IBDP Mathematics Applications & Interpretations SL Chapter 4 Notes

Dividing up space : coordinate geometry , lines , Voronoi diagrams

STUDY NOTES FOR MATHEMATICS – CHAPTER 4 – Dividing up space : coordinate geometry , lines , Voronoi diagrams

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All DP mathematics courses serve to accommodate the diverse range of needs, interests and abilities of students, and to fulfill the requirements of various university and career aspirations. The aims of these courses are to enable students to: develop mathematical knowledge, concepts, principles, logical, critical and creative thinking, employ and refine their powers of abstraction and generalization. Students are also encouraged to appreciate the international dimensions of mathematics and the multiplicity of its cultural and historical perspectives.

In this chapter, under coordinate geometry, we learn to find the distance between two points in 3-D space, the midpoint of a line segment in 3-D space, the volume and surface area of solids and the size of angles between lines. In a Voronoi diagram, the important locations are called sites and every site is surrounded by a cell which consists of the points that are closer to that site than any other site. Edges are the lines separating the cells. Perpendicular bisectors are useful when it comes to constructing Voronoi diagrams. Interpolation is the process of using the values of a variable at known points to estimate the value of the variable at other points. Nearest neighbour interpolation is a simple method of interpolation. By using the value of the variable at the nearest known point, the value of a variable at any point can be estimated. To do so, a Voronoi diagram can be constructed with the known data points as sites. Then, the nearest known data point can be identified to any given point. In the case, the given point lies on an edge or at a vertex, the average of the closest known data points is calculated. The largest empty circle problem is the problem of finding the largest circle centered within the convex hull, whose interior does not contain any sites. This problem can be solved by drawing Voronoi diagrams for the sites.