J. W. Bruce

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20232025

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math.DG2025

On vertices and inflections of singular plane curves

James William Bruce, Marco Antônio do Couto Fernandes, Farid Tari

Given the germ of a smooth plane curve $({f (x, y) = 0}, 0) \subset (K^{2}, 0), K = R, C$ , with an isolated singularity, we define two invariants $I_{f}$ a…

math.DG2024

On geometric invariants of singular plane curves

J. W. Bruce, M. A. C. Fernandes, F. Tari

Given a germ of a smooth plane curve $({f (x, y) = 0}, 0) \subset (K^{2}, 0), K = R, C$ , with an isolated singularity, we define two invariants $I_{f}$ and $V_…

math.DG2023

On the affine geometry of congruences of lines

J. W. Bruce, F. Tari

Congruences, or $2$ -parameter families of lines in $3$ -space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geom…

math.DG2023

Binary differential equations associated to congruences of lines in Euclidean 3-space

J. W. Bruce, F. Tari

We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classic…