are no more intersections.
        v) Shade your regions
        
        The toxic waste dump problem
        
        This problem is to find the point on the Voronoi diagram that is as far as possible from any of the sites.
        
        This problem can also be used to consider where shops, businesses, etc. should be placed so that it is the centre of all surrounding sites.
        
        Another way of thinking about this problem is that it is basically the vertex, refer to vertex definition in key definitions section.
        
        the solution to this problem will be at an intersection of cell boundaries or on the boundary of the diagram
      - Perpendicular bisector
        
        The perpendicular bisector is the line that passes through the midpoint of the line segment joining two points and is perpendicular to it.
- - - - Simultaneous equation to find (x,y) of intersection
  - - - y1=mx1+c1 is parallel to y2=mx2+c2 as m=m
- - - - Vector can be written as:
        
        Column Vector
        
        Base Vector
      - Addition of Vectors
        
        Parallel Vector
        
        a and b are parallel if and only if
      - Position Vector (with example)
        
        A position vector gives a point's displacement from the origin
        
        The point A with coordinates (4,2) has position vector
        
        The point B with coordinates (-2,3) has position vector
        
        Therefore, we can write these vector as
        
        This is normally written as
      - The magnitude and direction of a vector
        
        The magnitude of a vector is shown by two vertical bars on either side of the vector:
        |a|
- - - - Where r is vector, written as
      - a is position position vector of a point on the line
      - b is a direction vector: a vector parallel to the line
      - parameter t: is a measure of how far the point is away from A as a multiple of the vector b
      - Angle between direction vectors: angle between two intersecting lines
    - - Speed: measure of distance covered in one unit of time
      - Velocity of an object: vector quantity which needs both a magnitude and a direction to define it
      - Position of a particle moving with velocity v at position vector when t=0 is can be given by equation
    - - Unit vector in the same direction as b is writtten as
      - If a particle has speed in the direction of b its velocity will be
      - Rescaling the vector: changing the magnitude of a vector while keeping its direction constant
      - Normalising the Vector: Rescaling to form a vector with a magnitude of 1
    - - If A has a position vector and B has a position vector , then the Relative Position of B from A is the