  I got a 6 for my Mathematics HL Internal Assessment. The topic was “Exploring the shape of roads that would allow the center of polygonal wheels to ride along a straight horizontal line” using the basic algebra of hyperbolic function, anti-derivatives of hyperbolic function, and step function.

1. Introduction
2. Background Information
2.1. Basic Algebra of Hyperbolic Function
2.2. Anti-derivatives of Hyperbolic Cosine
2.3. Step Function
2.4. Movement of the Center of the Wheel
3. Assumptions
4. Optimal Function of “a bump” for Square Wheels
4.1. Deriving General Equation for Sqaure Wheels
4.2. Sample Calculation
4.2. Modelled Square Wheel and Road
5. General Formula of Optimal Functions for “a bump” for n-sided Wheels
6. General Formula of functions for “bumps” for regular n-sided wheels
6.1. Modelled Polygonal Wheel and their Corresponding Road Function
7. Conclusion
8. Resource and References

[Sample excerpts]

## Exploring the shape of roads that would allow the center of polygonal wheels to ride along a straight horizontal line

### 2. Introduction

2.1. Basic Algebra of Hyperbolic Function

Hyperbolic function is a function of an angle expressed as a relation between the distance from a point on a hyperbola to the origin and to the coordinate axes (Webster, 2010). This relation is commonly applied to identify the function of a curve that has the shape of an inverted catenary (Datolli, 2003).
The movement of a regular polygonal wheel would be ‘bumpy’ on a flat horizontal road. So when a dot is marked on one vertex of the square and was traced for one revolution, the trace would be in a shape of an inverted catenary. To derive a function that would allow the center of mass of any polygonal wheels to ride smoothly along a straight horizontal line, the road should have the shape of an inverted catenary as well. Thus hyperbolic functions, more specifically hyperbolic cosine, was used to describe the function of the inverted catenary shaped road, f(x)…

• Total number of pages: 15 pages
• Topic: Exploring the shape of roads that would allow the center of polygonal wheels to ride along a straight horizontal line
• Subject: Math
• The file is in PDF format.