Roots of Polynomials and Umbilics of Surfaces

Guilfoyle, Brendan; Klingenberg, Wilhelm

doi:10.1007/s00025-023-02003-4

Results Math

2023

DOI: 10.1007/s00025-023-02003-4

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Roots of Polynomials and Umbilics of Surfaces

Brendan Guilfoyle,

Wilhelm Klingenberg

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2024

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A note on umbilic points at infinity

Guilfoyle

2024

Beitr Algebra Geom

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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 in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric interpretation for the 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.

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A note on umbilic points at infinity

Guilfoyle

2024

Beitr Algebra Geom

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Roots of Polynomials and Umbilics of Surfaces

Cited by 1 publication

References 16 publications

A note on umbilic points at infinity

A note on umbilic points at infinity

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