# Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces

@article{Guilfoyle2018UmbilicPO, title={Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces}, author={Brendan Guilfoyle and Adriana Ortiz-Rodr'iguez}, journal={arXiv: Differential Geometry}, year={2018} }

The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function $f$ on the plane are studied. We provide a Poincar\'e-Hopf type formula where the sum over all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of $f$. Moreover, we study the projective extension of these fields and prove, under generic conditions, that every umbilic point at…

## One Citation

A Note on Umbilic Points at Infinity

- Mathematics
- 2021

In this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than…

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