This repository contains the mathematical modeling and topographic analysis for Project 1 of the Calculus II course. The project explores the practical application of multivariable functions to simulate a mountainous amusement park environment.
Developed during the 1st semester of 2026 as part of the Computer Engineering program at the Pontifical Catholic University of Campinas (PUC-Campinas), this study bridges theoretical calculus and engineering practice. It focuses on solving real-world challenges such as land altitude superposition, drainage path optimization, and infrastructure placement based on terrain constraints.
This repository serves both as academic documentation and engineering portfolio material.
GeoGebra is a dynamic mathematics software used for visualizing and analyzing complex geometric and algebraic structures. In the context of Multivariable Calculus, it serves as an essential tool for 3D spatial reasoning.
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Visualize Surfaces: Render complex multivariable functions, such as
$f(x,y)$ , to understand terrain behavior. - Map Contour Lines: Generate 2D representations of 3D altitudes (level curves) to identify slopes and ridges.
- Trace Trajectories: Model the exact paths of attractions and infrastructure across irregular elevations.
- Verify Critical Points: Visually confirm the location of peaks, valleys, and saddle points identified through calculus.
The analysis in this project demonstrates:
- Mathematical Surface Modeling: Using superposition of different functions to represent natural landforms.
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Constraint Analysis: Applying domain restrictions (e.g.,
$z \ge 0.5$ ) for safety and aesthetic requirements. - Optimization & Extrema: Finding and justifying absolute and relative maximum and minimum points on a surface.
- Vector Analysis & Slopes: Utilizing gradients to determine directions of steepest descent for drainage systems.
- Applied Geometry: Modeling tracks and trails using parametric curves and elliptical projections.
| Category | Detail |
|---|---|
| Mathematical Tool | GeoGebra (2D & 3D) |
| Methodology | Multivariable Analysis |
| Application | Civil & Computer Engineering Foundations |
| Documentation | Markdown |
| Precision | 3 Decimal Places |
- Multivariable Functions: Combining exponential and trigonometric functions to model relief.
- Topographic Maps: Creating contour maps with specific intervals to study terrain height.
- Gradient Vectors: Identifying the path of water flow for drainage pipe installation.
- Saddle Points: Locating points of equilibrium (Point A) for support structures.
- Distance & Length Estimation: Calculating approximate lengths of segmented paths on a surface.
- Directional Rates of Change: Analyzing if a chosen trail starts with an ascent or descent.
| Detail | Value |
|---|---|
| Course | Calculus II / Calculus 3 (Cálculo II/III) |
| Institution | Pontifical Catholic University of Campinas (PUC-Campinas) |
| Program | Computer Engineering |
| Semester | 1st Semester 2026 |
- Access the provided technical report to see the full mathematical derivation.
- Use GeoGebra to interact with the 3D terrain models and level curves.
- Review the analysis of the drainage path and panoramic tracks directly on the rendered surfaces.
- All calculations and results follow a strict requirement of 3 decimal places for engineering precision.
