Mathematical Modeling and Volume Calculation of Unicorn Horns Using Volumes of Revolution and Rosette-Based Integration

This exploration investigates the mathematical modeling of unicorn horn volumes to optimize material estimation for 3D printing. The author develops two distinct approaches: a volume of revolution model utilizing a damped cosine function and a modular model based on the integration of tapered circular rosette cross-sections.

The methodology employs advanced calculus, including integration by parts and trigonometric identities, to derive closed-form analytical expressions. Results are validated using Desmos and 3D volumetric analysis software. The paper concludes that while the first model excels in biological fidelity, the second provides superior modularity through a parametric "Rosette Multiplier," allowing for precise control over the horn's geometric complexity.