#### Project Title

### Double Crescent Moon Reflection in Coffee Mugs

#### Disciplines

Mathematics

#### Abstract (300 words maximum)

A phenomenon referred to as the double crescent moon reflection can be seen at the bottom of a circular mug when a light source is set to shine at the mug from above it at an angle. Using basic light properties, assumptions from physical laws, and calculus, a set of mathematical equations are introduced to describe the phenomenon through multivariable and parametric functions. The set of equations serve as a tool for the discovery of the conditions that lead to an experimentally verified relationship between incoming light rays and their location of reflection at the bottom of the mug. A second set of equations are introduced to describe a parabolically shaped mug where a light source shines at an angle and reflects onto a flat surface. Using the same methodology and assumptions, a final relationship is again identified. The complete results are represented by two equations, for the two different shapes, that express the aforementioned relationship between incoming light ray paths and final reflection location at the bottom of the mug. This talk will cover the process of finding these results, as well as the intent to discover a more general model for the situation.

#### Academic department under which the project should be listed

CSM - Mathematics

#### Primary Investigator (PI) Name

Tsz Chan

Double Crescent Moon Reflection in Coffee Mugs

A phenomenon referred to as the double crescent moon reflection can be seen at the bottom of a circular mug when a light source is set to shine at the mug from above it at an angle. Using basic light properties, assumptions from physical laws, and calculus, a set of mathematical equations are introduced to describe the phenomenon through multivariable and parametric functions. The set of equations serve as a tool for the discovery of the conditions that lead to an experimentally verified relationship between incoming light rays and their location of reflection at the bottom of the mug. A second set of equations are introduced to describe a parabolically shaped mug where a light source shines at an angle and reflects onto a flat surface. Using the same methodology and assumptions, a final relationship is again identified. The complete results are represented by two equations, for the two different shapes, that express the aforementioned relationship between incoming light ray paths and final reflection location at the bottom of the mug. This talk will cover the process of finding these results, as well as the intent to discover a more general model for the situation.