Spring 2015, Thursdays, Blocker 506A, 4:00-4:50 PM

Date: April 16

"  Osculating Curves, and Cubics that Intersect only Once "
  Taylor Brysiewicz  


The tangent line is the best linear approximation of a planar curve at a smooth point. Intersection multiplicity gives us a way to talk about the best degree d approximation of a planar curve at a point. We call this curve "the" osculating curve of degree d. These osculating curves can be defined implicitly, which means that they are more than just the truncation of a local power series expansion. While the tangent line and osculating conic are unique when they exist, the osculating cubic may not be. It is very special when this happens since it means that there exists a pair of cubics that intersect at precisely one point with multiplicity 9. In this talk we will give a formula for the osculating conic at a point and describe necessary and sufficient geometric conditions for two cubics to intersect only once.