Dynamical System Modeling and Stability Investigation�DSMSI-2025
May 08-10, 2025, Kyiv, Ukraine
Gradient zoning and density-controlled procedural urban planning using Voronoi diagrams�
Kyryl Petrachyk, Iryna Yurchuk, Anastasiia Nikolaienko
Taras Shevchenko National University of Kyiv
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Introduction
Procedural generation is widely used in games, simulation, and urban planning.
Dynamical System Modeling and Stability Investigation, DSMSI-2025
There's a need for a more organic and multifunctional method that mimics real-world city growth.
Existing approaches often rely on rigid zoning, resulting in unrealistic, monotonous layouts.
Aim & Objectives
To develop a flexible city generation method that supports mixed-use zones, density variation, and organic growth.
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Combine Voronoi diagrams, fuzzy logic, and gradient waves.
Integrate constraints and dynamic density modeling.
Enable realistic simulation of urban sprawl and multifunctionality.
Methodology
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Voronoi Diagrams
Divide the map into irregular, sector-like areas
Fuzzy Logic
Allow overlapping functions within each sector
Gradient Waves
Simulate influence and score propagation from key points
Density Function
Control zoning granularity and spatial intensity
Density Factor
Models population distribution changes over time
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Supports growth simulation, zoning variability, and infrastructure planning
Constraints & Suitability Grid
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Each plot is evaluated for terrain, slope, soil, and other factors.
Enables dynamic adaptation to context without rigid rules
Voronoi Diagrams & Area Prep
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Weighted placement of seed points reflects real-world irregularity.
Each sector gets fuzzy influence values instead of fixed types.
Gradient-Based Scoring System
Dynamical System Modeling and Stability Investigation, DSMSI-2025
FIRST WAVE
starts from the edge of buildable zones
↓
increasing scores with distance from constraints
why?
helps locate natural city centers
an example of grading sectors to define centers
Gradient-Based Scoring System
Dynamical System Modeling and Stability Investigation, DSMSI-2025
SECOND WAVE
spreads from centers identified earlier
↓
increasing scores with distance
why?
additional hubs formed using score peaks
an example of grading sectors to define centers
Influence Propagation & Coefficients
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Different decay rates per zone type
Enables smooth, realistic transitions
Comparison with WFC
Dynamical System Modeling and Stability Investigation, DSMSI-2025
WFC uses rigid constraints, leading to dead-ends
More expressive, modular, and adaptable
Our method uses fuzzy influence and avoids generation failure
In WFC, each grid element initially holds all possible variants, and a specialized selection function chooses one based on compatibility rules with neighboring elements
Challenges & Optimization
Dynamical System Modeling and Stability Investigation, DSMSI-2025
LARGE MAPS
HIGH COMPLEXITY
use multiresolution
grids
apply hierarchical
zoning
Performance: cache frequent operations, local updates only.
Conclusions
Dynamical System Modeling and Stability Investigation, DSMSI-2025
Combines Voronoi diagrams, fuzzy logic, and gradient scoring.
Supports density-aware, multifunctional, and scalable urban layouts.
Avoids WFC-like rigidity, supports real-world inspired city growth.
Applicable to games, simulations, and urban research.