Voronoi Diagrams Lecture Notes

Overview

• Voronoi diagrams are part of the AI course SL and HL under geometry and trigonometry.
• They resemble a map and help determine proximity to specific points or sites.

Terminology

• Sites: Previously known as coordinates (e.g., a, b, c, d).
• Edges: Lines separating sites.
• Regions/Cells: Areas created by edges.
• Intersections: Points where edges intersect (e.g., P, Q).

Purpose of Voronoi Diagrams

• Used to determine proximity within a region to a specific site.
• Example: Identifying the closest pizza shop from a given location.
  • If within a region of site D, then closest to the pizza shop at site D.
  • If standing on an edge, equally close to sites on either side of the edge.

Key Concepts

Construction of Voronoi Diagrams

• Constructed using perpendicular bisectors.
• Example: The line separating regions of sites D and C is a perpendicular bisector.
• Perpendicular bisectors are halfway and perpendicular to the line segments joining two sites.
• In exams, creating an entire Voronoi diagram is uncommon; focus on finding missing edges or nearest sites.

Nearest Neighbor Interpolation

• Used for estimating values based on proximity to known data points.
• Example: Estimating pizza sales for a new restaurant at a location based on the nearest existing restaurant's sales.
  • If new restaurant is closest to site B with 1752 sales per week, estimate similar sales for new restaurant.

Largest Empty Circle Problem

• Identifying the largest possible circle that can be drawn without containing any sites within it.
• Application: Positioning a toxic waste dump or new restaurant far from existing sites.
• The center of the largest empty circle always occurs at an intersection.
• Determining suitable intersection involves measuring distances to nearest sites.

Practical Applications

• Useful for determining optimal locations based on proximity and distance constraints.
• Commonly used in geographic and spatial analysis scenarios.

Recommendations

• Practice problems from the question bank to gain familiarity and proficiency.
• Focus on understanding the context and applications of Voronoi diagrams in real-world situations.