Metrical structure and curvature of a parabola | Differential Geometry 12 | NJ Wildberger

Описание: In this lecture we introduce a general approach to metrical structure, via a symmetric bilinear form in either an affine or projective setting, and then begin studying the crucially important concept of curvature, first of all for a parabola of the form y=ax^2.

Metrical structures are usually associated with quadratic forms, or the corresponding symmetric bilinear forms, also known as dot products or inner products, and are represented mathematically by symmetric matrices. These are also naturally linked to conics. The essential equivalence between these seemingly different objects is an important understanding.

For applications to physics, in particular relativistic geometries, it is useful to have a flexible attitude here about metrical structure--the Euclidean one is not the only one to consider. This is rather a departure from classical courses in differential geometry, but the student will gain much by adopting such a more flexible and general point of view. We discuss two-dimensional relativistic geometries, and explain why circles in these contexts are what we would usually call rectangular hyperbolas.

For those particularly interested in these metrical connections between 2D and 3D relativistic geometries, I suggest the video in my MathSeminars series called
Hyperbolic Geometry is Projective Relativistic Geometry.

In the last part of the lecture we begin studying the parabola y=ax^2 at the point [0,0], with a view of determining the relations between the focus of the parabola, the center and radius of curvature, and the curvature itself. We are following here the approach of Zvi Har'El, explained in a paper called: Curvature of Curves and Surfaces: a Parabolic Approach (1995).

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