#### Presentation Title

Finding Osculating Circles with Rational Center and Radius

#### Faculty Mentor

Christopher Lyons

#### Start Date

23-11-2019 12:45 PM

#### End Date

23-11-2019 1:00 PM

#### Location

Markstein 101

#### Session

oral 3

#### Type of Presentation

Oral Talk

#### Subject Area

physical_mathematical_sciences

#### Abstract

Curvature in calculus measures how fast a curve is changing direction at a given point. Similarly, the circle that best approximates the curvature at a point on the given curve is called the osculating circle. The osculating circle’s two defining attributes are its center and its radius, and this is important because number theory asks when are its center and radius rational. This matters because when we are solving for the radius of the osculating circle, we sometimes obtain irrational solutions. This brings me to the question I am raising and research we are currently working on: Given a rational point on the graph of a polynomial function with rational coefficients and degree greater than 1, when are the center of the osculating circle and its radius rational?

Finding Osculating Circles with Rational Center and Radius

Markstein 101

Curvature in calculus measures how fast a curve is changing direction at a given point. Similarly, the circle that best approximates the curvature at a point on the given curve is called the osculating circle. The osculating circle’s two defining attributes are its center and its radius, and this is important because number theory asks when are its center and radius rational. This matters because when we are solving for the radius of the osculating circle, we sometimes obtain irrational solutions. This brings me to the question I am raising and research we are currently working on: Given a rational point on the graph of a polynomial function with rational coefficients and degree greater than 1, when are the center of the osculating circle and its radius rational?