A Minimal Mathematical Model for Conducting Patterns

2026-04-11

AI Summary

This work addresses the challenge of jointly modeling spatial trajectories and temporal dynamics in conducting gestures by proposing a decoupled, parameterized schematic representation. It employs cubic Hermite curves with horizontal tangent constraints to accurately capture the geometric path from preparation point to beat point, while a quintic temporal function uniformly governs both constant-velocity motion and expressive accelerations or decelerations. The approach achieves high expressiveness with only a single time parameter, maintaining exceptional model simplicity. This compact representation has been successfully integrated into the Wolfram Demonstration “Conducting Patterns” and the Crusis web application, demonstrating its effectiveness and practical utility.

Technology Category

Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningPlanning, Routing, and Scheduling: Optimization of Spatio-temporal SystemsComputer Vision: Motion & Tracking

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsUser Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systemsSearch and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We present a minimal mathematical model for conducting patterns that separates geometric trajectory from temporal parametrization. The model is based on a cyclic sequence of preparation and ictus points connected by cubic Hermite segments with constrained horizontal tangents, combined with a quintic timing law controlling acceleration and deceleration. A single parameter governs the balance between uniform motion and expressive emphasis. The model provides a compact yet expressive representation of conducting gestures. It is implemented as the interactive Wolfram Demonstration "Conducting Patterns" and is used in the Crusis web app.

Problem

• conducting patterns
• mathematical model
• geometric trajectory
• temporal parametrization
• gesture representation

Innovation

• minimal mathematical model
• cubic Hermite segments
• quintic timing law
• conducting patterns
• trajectory–timing separation