[br][br][table][br][tr][/tr][br][tr][br][td]Voronoi diagrams[/td][br][td]ボロノイ図[/td][br][td]보로노이 다이어그램[/td][br][td]沃罗诺伊图[/td][br][/tr][br][tr][br][td]Partitioning a plane[/td][br][td]平面の分割[/td][br][td]평면 분할[/td][br][td]平面分割[/td][br][/tr][br][tr][br][td]Distances[/td][br][td]距離[/td][br][td]거리[/td][br][td]距离[/td][br][/tr][br][tr][br][td]Voronoi regions[/td][br][td]ボロノイ領域[/td][br][td]보로노이 영역[/td][br][td]沃罗诺伊区域[/td][br][/tr][br][tr][br][td]Perpendicular bisectors[/td][br][td]垂直二等分線[/td][br][td]수직 이등분선[/td][br][td]垂直平分线[/td][br][/tr][br][tr][br][td]Delaunay triangulation[/td][br][td]ドロネー三角測量[/td][br][td]델로네 삼각분할[/td][br][td]Delaunay三角剖分[/td][br][/tr][br][/table][br]

[table][br][tr][br][td][b]Factual Questions[/b][/td][br][td][b]Conceptual Questions[/b][/td][br][td][b]Debatable Questions[/b][/td][br][/tr][br][tr][br][td]How are Voronoi regions constructed from a given set of points in a plane?[/td][br][td]Why are Voronoi diagrams important for understanding spatial relationships and area partitioning?[/td][br][td]Can Voronoi diagrams be considered an effective tool for all types of spatial optimization problems, or are there limitations to their applicability?[/td][br][/tr][br][tr][br][td]What geometric properties do the perpendicular bisectors have in relation to Voronoi diagrams?[/td][br][td]In what ways can the concept of Voronoi diagrams be used to optimize real-life problems like urban planning or resource distribution?[/td][br][td]How might the reliance on Voronoi diagrams in certain fields, like telecommunications, influence the development of infrastructure?[/td][br][/tr][br][tr][br][td]How does altering the position of a single point affect the layout of its Voronoi region?[/td][br][td]How does the Delaunay triangulation relate to its dual, the Voronoi diagram, in terms of graphical representation?[/td][br][td]Is the use of Voronoi diagrams and Delaunay triangulation in computational geometry always the most efficient method for solving problems involving proximity or coverage?[/td][br][/tr][br][/table][br]

Mini-Investigation: Voronoi Diagrams[br][br]Objective:[br]Explore the construction and applications of Voronoi diagrams, which partition a plane based on distances to a specific set of objects.[br][br]Questions:[br]1. How does the placement of points affect the shape and size of the corresponding Voronoi regions?[br]2. What happens to the Voronoi diagram when two points are very close together compared to the others?[br]3. Can you create a situation where a Voronoi region is completely enclosed by another? Try moving the points to achieve this.[br]4. Investigate the role of the perpendicular bisectors in the creation of Voronoi diagrams. Why are they important?[br]5. Experiment with adding more points to the diagram. How does the number of points affect the complexity of the Voronoi diagram?[br]6. Identify a real-life situation where a Voronoi diagram could be useful (e.g., cell phone towers, distribution centers). Discuss how it could be applied.[br]7. Challenge: Try to create a Voronoi diagram with regions of equal area. Is this possible? What does it tell us about the distribution of points?[br]8. Explore the concept of the dual graph to the Voronoi diagram, the Delaunay triangulation. Can you infer its properties by looking at the Voronoi diagram?[br][br]Activity:[br]Using the applet, create a Voronoi diagram representing different shops in a city. Discuss which areas would be served by each shop and how the Voronoi diagram helps in planning delivery routes or service areas.[br]