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A Note on Umbilic Points at Infinity

Brendan Guilfoyle

Abstract

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 that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed by the authors in a previous paper. A geometric interpretation for our definition is that an umbilic point at infinity occurs when the tangent to the level set at infinity is also an asymptotic direction at infinity. We prove that a homogeneous polynomial graph must have an umbilic point, albeit at infinity.

A Note on Umbilic Points at Infinity

Abstract

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 that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed by the authors in a previous paper. A geometric interpretation for our definition is that an umbilic point at infinity occurs when the tangent to the level set at infinity is also an asymptotic direction at infinity. We prove that a homogeneous polynomial graph must have an umbilic point, albeit at infinity.

Paper Structure

This paper contains 6 sections, 10 theorems, 50 equations.

Table of Contents

  1. Isolated Umbilic Points
  2. Background
  3. Smooth Graphs
  4. Umbilic points at infinity
  5. Homogeneous Polynomial Graphs
  6. Paraboloids

Key Result

Theorem 1

The graph of any homogeneous polynomial has a finite, even number of umbilic points at infinity. If the polynomial has degree $n$, then the number of umbilic points at infinity is no more than $6n-8$.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 7 more